We are IntechOpen, the world's leading publisher of Open Access books Built by scientists, for scientists

3,700+ Open access books available

116,000+

International authors and editors

119M+

Downloads

151 Countries delivered to Our authors are among the

Top 1% most cited scientists

12.2%

Contributors from top 500 universities

Selection of our books indexed in the Book Citation Index in Web of Science™ Core Collection (BKCI)

## Interested in publishing with us? Contact book.department@intechopen.com

Numbers displayed above are based on latest data collected. For more information visit www.intechopen.com

## **Meet the editor**

Pedro Pinho was born in Vale de Cambra, Portugal, in 1974. He received *Licenciado* and Master's degrees in Electrical and Telecommunications Engineering, and a Ph.D. degree in Electrical Engineering from the University of Aveiro in 1997, 2000, and 2004, respectively. He is currently an assistant professor in the Electronics, Telecommunications and Computers Engineering Depart-

ment in Instituto Superior de Engenharia de Lisboa and a senior member of the research staff in Instituto de Telecomunicações, Aveiro, Portugal. Dr. Pinho is also a senior member of the IEEE, serves on the technical program committee in several conferences, and is a reviewer of several IEEE journals. He has authored or co-authored one book, 10 book chapters, and more than 100 papers for conferences and international journals. He participated as principal investigator or coordinator in projects with scientific and/or industry focus, both at a national and international level. To date he has led and leads five Ph.D. students and 42 M.Sc. students. His current research interest is antennas and radio propagation.

Contents

**Preface VII**

**Section 1 Antennas and Wave Propagation 1**

Andrey Viana Pires

Thomas Vaupel

Chapter 1 **Numerical Analysis of Broadband Dipole-Loop Graphene**

Chapter 2 **A Combined Electric/Magnetic Field Surface Volume Integral**

Chapter 3 **Time-Domain Analysis of Modified Vivaldi Antennas 39** Sultan Aldırmaz Çolak and Nurhan Türker Tokan

Chapter 4 **Teaching Transmission Line Propagation and Plane Wave**

Chapter 5 **Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks 77**

Chapter 7 **Ultra Wideband Transient Scattering and Its Applications to**

Hoi-Shun Lui, Faisal Aldhubaib, Stuart Crozier and Nicholas V.

**Reflection Using Software Tools 57** Susana Mota and Armando Rocha

Chapter 6 **Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver 115** Jan M. Kelner and Cezary Ziółkowski

**Automated Target Recognition 143**

Wojciech Jan Krzysztofik

Shuley

**Antenna for Applications in Terahertz Communications 3** Karlo Queiroz da, Gleida Tayanna Conde de, Gabriel Silva Pinto and

**Equation Approach for the Fast Characterization of Microstrip/ Substrate Integrated Waveguide Structures and Antennas 19**

## Contents

## **Preface XI**


Chapter 8 **Anisotropic Propagation of Electromagnetic Waves 167** Gregory Mitchell

Preface

communication system.

which can improve the learning outputs.

Over the past few years, wireless data traffic has drastically increased due to a change in the way today's society creates, shares, and consumes information, causing profound changes in our world and in our ways of life. This rapid increase in mobile data growth and the use of smartphones are creating unprecedented challenges, and new frequency bands are need‐ ed to explore and support new wireless systems architectures. Millimeter-wave and tera‐ hertz band communications are now being investigated by major research institutions and industry. In this context, new challenges in antennas and radio propagation modeling fields have started to meet the expected need for higher data rates for the future. Considering all

This book aims to introduce and treat a series of advanced and emerging topics in the field of antennas and radio propagation. One of the main subjects of this book is related to the trend to use higher frequencies, and an increasing demand for much higher-speed wireless communication anywhere and anytime. Consequently, new spectral bands will be required to support these extremely high data rates and new transceivers are necessary. Moreover, some aspects related to radio propagation modeling are presented; in addition, teaching transmission line propagation and a software tool for plane wave reflection are shown.

The book is organized into nine chapters. In the first chapter, a broadband dipole-loop gra‐ phene antenna for terahertz band communication is presented. The second chapter is dedi‐ cated to the characterization of substrate-integrated waveguide components and antennas to implement applications with low losses using standard printed board technologies. The third chapter presents a time domain analysis of modified Vivaldi antennas. Besides fre‐ quency analysis, a time domain analysis is required to characterize the transient behavior of ultra-wideband antennas (UWB) for pulsed operations, since pulse distortion of the UWB antenna reduces the system performance and decreases the signal-to-noise ratio of the UWB

Teaching transmission line and wave propagation is a challenging task because it involves quantities that are not easily observable and because the underlying mathematical equations involve time and distance variables and complex numbers. In such a context, the fourth chapter presents a set of tools with a strong visualization and easy student interaction,

The fifth chapter presents state-of-the-art radio network planning and propagation models for urban and indoor wireless communication networks. The sixth chapter presents a geo‐ metric-statistical propagation model that defines three groups of received signal compo‐ nents: direct path, delayed, and local scattering components. The multielliptical propagation model, which represents the geometry of scatterer locations, is the basis for determining the

this, one can say that there are endless possibilities for wireless communications.

#### Chapter 9 **Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium 185** Saeed Ahmed and Mona Lisa

## Preface

Chapter 8 **Anisotropic Propagation of Electromagnetic Waves 167**

Chapter 9 **Magnetic Line Source Diffraction by a PEMC Step in**

Gregory Mitchell

**VI** Contents

**Lossy Medium 185**

Saeed Ahmed and Mona Lisa

Over the past few years, wireless data traffic has drastically increased due to a change in the way today's society creates, shares, and consumes information, causing profound changes in our world and in our ways of life. This rapid increase in mobile data growth and the use of smartphones are creating unprecedented challenges, and new frequency bands are need‐ ed to explore and support new wireless systems architectures. Millimeter-wave and tera‐ hertz band communications are now being investigated by major research institutions and industry. In this context, new challenges in antennas and radio propagation modeling fields have started to meet the expected need for higher data rates for the future. Considering all this, one can say that there are endless possibilities for wireless communications.

This book aims to introduce and treat a series of advanced and emerging topics in the field of antennas and radio propagation. One of the main subjects of this book is related to the trend to use higher frequencies, and an increasing demand for much higher-speed wireless communication anywhere and anytime. Consequently, new spectral bands will be required to support these extremely high data rates and new transceivers are necessary. Moreover, some aspects related to radio propagation modeling are presented; in addition, teaching transmission line propagation and a software tool for plane wave reflection are shown.

The book is organized into nine chapters. In the first chapter, a broadband dipole-loop gra‐ phene antenna for terahertz band communication is presented. The second chapter is dedi‐ cated to the characterization of substrate-integrated waveguide components and antennas to implement applications with low losses using standard printed board technologies. The third chapter presents a time domain analysis of modified Vivaldi antennas. Besides fre‐ quency analysis, a time domain analysis is required to characterize the transient behavior of ultra-wideband antennas (UWB) for pulsed operations, since pulse distortion of the UWB antenna reduces the system performance and decreases the signal-to-noise ratio of the UWB communication system.

Teaching transmission line and wave propagation is a challenging task because it involves quantities that are not easily observable and because the underlying mathematical equations involve time and distance variables and complex numbers. In such a context, the fourth chapter presents a set of tools with a strong visualization and easy student interaction, which can improve the learning outputs.

The fifth chapter presents state-of-the-art radio network planning and propagation models for urban and indoor wireless communication networks. The sixth chapter presents a geo‐ metric-statistical propagation model that defines three groups of received signal compo‐ nents: direct path, delayed, and local scattering components. The multielliptical propagation model, which represents the geometry of scatterer locations, is the basis for determining the delayed components. In the seventh chapter, a review of the background and state of the art of resonance-based target recognition is introduced. This chapter covers recent develop‐ ments in using a polarimetric signature for target recognition, as well as using natural reso‐ nant frequencies for subsurface sensing applications. The chapter concludes with some highlights of the ongoing challenges in the field.

The eighth chapter is dedicated to anisotropic propagation of electromagnetic waves. Engi‐ neered anisotropic media provide unique electromagnetic properties, including a higher effec‐ tive refractive index, high permeability with relatively low magnetic loss tangent at microwave frequencies, and lower density and weight when compared with traditional media.

The last chapter covers the diffraction problem in detail and investigates magnetic line source diffraction by a perfect electromagnetic conductor (PEMC) step for a lossy medium. The PEMC step is assumed to be placed in lossy medium.

The editor and authors would like to express their gratitude to the publisher for the as‐ signed time, invaluable experience, efforts, and staff who successfully contributed toward enriching the final overall quality of the book.

To Carla Lourenço, Íris Pinho and Petra Pinho.

## **Professor Pedro Pinho**

**Section 1**

**Antennas and Wave Propagation**

Assistant Professor Electronics, Telecommunications and Computers Engineering Department Instituto Superior de Engenharia de Lisboa Lisbon, Portugal

> Senior Member Instituto de Telecomunicações Aveiro, Portugal

**Antennas and Wave Propagation**

delayed components. In the seventh chapter, a review of the background and state of the art of resonance-based target recognition is introduced. This chapter covers recent develop‐ ments in using a polarimetric signature for target recognition, as well as using natural reso‐ nant frequencies for subsurface sensing applications. The chapter concludes with some

The eighth chapter is dedicated to anisotropic propagation of electromagnetic waves. Engi‐ neered anisotropic media provide unique electromagnetic properties, including a higher effec‐ tive refractive index, high permeability with relatively low magnetic loss tangent at microwave

The last chapter covers the diffraction problem in detail and investigates magnetic line source diffraction by a perfect electromagnetic conductor (PEMC) step for a lossy medium.

The editor and authors would like to express their gratitude to the publisher for the as‐ signed time, invaluable experience, efforts, and staff who successfully contributed toward

Electronics, Telecommunications and Computers Engineering Department

To Carla Lourenço, Íris Pinho and Petra Pinho.

Instituto Superior de Engenharia de Lisboa

**Professor Pedro Pinho** Assistant Professor

Instituto de Telecomunicações

Lisbon, Portugal Senior Member

Aveiro, Portugal

frequencies, and lower density and weight when compared with traditional media.

highlights of the ongoing challenges in the field.

VIII Preface

enriching the final overall quality of the book.

The PEMC step is assumed to be placed in lossy medium.

**Chapter 1**

**Provisional chapter**

**Numerical Analysis of Broadband Dipole-Loop**

**Numerical Analysis of Broadband Dipole-Loop** 

**Communications**

**Communications**

Costa Karlo Queiroz da,

and Andrey Viana Pires

**Abstract**

**1. Introduction**

Karlo Queiroz da Costa,

Sousa Gleida Tayanna Conde de,

http://dx.doi.org/10.5772/intechopen.74936

Gabriel Silva Pinto and Andrey Viana Pires

Gleida Tayanna Conde de Sousa, Gabriel Silva Pinto

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Graphene Antenna for Applications in Terahertz**

**Graphene Antenna for Applications in Terahertz** 

DOI: 10.5772/intechopen.74936

Graphene possesses good properties as unusually high electron mobility, atomic layer thickness, and unique mechanical flexibility, which made it one promising material in the design of terahertz antennas. In this book chapter, we present a numerical analysis of a broadband dipole-loop graphene antenna for application in terahertz communications. The bidimensional method of moments (MoM-2D), with equivalent surface impedance of graphene, is used for numerical analysis. First, we review the principal characteristics of the conventional rectangular graphene dipole. Then, we consider the broadband graphene antenna, composed by one rectangular dipole placed near and parallel to a circular-loop graphene element, where only the dipole is feed. In this analysis, we investigated the effects of the geometrical parameters and the chemical potential, of the graphene material, on the overall characteristics of the compound antenna. Some results are compared with simulations performed with software based on finite element method. The results show that this simple compound graphene antenna can be used for broadband communications in the terahertz band.

**Keywords:** graphene antenna, broadband dipole-loop antenna, terahertz radiation,

Over the last few years, wireless data traffic has drastically increased due to a change in the way today's society creates, shares, and consumes information. This change has been accompanied

method of moment (MoM), graphene surface impedance

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

#### **Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz Communications Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz Communications**

DOI: 10.5772/intechopen.74936

Costa Karlo Queiroz da, Sousa Gleida Tayanna Conde de, Gabriel Silva Pinto and Andrey Viana Pires Karlo Queiroz da Costa, Gleida Tayanna Conde de Sousa, Gabriel Silva Pinto and Andrey Viana Pires

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74936

#### **Abstract**

Graphene possesses good properties as unusually high electron mobility, atomic layer thickness, and unique mechanical flexibility, which made it one promising material in the design of terahertz antennas. In this book chapter, we present a numerical analysis of a broadband dipole-loop graphene antenna for application in terahertz communications. The bidimensional method of moments (MoM-2D), with equivalent surface impedance of graphene, is used for numerical analysis. First, we review the principal characteristics of the conventional rectangular graphene dipole. Then, we consider the broadband graphene antenna, composed by one rectangular dipole placed near and parallel to a circular-loop graphene element, where only the dipole is feed. In this analysis, we investigated the effects of the geometrical parameters and the chemical potential, of the graphene material, on the overall characteristics of the compound antenna. Some results are compared with simulations performed with software based on finite element method. The results show that this simple compound graphene antenna can be used for broadband communications in the terahertz band.

**Keywords:** graphene antenna, broadband dipole-loop antenna, terahertz radiation, method of moment (MoM), graphene surface impedance

## **1. Introduction**

Over the last few years, wireless data traffic has drastically increased due to a change in the way today's society creates, shares, and consumes information. This change has been accompanied

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

by an increasing demand for a much higher speed wireless communication anywhere, anytime. Advanced physical layer solutions and, more importantly, new spectral bands will be required to support these extremely high data rates [1].

In this context, terahertz (THz) band communication [2–8] is envisioned as a key wireless technology to satisfy this demand; it is an alternative to spectrum demand and capacity limitations of current wireless systems, allowing multitude of applications. The THz band is the spectral band that spans the frequencies between 0.1 and 10 THz. While the frequency regions immediately below and above this band (the microwaves and the far infrared, respectively) have been extensively investigated, this is still one of the least-explored frequency bands for communication [1].

Therefore, there is a need to develop new transceiver antennas that are able to operate in THz frequencies in a very large operating bandwidth. Different technologies are actually in development in literature. In this chapter, we focus on graphene technology for the design of broadband terahertz antennas.

Graphene is a monolayer of carbon atoms arranged in a two-dimensional hexagonal honeycomb lattice [9]. The exceptional properties of graphene like unusually high electron mobility, atomic layer thickness, and possibility of miniaturizing antennas based on this material and many other properties made it one promising material in many areas ranging from solar cells [10] to ultra-high-speed transistors [11].

The two elements are separated by a height *H*, as shown in **Figure 1**. In the analysis, the geometry of antennas was maintained fixed, that is, the dipole's dimensions and the values of ring's radii (R1 and R2) and height (*H*) between antennas were fixed, and the chemical potential of

**Figure 1.** Geometry of the rectangular planar dipole graphene coupled to a circular-loop antenna of the same material:

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz…

http://dx.doi.org/10.5772/intechopen.74936

5

The size of the dipole graphene has only one planar dimension (e.g., *W* and *L*) because the graphene thickness is considered very small. This antenna is fed by an equivalent ideal voltage source called photomixer [13] with width *W* and gap length *G* in the middle of the dipole shown in **Figure 1**. In MoM, we call it voltage source and, in FEM, we used a lumped port [17].

In this section, the model used for the surface conductivity of graphene, a summary of the

The experimental results show that edge effects on the graphene conductivity can be disregarded in the micrometer scale [15]. Therefore, one can use the electrical conductivity model developed for infinite graphene sheet. In this chapter, we use the Drude model for graphene

*<sup>π</sup>h*<sup>2</sup> ln[2cosh(

*<sup>μ</sup>*\_\_\_\_*<sup>C</sup>* 2 *kB T*)] <sup>−</sup>*<sup>j</sup>* \_\_\_\_\_

*<sup>ω</sup>* <sup>−</sup> *<sup>j</sup> <sup>τ</sup>*<sup>−</sup><sup>1</sup> (1)

MoM-2D model used in the analysis, and details of the Comsol model are presented.

graphene circular loop was changed to obtain a broadband operation.

**3. Theoretical development**

(a) top view and (b) side view.

**3.1. Graphene surface conductivity**

surface conductivity in the range of 0.5–2 THz

*<sup>σ</sup>*(*ω*) <sup>=</sup> <sup>2</sup> *<sup>e</sup>* <sup>2</sup> *kB <sup>T</sup>* \_\_\_\_\_\_

Significant benefits can be obtained for graphene antennas in telecommunications applications such as monolithic integration with nanoelectronic graphene radio frequency (RF), efficient dynamic adjustment through chemical potential, relatively low loss in the band of terahertz (THz), and the possibility of miniaturization of antennas due to common plasmon effect in metamaterials [12, 13]. On the other side, there are few alternatives and works in literature about broadband graphene antennas [14, 15].

In this chapter, a theoretical analysis was made in a broadband graphene antenna composed by a rectangular dipole and a circular loop. The analysis is made using the two-dimensional method of moments (MoM 2D) with surface impedance [15, 16]. It was calculated by input impedance, reflection coefficient, and bandwidth from antennas with different geometrical parameters and values of chemical potential in the range of 0.5–2 THz. Some results were obtained by finite element method (FEM) with the Comsol software for comparison [17].

## **2. Antenna geometry**

**Figure 1** shows the geometry of the proposed broadband graphene antenna. This antenna is composed of two elements: a rectangular planar dipole with dimensions *L* and *W*, with same values used in [13] for comparison, and a circular passive ring (or circular loop) with inner radius R1 and outer radius R2. The environment in which the elements are inserted has a relative permittivity *ε<sup>r</sup>* = 2.4, which is the average of air and substrate permittivity used in [13], where in this reference, the substrate is in *z* < 0. In other words, here we use an equivalent effective permittivity for the whole medium, which is considered homogeneous with no substrate.

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz… http://dx.doi.org/10.5772/intechopen.74936

**Figure 1.** Geometry of the rectangular planar dipole graphene coupled to a circular-loop antenna of the same material: (a) top view and (b) side view.

The two elements are separated by a height *H*, as shown in **Figure 1**. In the analysis, the geometry of antennas was maintained fixed, that is, the dipole's dimensions and the values of ring's radii (R1 and R2) and height (*H*) between antennas were fixed, and the chemical potential of graphene circular loop was changed to obtain a broadband operation.

The size of the dipole graphene has only one planar dimension (e.g., *W* and *L*) because the graphene thickness is considered very small. This antenna is fed by an equivalent ideal voltage source called photomixer [13] with width *W* and gap length *G* in the middle of the dipole shown in **Figure 1**. In MoM, we call it voltage source and, in FEM, we used a lumped port [17].

## **3. Theoretical development**

by an increasing demand for a much higher speed wireless communication anywhere, anytime. Advanced physical layer solutions and, more importantly, new spectral bands will be required

In this context, terahertz (THz) band communication [2–8] is envisioned as a key wireless technology to satisfy this demand; it is an alternative to spectrum demand and capacity limitations of current wireless systems, allowing multitude of applications. The THz band is the spectral band that spans the frequencies between 0.1 and 10 THz. While the frequency regions immediately below and above this band (the microwaves and the far infrared, respectively) have been extensively investigated, this is still one of the least-explored frequency bands for communication [1]. Therefore, there is a need to develop new transceiver antennas that are able to operate in THz frequencies in a very large operating bandwidth. Different technologies are actually in development in literature. In this chapter, we focus on graphene technology for the design of

Graphene is a monolayer of carbon atoms arranged in a two-dimensional hexagonal honeycomb lattice [9]. The exceptional properties of graphene like unusually high electron mobility, atomic layer thickness, and possibility of miniaturizing antennas based on this material and many other properties made it one promising material in many areas ranging from solar cells

Significant benefits can be obtained for graphene antennas in telecommunications applications such as monolithic integration with nanoelectronic graphene radio frequency (RF), efficient dynamic adjustment through chemical potential, relatively low loss in the band of terahertz (THz), and the possibility of miniaturization of antennas due to common plasmon effect in metamaterials [12, 13]. On the other side, there are few alternatives and works in

In this chapter, a theoretical analysis was made in a broadband graphene antenna composed by a rectangular dipole and a circular loop. The analysis is made using the two-dimensional method of moments (MoM 2D) with surface impedance [15, 16]. It was calculated by input impedance, reflection coefficient, and bandwidth from antennas with different geometrical parameters and values of chemical potential in the range of 0.5–2 THz. Some results were obtained by finite element method (FEM) with the Comsol software for comparison [17].

**Figure 1** shows the geometry of the proposed broadband graphene antenna. This antenna is composed of two elements: a rectangular planar dipole with dimensions *L* and *W*, with same values used in [13] for comparison, and a circular passive ring (or circular loop) with inner radius R1 and outer radius R2. The environment in which the elements are inserted has a relative permittivity *ε<sup>r</sup>* = 2.4, which is the average of air and substrate permittivity used in [13], where in this reference, the substrate is in *z* < 0. In other words, here we use an equivalent effective permittivity for the whole medium, which is considered homogeneous with no substrate.

to support these extremely high data rates [1].

broadband terahertz antennas.

4 Antennas and Wave Propagation

**2. Antenna geometry**

[10] to ultra-high-speed transistors [11].

literature about broadband graphene antennas [14, 15].

In this section, the model used for the surface conductivity of graphene, a summary of the MoM-2D model used in the analysis, and details of the Comsol model are presented.

#### **3.1. Graphene surface conductivity**

The experimental results show that edge effects on the graphene conductivity can be disregarded in the micrometer scale [15]. Therefore, one can use the electrical conductivity model developed for infinite graphene sheet. In this chapter, we use the Drude model for graphene surface conductivity in the range of 0.5–2 THz

$$\sigma(\omega) = \frac{2e^2 k\_s T}{\pi h^2} \text{Im} \left[ 2 \cosh \left( \frac{\mu\_c}{2k\_s T} \right) \right] \frac{-j}{\omega - j \cdot \tau^{-1}} \tag{1}$$

5

where *τ* = 10−12 s is the relaxation time, *T* is the temperature, and *μC* is the chemical potential, which is a function, for example, of a DC voltage applied in graphene sheet [18]. **Figure 2** shows examples of *σ* for different values of chemical potential with *T* = 300 K.

#### **3.2. Method of moment model**

The boundary condition on the antenna surface produces the following integral equation of electric field in frequency domain with temporal dependence exp(*jωt*):

$$\left[ \left( \mathbb{E}\_s + \mathbb{E}\_l \right) \cdot \overline{a}\_l \right] \overline{a}\_l = Z\_s \overline{f} \tag{2}$$

**3.3. Finite element method model**

lumped port element.

**4. Numerical results**

geometry and chemical potential of the loop.

method requires a smaller computational cost than the FEM.

**Table 1.** Parameters of conventional graphene dipole antennas.

**Antenna** *μC L W* 1 0.13 *e*V 17 *μ*m 10 *μ*m 2 0.25 *e*V 23 *μ*m 20 *μ*m

**4.1. Conventional graphene dipole**

The Comsol software [17], which is based on the finite element method, was used to simulate examples of graphene antennas to compare our MoM model. The graphene sheet is modeled by an equivalent volumetric electrical conductivity, where it is defined by the surface conductivity (1) divided by the graphene thickness, which was considered finite in the Comsol model *σV*(*ω*) = *σ*(*ω*)/Δ, where Δ is the thickness of the antenna. The domain used is a spherical volume with *ε<sup>r</sup>* = 2.4, where a perfect matched layer (PML) is placed in the outer boundary to absorb the radiated waves. In this model, the dipole is excited by a voltage source with a

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz…

http://dx.doi.org/10.5772/intechopen.74936

7

In this section, we first present the results of two examples of conventional graphene dipole with different sizes. The principal characteristics of this antenna are reviewed. The results are obtained by MoM and FEM models and compared with data of literature [13]. After that, we present the results for the broadband graphene dipole loop. In this case, we analyze the dependence of the radiation and broadband properties of this antenna as a function of the

For comparison of our models, this section presents the analysis of the two graphene antennas of the study [13]. The parameters of these antennas are presented in **Table 1**, where we named them Antennas 1 and 2. These two antennas were simulated by MoM and Comsol. The discretization details used in these models are shown in **Figure 3**, where **Figure 3a** and **b** show the meshes used in the MoM method and **Figure 3c** and **d** show the meshes used in the FEM. Note that the meshes in MoM method are only in the surface's antennas, while in the FEM, the meshes are also in a spherical volume around the antennas. This is why the MoM method presents small number of unknowns than the FEM, and consequently, the MoM

The input impedance obtained for both antennas is presented in **Figure 4** and the results of input resistance *Rin* and input reactance *Xin* between MoM, simulation Comsol and data from [12] are compared. In general, a good agreement of the results is observed in these figures; the little differences are due to the differences in models and discretizations. The antennas present dipolar resonances similar to conventional RF-microwave antennas, where the fundamental

where *E*¯*<sup>S</sup>* (V/m) is the scattering field from the antenna, *E*¯*<sup>i</sup>* (V/m) is the incident electric field from the voltage source, *a* ¯*t* is a unitary vector tangential to the antenna's surface, *J*¯(A/m) is the surface current density of the antenna, and *ZS* = 1/*σ* is the surface impedance of graphene. The scattered field is

$$\overline{E}\_{\rm s} = -j\,\alpha\mu\_{\rm o} \iint \overline{f} \, \frac{e^{-j\lambda t}}{4\pi R} \, ds + \nabla \left[ \frac{1}{j\,\alpha\varepsilon\_{\rm o}} \iint \nabla \cdot \overline{f} \, \frac{e^{-j\lambda t}}{4\pi R} \, dS \right] \tag{3}$$

where *j* is the imaginary unit, *k* = *ω*(*μ*<sup>0</sup> *ε<sup>r</sup> ε*0 ) 1/2, *ω* is the angular frequency (rad/s), *μ*<sup>0</sup> and *ε*<sup>0</sup> are the magnetic permeability and electrical permittivity, respectively, in free space, *ε<sup>r</sup>* = 2.4 in the present analysis, and *R* is the distance between source points and observation points, both on the antenna surface *S*.

The numerical solution of Eq. (2) by MoM consists in to approximate the surface current on the antenna by a linear combination in a given set of basis function and performs the conventional test procedure with a given set of test function [16]. With this approximation, we transform the integral Eq. (2) in an algebraic linear system which is numerically solved.

**Figure 2.** Surface conductivity of graphene versus frequency for different values of *μC*, with *T* = 300 K.

## **3.3. Finite element method model**

where *τ* = 10−12 s is the relaxation time, *T* is the temperature, and *μC* is the chemical potential, which is a function, for example, of a DC voltage applied in graphene sheet [18]. **Figure 2**

The boundary condition on the antenna surface produces the following integral equation of

¯*t*] *a*

the surface current density of the antenna, and *ZS* = 1/*σ* is the surface impedance of graphene.

\_\_\_\_ 1 *j* <sup>0</sup> ∬

*S*

<sup>4</sup>*<sup>R</sup> ds*' <sup>+</sup> <sup>∇</sup>[

the magnetic permeability and electrical permittivity, respectively, in free space, *ε<sup>r</sup>* = 2.4 in the present analysis, and *R* is the distance between source points and observation points, both on

The numerical solution of Eq. (2) by MoM consists in to approximate the surface current on the antenna by a linear combination in a given set of basis function and performs the conventional test procedure with a given set of test function [16]. With this approximation, we transform the integral Eq. (2) in an algebraic linear system which is numerically solved.

 *ε*0 )

**Figure 2.** Surface conductivity of graphene versus frequency for different values of *μC*, with *T* = 300 K.

¯*<sup>t</sup>* = *ZSJ*¯ (2)

is a unitary vector tangential to the antenna's surface, *J*¯(A/m) is

∇⋅ *<sup>J</sup>*¯ *<sup>e</sup>* <sup>−</sup>*jkR* \_\_\_\_ <sup>4</sup>*<sup>R</sup> dS*'

1/2, *ω* is the angular frequency (rad/s), *μ*<sup>0</sup>

(V/m) is the incident electric field

] (3)

and *ε*<sup>0</sup>

are

shows examples of *σ* for different values of chemical potential with *T* = 300 K.

electric field in frequency domain with temporal dependence exp(*jωt*):

(V/m) is the scattering field from the antenna, *E*¯*<sup>i</sup>*

∬

*S J*¯ *<sup>e</sup>* <sup>−</sup>*jkR* \_\_\_\_

**3.2. Method of moment model**

6 Antennas and Wave Propagation

from the voltage source, *a*

*E*¯*<sup>S</sup>* = −*j* <sup>0</sup>

where *j* is the imaginary unit, *k* = *ω*(*μ*<sup>0</sup> *ε<sup>r</sup>*

The scattered field is

the antenna surface *S*.

where *E*¯*<sup>S</sup>*

[(*E*¯*<sup>S</sup>* + *E*¯*i*) ⋅ *a*

¯*t*

The Comsol software [17], which is based on the finite element method, was used to simulate examples of graphene antennas to compare our MoM model. The graphene sheet is modeled by an equivalent volumetric electrical conductivity, where it is defined by the surface conductivity (1) divided by the graphene thickness, which was considered finite in the Comsol model *σV*(*ω*) = *σ*(*ω*)/Δ, where Δ is the thickness of the antenna. The domain used is a spherical volume with *ε<sup>r</sup>* = 2.4, where a perfect matched layer (PML) is placed in the outer boundary to absorb the radiated waves. In this model, the dipole is excited by a voltage source with a lumped port element.

## **4. Numerical results**

In this section, we first present the results of two examples of conventional graphene dipole with different sizes. The principal characteristics of this antenna are reviewed. The results are obtained by MoM and FEM models and compared with data of literature [13]. After that, we present the results for the broadband graphene dipole loop. In this case, we analyze the dependence of the radiation and broadband properties of this antenna as a function of the geometry and chemical potential of the loop.

#### **4.1. Conventional graphene dipole**

For comparison of our models, this section presents the analysis of the two graphene antennas of the study [13]. The parameters of these antennas are presented in **Table 1**, where we named them Antennas 1 and 2. These two antennas were simulated by MoM and Comsol. The discretization details used in these models are shown in **Figure 3**, where **Figure 3a** and **b** show the meshes used in the MoM method and **Figure 3c** and **d** show the meshes used in the FEM. Note that the meshes in MoM method are only in the surface's antennas, while in the FEM, the meshes are also in a spherical volume around the antennas. This is why the MoM method presents small number of unknowns than the FEM, and consequently, the MoM method requires a smaller computational cost than the FEM.

The input impedance obtained for both antennas is presented in **Figure 4** and the results of input resistance *Rin* and input reactance *Xin* between MoM, simulation Comsol and data from [12] are compared. In general, a good agreement of the results is observed in these figures; the little differences are due to the differences in models and discretizations. The antennas present dipolar resonances similar to conventional RF-microwave antennas, where the fundamental


**Table 1.** Parameters of conventional graphene dipole antennas.

**Figure 3.** Discretization mesh of graphene antenna used in simulations. (a) Antenna 1—MoM model. (b) Antenna 2— MoM model. (c) Antenna 1—Comsol model. (d) Antenna 2—Comsol model.

resonance is dipolar λ/2 with lower *Rin* and the second resonance is λ with higher *Rin*. The values of these resonant frequencies are presented in **Table 2**. Antenna 1 possesses a smaller length *L* than Antenna 2 but the resonances of Antenna 2 are higher than those of Antenna 1; this occur because the chemical potential of Antenna 2 is higher than that of Antenna 1 and this parameter shifts the input impedance to higher frequencies.

**Figure 5** shows the reflection coefficient of Antennas 1, when this antenna is matched with a transmission line with characteristic impedance of *Zc* = 100 Ω. In this case, the fractional bandwidth is *B* = 11.44%, for a reference reflection level of −10 dB. For Antenna 2, this fractional bandwidth is *B* = 8.88%. These results show that conventional graphene dipoles possess a smaller bandwidth. The next sections present the analysis of broadband graphene dipoles with bandwidths higher than those presented in this section.

*4.2.1. Parametric analysis with geometry*

**Figure 4.** Input impedance: (a) Antenna 1 and (b) Antenna 2.

loop element: inner radius R1, outer radius R2, and distance H.

**Table 2.** Resonant frequencies of Antennas 1 and 2 calculated by different methods.

MoM F1 = 0.89 THz F2 = 1.34 THz Comsol F1 = 0.97 THz F2 = 1.33 THz Tamagnone [13] F1 = 1.02 THz F2 = 1.35 THz

A parametric analysis of graphene dipole loop of **Figure 1** is presented in this section. We investigate the variation of the characteristics of antenna as a function of the loop's geometry element. In all the analysis, we fixed the size and chemical potential of the dipole with those values of Antennas 1 presented in **Table 1**. In addition, we varied the following parameters of

**First resonance Second resonance**

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz…

http://dx.doi.org/10.5772/intechopen.74936

9

#### **4.2. Graphene dipole loop**

In this section, we present the numerical results for the broadband graphene dipole-loop antennas of **Figure 1**. First, we make a parametric analysis of geometry and then the effect of chemical potential of loop on the bandwidth and radiation characteristics. The results presented are input impedance, reflections coefficient, bandwidth, and radiation diagram.

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz… http://dx.doi.org/10.5772/intechopen.74936 9

**Figure 4.** Input impedance: (a) Antenna 1 and (b) Antenna 2.

resonance is dipolar λ/2 with lower *Rin* and the second resonance is λ with higher *Rin*. The values of these resonant frequencies are presented in **Table 2**. Antenna 1 possesses a smaller length *L* than Antenna 2 but the resonances of Antenna 2 are higher than those of Antenna 1; this occur because the chemical potential of Antenna 2 is higher than that of Antenna 1 and

**Figure 3.** Discretization mesh of graphene antenna used in simulations. (a) Antenna 1—MoM model. (b) Antenna 2—

**Figure 5** shows the reflection coefficient of Antennas 1, when this antenna is matched with a transmission line with characteristic impedance of *Zc* = 100 Ω. In this case, the fractional bandwidth is *B* = 11.44%, for a reference reflection level of −10 dB. For Antenna 2, this fractional bandwidth is *B* = 8.88%. These results show that conventional graphene dipoles possess a smaller bandwidth. The next sections present the analysis of broadband graphene dipoles

In this section, we present the numerical results for the broadband graphene dipole-loop antennas of **Figure 1**. First, we make a parametric analysis of geometry and then the effect of chemical potential of loop on the bandwidth and radiation characteristics. The results presented are input impedance, reflections coefficient, bandwidth, and radiation diagram.

this parameter shifts the input impedance to higher frequencies.

MoM model. (c) Antenna 1—Comsol model. (d) Antenna 2—Comsol model.

with bandwidths higher than those presented in this section.

**4.2. Graphene dipole loop**

8 Antennas and Wave Propagation


**Table 2.** Resonant frequencies of Antennas 1 and 2 calculated by different methods.

#### *4.2.1. Parametric analysis with geometry*

A parametric analysis of graphene dipole loop of **Figure 1** is presented in this section. We investigate the variation of the characteristics of antenna as a function of the loop's geometry element. In all the analysis, we fixed the size and chemical potential of the dipole with those values of Antennas 1 presented in **Table 1**. In addition, we varied the following parameters of loop element: inner radius R1, outer radius R2, and distance H.

**Figure 5.** Reflection coefficient of Antenna 1 matched with an impedance feed of *Zc* = 100 Ω.

The values of H are varied from 1 to 5 *μ*m. The outer radius R2 presents eight values from 3 to 10 mm, and the inner radius R1 with three fractions of R2, that is, R1 = 0.4 × R2, R1 = 0.6 × R2, and R1 = 0.8 × R2. A total of 120 simulations were done with the MoM code developed. For both elements (dipole and loop), we fixed the potential of *μC* = 0.13 eV (**Table 1**). **Figure 6** shows some examples of graphene dipole loop analyzed with different loop's size and the correspondent mesh used in the MoM model.

For each simulation, we plot the input impedance *Zin* of the antenna. First, we noted that for higher values of H and R1, the electromagnetic coupling between the dipole and the loop element is smaller. Therefore, the parameters where we obtained best coupling and matching bandwidth are H = 1 mm and R1 = 0.4 × R2. The geometries for some of these cases are presented in **Figure 5d**–**f**. **Figure 7** shows the input impedances for these cases R2 equal to 3, 4, 5, and 6 *μ*m, and **Figure 8** the *Zin* for the cases 7, 8, 9, and 10 *μ*m.

**Figure 6.** Geometry and mesh of some analyzed examples of graphene dipole-loop antenna with different values of inner radius R1: (a) R1 = 0.4 × R2, (b) R1 = 0.6 × R2, and (c) R1 = 0.8 × R2, and with different values of outer radius R2,

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz…

http://dx.doi.org/10.5772/intechopen.74936

11

**Figure 7.** Input impedance for antennas with H = 1 *μ*m, where R1 = 0.4 × R2, R2 = 3, 4, 5, and 6 *μ*m: (a) input resistance

where R1 = 0.4 × R2: (d) R2 = 4 μm, (e) R2 = 7 μm, and (f) R2 = 10 μm.

(Re(Zin)) and (b) input reactance (Im(*Zin*)).

In these figures, we can see the effect of the loop and the dipole in the total input impedance. For example, for the case of R2 = 9 mm in **Figure 9**, the resonances of the loop and the dipole are approximately in 0.75 and 1.0 THz, respectively. Also, the loop resonance is shifted to lower frequencies for higher values of R2, and the dipole resonance remains approximately constant. This happens because we are varying only the size of the loop in these simulations, and this size modifies the loop's resonance more strongly.

This behavior of multi-resonance is common for antennas with multiresonant elements coupled electromagnetically, which is the case of the dipole-loop antenna. This analysis of **Figures 7** and **8** shows that we can control the total input impedance so that it presents a broadband characteristic. For this purpose, we choose the loop's size in such a way as to couple the loop and the dipole resonance near to each other to obtain a broader resonance.

To observe this statement, we plot the reflection coefficient of these antennas when they are connected to a transmission line impedance characteristic of *Zc* in **Figure 9**. This parameter is calculated by Γ = |(*Zin*-*Zc* )/(*Zin* + *Zc* )|. A wider bandwidth was obtained when we choose *Zc* = 80 Ω for the cases R2 = 3, 4, 5 and 6 *μ*m, and *Zc* = 150 Ω for the cases R2 = 7, 8, 9 and 10 *μ*m.

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz… http://dx.doi.org/10.5772/intechopen.74936 11

The values of H are varied from 1 to 5 *μ*m. The outer radius R2 presents eight values from 3 to 10 mm, and the inner radius R1 with three fractions of R2, that is, R1 = 0.4 × R2, R1 = 0.6 × R2, and R1 = 0.8 × R2. A total of 120 simulations were done with the MoM code developed. For both elements (dipole and loop), we fixed the potential of *μC* = 0.13 eV (**Table 1**). **Figure 6** shows some examples of graphene dipole loop analyzed with different loop's size and the

For each simulation, we plot the input impedance *Zin* of the antenna. First, we noted that for higher values of H and R1, the electromagnetic coupling between the dipole and the loop element is smaller. Therefore, the parameters where we obtained best coupling and matching bandwidth are H = 1 mm and R1 = 0.4 × R2. The geometries for some of these cases are presented in **Figure 5d**–**f**. **Figure 7** shows the input impedances for these cases R2 equal to 3,

In these figures, we can see the effect of the loop and the dipole in the total input impedance. For example, for the case of R2 = 9 mm in **Figure 9**, the resonances of the loop and the dipole are approximately in 0.75 and 1.0 THz, respectively. Also, the loop resonance is shifted to lower frequencies for higher values of R2, and the dipole resonance remains approximately constant. This happens because we are varying only the size of the loop in these simulations,

This behavior of multi-resonance is common for antennas with multiresonant elements coupled electromagnetically, which is the case of the dipole-loop antenna. This analysis of **Figures 7** and **8** shows that we can control the total input impedance so that it presents a broadband characteristic. For this purpose, we choose the loop's size in such a way as to couple the loop and the dipole

To observe this statement, we plot the reflection coefficient of these antennas when they are

*Zc* = 80 Ω for the cases R2 = 3, 4, 5 and 6 *μ*m, and *Zc* = 150 Ω for the cases R2 = 7, 8, 9 and 10 *μ*m.

in **Figure 9**. This parameter

)|. A wider bandwidth was obtained when we choose

correspondent mesh used in the MoM model.

10 Antennas and Wave Propagation

4, 5, and 6 *μ*m, and **Figure 8** the *Zin* for the cases 7, 8, 9, and 10 *μ*m.

**Figure 5.** Reflection coefficient of Antenna 1 matched with an impedance feed of *Zc* = 100 Ω.

and this size modifies the loop's resonance more strongly.

resonance near to each other to obtain a broader resonance.

is calculated by Γ = |(*Zin*-*Zc*

connected to a transmission line impedance characteristic of *Zc*

)/(*Zin* + *Zc*

**Figure 6.** Geometry and mesh of some analyzed examples of graphene dipole-loop antenna with different values of inner radius R1: (a) R1 = 0.4 × R2, (b) R1 = 0.6 × R2, and (c) R1 = 0.8 × R2, and with different values of outer radius R2, where R1 = 0.4 × R2: (d) R2 = 4 μm, (e) R2 = 7 μm, and (f) R2 = 10 μm.

**Figure 7.** Input impedance for antennas with H = 1 *μ*m, where R1 = 0.4 × R2, R2 = 3, 4, 5, and 6 *μ*m: (a) input resistance (Re(Zin)) and (b) input reactance (Im(*Zin*)).

**Figure 8.** Input impedance for antennas with H = 1 *μ*m, where R1 = 0.4 × R2, R2 = 7, 8, 9, and 10 *μ*m: (a) input resistance (Re(Zin)) and (b) input reactance (Im(*Zin*)).

**Figure 9.** Reflection coefficient of antennas of **Figures 6** and **7**. (a) R2 = 3, 4, 5, and 6 *μ*m (*Zc* = 80 Ω). (b) R2 = 7, 8, 9, and 10 *μ*m (*Zc* = 150).

To compare the bandwidth of these antennas, we calculated the fractional bandwidth defined by *B*(%) = 200 × (*f H*−*f L* )/(*f <sup>H</sup>* + *f L* ), where *f <sup>H</sup>* and *f L* are the superior and inferior, respectively, limits of the band for a level of −10 dB. The results are presented in **Table 3**, where the fractional bandwidth for all simulations for the case H = 1 mm is presented. We observe that the best case is R2 = 7 mm with R1 = 0.4 × R2, where *B* = 21.7%.

*4.2.2. Effect of chemical potential*

**Fractional bandwidth,** *B***(%)**

R1

**Table 3.** Fractional bandwidth of dipole-loop antennas with H = 1 *μ*m.

R2 0.4 × R2 0.6 × R2 0.8 × R2 μm 12.1 14.7 11.2 μm 13.7 10.7 10.7 μm 15.4 6.9 10.1 μm 15.4 4.6 10.2 μm 21.7 11.7 15.3 μm 9.0 12.6 14.4 μm 2.2 18.7 14.5 μm 9.9 17.7 14.6

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz…

http://dx.doi.org/10.5772/intechopen.74936

13

the geometry and MoM mesh for this antenna.

*Zc* = 150 Ω. These values of the Zc produce a better bandwidth.

in the middle of the bandwidth. (a) 2D diagram at plane *xz*. (b) 3D diagram.

In this section, we present the variation of the characteristic of the antennas as a function of the chemical potential of the loop. The dimensions of the loop and the dipole were fixed as *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2. The chemical potential of the dipole is *μC* = 0.13 eV and the loop is varied *μC* = 0, 0.03, 0.07, 0.1 and 0.13 eV. **Figure 11** shows

**Figure 10.** Normalized radiation diagram of antennas with a higher bandwidth (R2 = 7 μm, R1 = 0.4 × R2) at F = 56THz,

**Figures 12** and **13** show the input impedance and reflection coefficient obtained, respectively. In **Figure 13**, we used *Zc* = 300 Ω, for *μC* = 0, 0.07, 0.1, and 0.13 eV, and for *μC* = 0.03 eV, we used

For the case with broad bandwidth of **Table 3** (R1 = 0.4 × R2 and R2 = 7 *μ*m), we plot the normalized radiation diagram at F = 0.56 THz in the middle of the bandwidth in **Figure 10**. The diagram is an asymmetric version of that diagram of an isolated dipole. The asymmetry is due to the asymmetric geometry of the antenna in the *xz* plane. This diagram radiates more energy in the –z direction, where in this case, the loop acts as a reflector element.

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz… http://dx.doi.org/10.5772/intechopen.74936 13


**Table 3.** Fractional bandwidth of dipole-loop antennas with H = 1 *μ*m.

**Figure 10.** Normalized radiation diagram of antennas with a higher bandwidth (R2 = 7 μm, R1 = 0.4 × R2) at F = 56THz, in the middle of the bandwidth. (a) 2D diagram at plane *xz*. (b) 3D diagram.

#### *4.2.2. Effect of chemical potential*

To compare the bandwidth of these antennas, we calculated the fractional bandwidth defined

**Figure 9.** Reflection coefficient of antennas of **Figures 6** and **7**. (a) R2 = 3, 4, 5, and 6 *μ*m (*Zc* = 80 Ω). (b) R2 = 7, 8, 9, and

**Figure 8.** Input impedance for antennas with H = 1 *μ*m, where R1 = 0.4 × R2, R2 = 7, 8, 9, and 10 *μ*m: (a) input resistance

of the band for a level of −10 dB. The results are presented in **Table 3**, where the fractional bandwidth for all simulations for the case H = 1 mm is presented. We observe that the best

For the case with broad bandwidth of **Table 3** (R1 = 0.4 × R2 and R2 = 7 *μ*m), we plot the normalized radiation diagram at F = 0.56 THz in the middle of the bandwidth in **Figure 10**. The diagram is an asymmetric version of that diagram of an isolated dipole. The asymmetry is due to the asymmetric geometry of the antenna in the *xz* plane. This diagram radiates more energy

are the superior and inferior, respectively, limits

*<sup>H</sup>* and *f L*

in the –z direction, where in this case, the loop acts as a reflector element.

by *B*(%) = 200 × (*f*

10 *μ*m (*Zc* = 150).

*H*−*f L* )/(*f <sup>H</sup>* + *f L*

(Re(Zin)) and (b) input reactance (Im(*Zin*)).

12 Antennas and Wave Propagation

), where *f*

case is R2 = 7 mm with R1 = 0.4 × R2, where *B* = 21.7%.

In this section, we present the variation of the characteristic of the antennas as a function of the chemical potential of the loop. The dimensions of the loop and the dipole were fixed as *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2. The chemical potential of the dipole is *μC* = 0.13 eV and the loop is varied *μC* = 0, 0.03, 0.07, 0.1 and 0.13 eV. **Figure 11** shows the geometry and MoM mesh for this antenna.

**Figures 12** and **13** show the input impedance and reflection coefficient obtained, respectively. In **Figure 13**, we used *Zc* = 300 Ω, for *μC* = 0, 0.07, 0.1, and 0.13 eV, and for *μC* = 0.03 eV, we used *Zc* = 150 Ω. These values of the Zc produce a better bandwidth.

**Figure 11.** Geometry and MoM mesh of dipole-loop antenna with 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2.

**Figure 12.** Input impedance variation with the chemical potential of loop. Geometry of dipole-loop antenna is *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2 (*μC* = 0.13 eV for dipole). Chemical potential of loop are: (a) *μC* = 0.0 eV; (b) *μC* = 0.03 eV; (c) *μC* = 0.07 eV; (d) *μC* = 0.1 eV; and (e) *μC* = 0.13 eV.

First, we observe a better agreement between MoM and Comsol. Also, we note that we can control the effect of the loop in the total input impedance by varying its chemical potential.

**Figure 14.** 2D radiation diagram of gain at plane *xy* and in frequencies (a) F = 0.69 THz (first resonance) and (b) F = 0.85 THz (second resonance). Data of antenna: *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2, *μC* = 0.13 eV (for

**Figure 13.** Variation of reflection coefficient with loop's chemical potential. Data of dipole-loop antenna: *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2 (*μC* = 0.13 eV for dipole), chemical potential of loop (a) *μC* = 0.0, 0.07,

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz…

http://dx.doi.org/10.5772/intechopen.74936

15

*μC* (eV*)* 0.0 0.03 0.07 0.1 0.13 *B* (%) 22.4 16.9 21.5 43.5 26.8

0.1, and 0.13 eV (*Zc* = 300 Ω), and (b) *μC* = 0.03 eV (*Zc* = 150 Ω).

**Table 4.** Fractional bandwidth for the antennas of **Figure 13**.

**Fractional bandwidth** *B***(%)**

We observe in **Figure 12** that only the resonance of loop is modified with chemical potential. For example, in **Figure 12c**, the loop resonance is near F = 0.75 THz, and in **Figure 12d**, it is

near F = 0.8 THz.

dipole) *μC* = 0.1 eV (for loop).

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz… http://dx.doi.org/10.5772/intechopen.74936 15

**Figure 13.** Variation of reflection coefficient with loop's chemical potential. Data of dipole-loop antenna: *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2 (*μC* = 0.13 eV for dipole), chemical potential of loop (a) *μC* = 0.0, 0.07, 0.1, and 0.13 eV (*Zc* = 300 Ω), and (b) *μC* = 0.03 eV (*Zc* = 150 Ω).


**Table 4.** Fractional bandwidth for the antennas of **Figure 13**.

**Figure 12.** Input impedance variation with the chemical potential of loop. Geometry of dipole-loop antenna is *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2 (*μC* = 0.13 eV for dipole). Chemical potential of loop are: (a) *μC* = 0.0 eV;

**Figure 11.** Geometry and MoM mesh of dipole-loop antenna with 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2.

14 Antennas and Wave Propagation

(b) *μC* = 0.03 eV; (c) *μC* = 0.07 eV; (d) *μC* = 0.1 eV; and (e) *μC* = 0.13 eV.

**Figure 14.** 2D radiation diagram of gain at plane *xy* and in frequencies (a) F = 0.69 THz (first resonance) and (b) F = 0.85 THz (second resonance). Data of antenna: *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2, *μC* = 0.13 eV (for dipole) *μC* = 0.1 eV (for loop).

First, we observe a better agreement between MoM and Comsol. Also, we note that we can control the effect of the loop in the total input impedance by varying its chemical potential.

We observe in **Figure 12** that only the resonance of loop is modified with chemical potential. For example, in **Figure 12c**, the loop resonance is near F = 0.75 THz, and in **Figure 12d**, it is near F = 0.8 THz.

**Acknowledgements**

**Conflict of interest**

**Author details**

Karlo Queiroz da Costa<sup>1</sup>

\*Address all correspondence to: karlo@ufpa.br

tions. Physics Communications. 2014;**12**:16-32

Journal of Applied Physics. 2010;**107**(11):111101

zine. 2011;**12**(4):108-116

Andrey Viana Pires<sup>2</sup>

**References**

The authors would like to thank the Mr. Mauro Roberto Collatto Junior Chief Executive Officer of the Junto Telecom Company for the financial and emotional support to this project.

Numerical Analysis of Broadband Dipole-Loop Graphene Antenna for Applications in Terahertz…

The authors declare that there is no conflict of interest regarding the publication of this work.

, Gabriel Silva Pinto<sup>2</sup>

http://dx.doi.org/10.5772/intechopen.74936

17

and

\*, Gleida Tayanna Conde de Sousa<sup>2</sup>

1 Department of Electrical Engineering, Federal University of Para, Tucuruí-PA, Brazil 2 Department of Electrical Engineering, Federal University of Para, Belém-PA, Brazil

[1] Akyildiz IF, Jornet JM, Han C. Terahertz band: Next frontier for wireless communica-

[2] Koch M. Terahertz communications: A 2020 vision. In: Miles R, Zhang XC, Eisele H, Krotkus A, editors. Terahertz Frequency Detection and Identification of Materials and Objects, NATO Security through Science Series. Vol. 19. Springer; 2007. pp. 325-338 [3] Piesiewicz R, Kleine-Ostmann T, Krumbholz N, Mittleman D, Koch M, Schoebel J, Kurner T. Short-range ultra-broadband terahertz communications: Concepts and per-

[4] Federici J, Moeller L. Review of terahertz and subterahertz wireless communications.

[5] Huang K-C, Wang Z. Terahertz terabit wireless communication. IEEE Microwave Maga-

[6] Kleine-Ostmann T, Nagatsuma T. A review on terahertz communications research.

[7] Song H, Nagatsuma T. Present and future of terahertz communications. IEEE Trans-

spectives. IEEE Antennas and Propagation Magazine. 2007;**49**(6):24-39

Journal of Infrared, Millimeter and Terahertz Waves. 2011;**32**:143-171

actions on Terahertz Science and Technology. 2011;**1**(1):256-263

**Figure 15.** 3D radiation diagram of gain at frequencies (a) F = 0.69 THz (first resonance) and (b) F = 0.85 THz (second resonance). Data of antenna: *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2, *μC* = 0.13 eV (for dipole) *μC* = 0.1 eV (for loop).

The results of **Figure 13** were obtained for *Zc* = 300 Ω; however for the case of *μC* = 0.03 eV, we used *Zc* = 150 Ω. These values of *Zc* presented a better bandwidth. The bandwidth obtained are presented in **Table 4**, where the broad bandwidth was found for the case with *μC* = 0.7 eV.

The resonant frequencies for the best bandwidth (*μC* = 0.1 eV) are F = 0.69 and 0.85 THz, as can be seen in **Figure 12d**, where the reactance is null. The radiation diagrams in these frequencies are presented in **Figures 14** and **15**. We observe that these diagrams are more symmetrical and similar to that of isolated dipole. This is because the size of the loop element is smaller than that of **Figure 10**; therefore, the effect of their surface current to produce far field is smaller.

## **5. Conclusions**

In this book chapter, we presented a numerical analysis of a broadband graphene dipole-loop antenna for terahertz application. In this antenna, only the dipole element is fed by a voltage source, while the loop element is electromagnetically coupled to the dipole. The bidimensional method of moment, with an equivalent surface impedance of graphene, was used for numerical calculations, and some results are obtained by finite element method for comparison. A good agreement between these two methods was obtained, but the method of moment is faster than the finite element method. In the results, we first presented a review of the principal characteristics of the conventional graphene dipole antenna. Then, we analyzed the broadband characteristics of the graphene dipole-loop antenna as a function of geometry and chemical potential of the loop. The results show that these parameters can be used to enhance the fractional bandwidth of this antenna, where a combination of these two parameters in the optimization process produces better results than one alone. The best antenna obtained presented a fractional bandwidth of 43.5% with a radiation diagram with linear polarization and good symmetry properties.

## **Acknowledgements**

The authors would like to thank the Mr. Mauro Roberto Collatto Junior Chief Executive Officer of the Junto Telecom Company for the financial and emotional support to this project.

## **Conflict of interest**

The authors declare that there is no conflict of interest regarding the publication of this work.

## **Author details**

**Figure 15.** 3D radiation diagram of gain at frequencies (a) F = 0.69 THz (first resonance) and (b) F = 0.85 THz (second resonance). Data of antenna: *W* = 17 *μ*m, *L* = 10 *μ*m, *H* = 1 *μ*m, R2 = 5 *μ*m, and R1 = 0.4 × R2, *μC* = 0.13 eV (for dipole)

The results of **Figure 13** were obtained for *Zc* = 300 Ω; however for the case of *μC* = 0.03 eV, we

are presented in **Table 4**, where the broad bandwidth was found for the case with *μC* = 0.7 eV. The resonant frequencies for the best bandwidth (*μC* = 0.1 eV) are F = 0.69 and 0.85 THz, as can be seen in **Figure 12d**, where the reactance is null. The radiation diagrams in these frequencies are presented in **Figures 14** and **15**. We observe that these diagrams are more symmetrical and similar to that of isolated dipole. This is because the size of the loop element is smaller than that of **Figure 10**; therefore, the effect of their surface current to produce far field is smaller.

In this book chapter, we presented a numerical analysis of a broadband graphene dipole-loop antenna for terahertz application. In this antenna, only the dipole element is fed by a voltage source, while the loop element is electromagnetically coupled to the dipole. The bidimensional method of moment, with an equivalent surface impedance of graphene, was used for numerical calculations, and some results are obtained by finite element method for comparison. A good agreement between these two methods was obtained, but the method of moment is faster than the finite element method. In the results, we first presented a review of the principal characteristics of the conventional graphene dipole antenna. Then, we analyzed the broadband characteristics of the graphene dipole-loop antenna as a function of geometry and chemical potential of the loop. The results show that these parameters can be used to enhance the fractional bandwidth of this antenna, where a combination of these two parameters in the optimization process produces better results than one alone. The best antenna obtained presented a fractional bandwidth of 43.5% with a radiation diagram with linear polarization and good symmetry properties.

presented a better bandwidth. The bandwidth obtained

*μC* = 0.1 eV (for loop).

16 Antennas and Wave Propagation

**5. Conclusions**

used *Zc* = 150 Ω. These values of *Zc*

Karlo Queiroz da Costa<sup>1</sup> \*, Gleida Tayanna Conde de Sousa<sup>2</sup> , Gabriel Silva Pinto<sup>2</sup> and Andrey Viana Pires<sup>2</sup>

\*Address all correspondence to: karlo@ufpa.br

1 Department of Electrical Engineering, Federal University of Para, Tucuruí-PA, Brazil

2 Department of Electrical Engineering, Federal University of Para, Belém-PA, Brazil

## **References**


[8] Kürner T, Priebe S. Towards THz communications-status in research, standardization and regulation. Journal of Infrared, Millimeter and Terahertz Waves. 2014;**35**(1):53-62

**Chapter 2**

Provisional chapter

**A Combined Electric/Magnetic Field Surface Volume**

DOI: 10.5772/intechopen.75062

A Combined Electric/Magnetic Field Surface Volume

**Characterization of Microstrip/Substrate Integrated**

In this contribution, a combined electric field/magnetic field surface/volume integral equation approach is presented with special features for the characterization of substrate integrated waveguide (SIW) components. Due to the use of a parallel-plate waveguide Green's function, only a small number of volume current basis functions are necessary to model the vias of the SIW sidewalls. The focal point is set on the specification of microstrip-SIW transitions using a via and a pad/antipad configuration for the coupling between the microstrip parts and the SIW and transitions with a two-stage ridged substrate integrated waveguide (SIW) where the SIW has a very thick substrate with regard to the microstrip line making it well suited for the design of a new class of compact end-fire SIW antennas for phased array applications which are partly characterized with CST Microwave Studio. An effective S-parameter extraction is used with both microstrip and

Keywords: integral equation, surface/volume discretization, microstrip-SIW transitions,

Due to the trend to higher frequencies, substrate integrated waveguide (SIW) components and antennas are once more well-suited building blocks to realize applications with low losses and compact design. A SIW consists of two periodic rows of metallic vias connecting metallic strips (or ground planes) on top and bottom of a dielectric substrate as a quasi-planar structure and

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

S-parameter extraction, via and slot modeling, horn, slot/leaky wave antennas

Characterization of Microstrip/Substrate Integrated

**Integral Equation Approach for the Fast**

Integral Equation Approach for the Fast

**Waveguide Structures and Antennas**

Waveguide Structures and Antennas

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75062

special SIW waveguide ports.

Thomas Vaupel

Abstract

1. Introduction

Thomas Vaupel


#### **A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization of Microstrip/Substrate Integrated Waveguide Structures and Antennas** A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization of Microstrip/Substrate Integrated Waveguide Structures and Antennas

DOI: 10.5772/intechopen.75062

Thomas Vaupel Thomas Vaupel

[8] Kürner T, Priebe S. Towards THz communications-status in research, standardization and regulation. Journal of Infrared, Millimeter and Terahertz Waves. 2014;**35**(1):53-62

[9] Geim A, Novoselov K. The rise of graphene. Nature Materials. March 2007;**6**(3):183-191 [10] Fang Z et al. Graphene-antenna sandwich photodetector. Nano Letters. June, 2012:

[12] Perruisseau-Carrier J. Graphene for antenna applications–Opportunities and challenges from microwaves to THz. Loughborough Antennas & Propagation Conference, UK. 2012

[13] Tamagnone M, Gómez-Díaz JS, Mosig JR, Perruisseau-Carrier J. Analysis and design of terahertz antennas based on plasmonic resonant graphene sheets. Journal of Applied

[14] Zhang H, Jiang Y, Wang J. A broadband terahertz antenna using graphene. 11th International Symposium on Antennas, Propagation and EM Theory (ISAPE), Antennas,

[15] da Costa KQ, Dimitriev V, Nascimento CM, Silvano GL. Theoretical analysis of graphene nanoantennas with different shapes. Microwave and Optical Technology Letters.

[16] da Costa KQ, Dimitriev V. Planar monopole UWB antennas with cuts at the edges and parasitic loops. InTech: Ultra Wideband Communications: Novel Trends—Antennas and

[18] Hanson GW. Dyadic Green's functions and guided surface waves for a surface conduc-

Propagation and EM Theory (ISAPE), Guilin, China. 18-21 October 2016

[17] COMSOL Multiphysic 4.4, COMSOL Inc., http://www.comsol.com

tivity model of Graphene. Journal of Applied Physics. 2008;**103**:064302

[11] Schwierz F. Graphene transistors. Nature Nanotechnology. May 2010

3808-3813

18 Antennas and Wave Propagation

Physics. December 2012;**112**

Propagation, 1st ed., pp. 143-145, 2011

May 2014;**56**(5)

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75062

#### Abstract

In this contribution, a combined electric field/magnetic field surface/volume integral equation approach is presented with special features for the characterization of substrate integrated waveguide (SIW) components. Due to the use of a parallel-plate waveguide Green's function, only a small number of volume current basis functions are necessary to model the vias of the SIW sidewalls. The focal point is set on the specification of microstrip-SIW transitions using a via and a pad/antipad configuration for the coupling between the microstrip parts and the SIW and transitions with a two-stage ridged substrate integrated waveguide (SIW) where the SIW has a very thick substrate with regard to the microstrip line making it well suited for the design of a new class of compact end-fire SIW antennas for phased array applications which are partly characterized with CST Microwave Studio. An effective S-parameter extraction is used with both microstrip and special SIW waveguide ports.

Keywords: integral equation, surface/volume discretization, microstrip-SIW transitions, S-parameter extraction, via and slot modeling, horn, slot/leaky wave antennas

## 1. Introduction

Due to the trend to higher frequencies, substrate integrated waveguide (SIW) components and antennas are once more well-suited building blocks to realize applications with low losses and compact design. A SIW consists of two periodic rows of metallic vias connecting metallic strips (or ground planes) on top and bottom of a dielectric substrate as a quasi-planar structure and

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

can thus very easily integrated and fabricated together with other microstrip/coplanar components using standard printed circuit board technologies and can be considered as a filled rectangular waveguide with the via rows as sidewalls [3]. Therefore, SIWs have the same low loss properties and first designs can be made using tools for the calculation of rectangular waveguide components. Thus, SIWs are increasingly used for antenna feeding networks with power dividers, filters, resonators, couplers, and phase shifters in micro- and mm-wave applications. For the full-wave characterization of such components, frequently mode matching techniques are proposed, but they are typically restricted to the modeling of the SIW components itself but without considering transitions to a microstrip/coplanar environment or radiating slot elements [2]. At the moment a strong interest is set on compact broadband horn-like SIW antennas with end-fire radiation characteristic [10, 11]. However, to reach the desired broadband properties with compact dimensions suited for phased array applications, SIWs on a thick substrate are needed. Simple microstrip-SIW transitions or interconnects like in [3, 6] cannot be used for the excitation of such thick SIWs; in this context, tapered transitions in vertical and lateral direction are proposed in [4]. In contrast to this, we have presented a twostage transition based on a Chebychef design in [5]. Since the fabrication of the ridged-SIW structures is quite elaborate, we have also investigated transitions using a shorted via between the microstrip line and the bottom of the SIW as given in [6]. Such structure can be easily fabricated, but the bandwidth becomes small using thicker substrates. For thick substrates we investigate transitions where the length of the coupling via is smaller than the SIW thickness like the typical probes in standard rectangular waveguides. For the characterization of these structures, we have extended an integral equation approach [7] with special features for the effective characterization of SIWs. Based on the Green's functions of multilayered media, a substrate-filled parallel-plate medium can be used in many cases; thus together with volume basis functions for vertical currents, only the vias of the SIWs must be discretized. The microstrip parts are modeled with electric surface currents, whereas the antipads and other apertures are modeled with magnetic surface currents allowing the characterization of a class of leaky wave antennas [8] with very low computational effort. A further task was the careful S-parameter determination comprising both microstrip- and SIW waveguide ports.

#### 2. Formulation with first applications

The round vias of SIW structures with diameter dcirc can be replaced for the simulations by quadratic vias with the side length lsquare with typically high accuracy using the formula [1]:

$$d\_{square} = \frac{d\_{circ}}{2} \left( 1 + \frac{1}{\sqrt{2}} \right) \tag{1}$$

The SIWs are modeled with parallel-plate media where the metallic plates are connected by the vias of the SIW. A typical scheme, here with SIW waveguide ports, is given in Figure 1. It consists of two SI waveguides on different levels and SIW ports with a slot coupling (blue) and optional slots for the radiation in the upper half space, which can be free space or a complex stratified medium. As indicated in the side view, the slots are modeled with corresponding magnetic surface currents (also in blue) at the upper and lower side of the corresponding metallic sheet with opposite signs leading to a magnetic field integral equation for the mag-

Figure 2. Used basis functions. Top: Planar (asymmetric) rooftop functions for electric and magnetic surface currents (x-

Figure 1. Structure with SI waveguides and ports on different levels with slot coupling as well as free space coupling (or a

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

21

For the matrix entries in the method of moments concerning magnetic surface current func-

!

lb,m kx; ky � � � <sup>F</sup>

!<sup>∗</sup>

lb,n kx; ky

� �dkxdky (2)

netic currents. A sketch of the used basis functions is given in Figure 2.

and y-orientation). Bottom: z-directed volume currents attached to x-directed surface currents.

tions n and m on one level lb, we get a spectral admittance representation:

M,lb, comp kx; ky; zlb � � � <sup>F</sup>

Ynmð Þ¼ lb; lb

ð

further stratified halfspace).

ð

ky G \$H

kx

The vias are then modeled with quadratic volume current basis functions (piecewise constant current in height) connecting the corresponding metallic sheets of one SIW waveguide with typically only one basis function per via. For multilayer microstrip-SIW transitions and other SIW components and circuits, we use a very effective modeling and discretization strategy [7]. A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization… http://dx.doi.org/10.5772/intechopen.75062 21

can thus very easily integrated and fabricated together with other microstrip/coplanar components using standard printed circuit board technologies and can be considered as a filled rectangular waveguide with the via rows as sidewalls [3]. Therefore, SIWs have the same low loss properties and first designs can be made using tools for the calculation of rectangular waveguide components. Thus, SIWs are increasingly used for antenna feeding networks with power dividers, filters, resonators, couplers, and phase shifters in micro- and mm-wave applications. For the full-wave characterization of such components, frequently mode matching techniques are proposed, but they are typically restricted to the modeling of the SIW components itself but without considering transitions to a microstrip/coplanar environment or radiating slot elements [2]. At the moment a strong interest is set on compact broadband horn-like SIW antennas with end-fire radiation characteristic [10, 11]. However, to reach the desired broadband properties with compact dimensions suited for phased array applications, SIWs on a thick substrate are needed. Simple microstrip-SIW transitions or interconnects like in [3, 6] cannot be used for the excitation of such thick SIWs; in this context, tapered transitions in vertical and lateral direction are proposed in [4]. In contrast to this, we have presented a twostage transition based on a Chebychef design in [5]. Since the fabrication of the ridged-SIW structures is quite elaborate, we have also investigated transitions using a shorted via between the microstrip line and the bottom of the SIW as given in [6]. Such structure can be easily fabricated, but the bandwidth becomes small using thicker substrates. For thick substrates we investigate transitions where the length of the coupling via is smaller than the SIW thickness like the typical probes in standard rectangular waveguides. For the characterization of these structures, we have extended an integral equation approach [7] with special features for the effective characterization of SIWs. Based on the Green's functions of multilayered media, a substrate-filled parallel-plate medium can be used in many cases; thus together with volume basis functions for vertical currents, only the vias of the SIWs must be discretized. The microstrip parts are modeled with electric surface currents, whereas the antipads and other apertures are modeled with magnetic surface currents allowing the characterization of a class of leaky wave antennas [8] with very low computational effort. A further task was the careful

S-parameter determination comprising both microstrip- and SIW waveguide ports.

lsquare <sup>¼</sup> dcirc

The round vias of SIW structures with diameter dcirc can be replaced for the simulations by quadratic vias with the side length lsquare with typically high accuracy using the formula [1]:

<sup>2</sup> <sup>1</sup> <sup>þ</sup>

The vias are then modeled with quadratic volume current basis functions (piecewise constant current in height) connecting the corresponding metallic sheets of one SIW waveguide with typically only one basis function per via. For multilayer microstrip-SIW transitions and other SIW components and circuits, we use a very effective modeling and discretization strategy [7].

1 ffiffiffi 2 p � �

(1)

2. Formulation with first applications

20 Antennas and Wave Propagation

Figure 1. Structure with SI waveguides and ports on different levels with slot coupling as well as free space coupling (or a further stratified halfspace).

Figure 2. Used basis functions. Top: Planar (asymmetric) rooftop functions for electric and magnetic surface currents (xand y-orientation). Bottom: z-directed volume currents attached to x-directed surface currents.

The SIWs are modeled with parallel-plate media where the metallic plates are connected by the vias of the SIW. A typical scheme, here with SIW waveguide ports, is given in Figure 1. It consists of two SI waveguides on different levels and SIW ports with a slot coupling (blue) and optional slots for the radiation in the upper half space, which can be free space or a complex stratified medium. As indicated in the side view, the slots are modeled with corresponding magnetic surface currents (also in blue) at the upper and lower side of the corresponding metallic sheet with opposite signs leading to a magnetic field integral equation for the magnetic currents. A sketch of the used basis functions is given in Figure 2.

For the matrix entries in the method of moments concerning magnetic surface current functions n and m on one level lb, we get a spectral admittance representation:

$$\begin{aligned} Y\_{nm}(lb, lb) &= \\ \int\_{\mathbf{k}\_x} \int\_{\mathbf{k}\_y} \widehat{G}^{H}\_{M, lb, comp} \left( \mathbf{k}\_{\mathbf{x}}, \mathbf{k}\_y, \mathbf{z}\_{lb} \right) \cdot \overrightarrow{F}\_{lb, m} \left( \mathbf{k}\_{\mathbf{x}}, \mathbf{k}\_y \right) \cdot \overrightarrow{F}^{\*}\_{lb, n} \left( \mathbf{k}\_x, \mathbf{k}\_y \right) dk\_x dk\_y \end{aligned} \tag{2}$$

with the composite Green's function:

$$
\stackrel{\rightharpoonup}{G}\_{\text{M, lb, \alphaump}}^{H} \left( k\_{\text{x}}, k\_{\text{y}}, z\_{\text{lb}} \right) = \stackrel{\leftrightarrow}{G}\_{\text{M, lb}}^{H} \left( k\_{\text{x}}, k\_{\text{y}}, z\_{\text{lb}} \right) + \stackrel{\leftrightarrow}{G}\_{\text{M, lb}-1}^{H} \left( k\_{\text{x}}, k\_{\text{y}}, z\_{\text{lb}} \right),
$$

which is the sum of the Green's functions G \$H M,lb above (radiation into a half-space or another parallel-plate medium) and below the metallic sheet lb (parallel-plate medium) G \$H M,lb�<sup>1</sup> and F ! lb,m,n kx; ky � � the Fourier transforms of the basis functions. For the couplings between volume currents, we get a combined space/spectral impedance expression:

$$\begin{split} \mathbf{Z}\_{nk}\Big(\mathbb{I}\mathbf{v},\mathbb{I}\mathbf{v}'\Big) &= \\ \displaystyle \int\_{k} \int\_{k} \int\_{z} \Big[ \bigoplus\_{l,k}^{x} \Big(\mathbf{k}\_{x},\mathbf{k}\_{y},z',z\big) \cdot \boldsymbol{\overrightarrow{F}}\_{\operatorname{lv},k}\Big(\mathbf{k}\_{x},\mathbf{k}\_{y},z'\Big) \cdot \boldsymbol{\overrightarrow{F}}\_{\operatorname{lv},n}^{\*}\Big(\mathbf{k}\_{x},\mathbf{k}\_{y},z\big) dz \,d\boldsymbol{z}'d\mathbf{k}\_{x}d\mathbf{k}\_{y} \end{split} \tag{3}$$

The SIW ports consist of an excitation via and four via (volume) functions here only used for field (voltage) monitoring in the center of the SIW to extract the forward and backward waves

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

To compute the voltages at the field monitoring vias, at first the complete coupling matrix with all via functions is computed. Afterward the matrix is compressed by deleting the matrix columns related to the monitor functions and by taking out the related line entries stored for later reuse and the voltage computation. |S11|,tl and |S12|,tl are the results for the homogeneous through line with nearly an ideal matching and transmission showing the good performance of the SIW port model and S-parameter extraction.|S11|,abs shows the broadband matching with the 5-via absorber at the right using 240 Ohm resistors connecting the vias to the ground plane. Such an absorber can be used as an effective numerical broadband-matched termination of a SIW, but it should also be possible to manufacture it using SMD resistors

For another test the SIW is fitted with two slots of length 2.34 mm in the middle. The separation and length of the slots are chosen in such a way that we get a so-called reflectioncanceling slot pair. |S11|,ds denotes the matching with the additional reflection-canceling slot

A good test for the reliability of the numerical solutions and to get a detailed insight into the power flow of a structure is the computation of the power balance. Thus the input power at the

Re U � I

n¼1

And the radiated space wave energy into the upper halfspace is determined with the help of

with the wavenumbers kxr ¼ �km sin ϑ<sup>r</sup> cosφ<sup>r</sup> and kyr ¼ �km sin ϑ<sup>r</sup> sinφr. Here km is the

\$E,FF

function for the electric field of magnetic (surface) currents radiating into layer m.

If the structures are analyzed with infinite low dielectric and metallic losses,

\$E,FF

<sup>M</sup> kxr; kyr � �XNs

n¼1 InFn !

jkyr <sup>r</sup> sin <sup>ϑ</sup><sup>r</sup> sin<sup>φ</sup>rG

<sup>∗</sup> f g (5)

http://dx.doi.org/10.5772/intechopen.75062

23

R Ij j <sup>n</sup> <sup>2</sup> (6)

<sup>M</sup> is the far-field evaluation of the Green's

kxr; kyr � � <sup>r</sup> sin <sup>ϑ</sup><sup>r</sup> <sup>d</sup>φrdϑ<sup>r</sup>

� � � � � � �

2

(7)

PIN <sup>¼</sup> <sup>1</sup> 2

Pabs <sup>¼</sup> <sup>1</sup> 2 X Nabs

The absorbed energy within the Nabs resistors with resistance R amounts to

in the SIW for S-parameter calculation.

connecting the vias with the ground planes.

the saddle point method leading to the integral

2ðπ

jkxr <sup>r</sup> sin <sup>ϑ</sup><sup>r</sup> cosφ<sup>r</sup> e

φr¼0 e

wavenumber of the topmost layer m and G

ðπ=2

ϑr¼0

excitation via (Figure 3) is

PRad <sup>¼</sup> jkm cos <sup>ϑ</sup><sup>r</sup> 2πr

� � � � � � �

pair showing a quite good matching around 36–37 GHz.

with a complete analytical treatment of the space-domain integrations, and for couplings between magnetic and volume currents, we get a matrix entry without dimension, denoted with ZY:

$$\begin{aligned} \boldsymbol{Z}\boldsymbol{Y}\_{\text{nl}}(\boldsymbol{l}\boldsymbol{v},\boldsymbol{l}\boldsymbol{b}) &= \\ \int\_{\boldsymbol{k}} \int\_{\boldsymbol{k}} \int\_{\boldsymbol{z}} \widehat{\boldsymbol{G}}\_{\text{l},\text{lb}}^{E} \left(\boldsymbol{k}\_{\text{r}},\boldsymbol{k}\_{\text{y}},\boldsymbol{z}\_{\text{lb}},\boldsymbol{z}'\right) \overrightarrow{\boldsymbol{F}}\_{\text{l}\boldsymbol{v},\boldsymbol{k}} \left(\boldsymbol{k}\_{\text{r}},\boldsymbol{k}\_{\text{y}},\boldsymbol{z}'\right) \cdot \overrightarrow{\boldsymbol{F}}\_{\text{lb},\text{n}}^{\*} \left(\boldsymbol{k}\_{\text{x}},\boldsymbol{k}\_{\text{y}},\boldsymbol{z}\_{\text{lb}}\right) d\boldsymbol{z}' d\boldsymbol{k}\_{\text{x}}d\boldsymbol{k}\_{\text{y}} \end{aligned} \tag{4}$$

The spectral domain integrations are carried out with two different strategies depending on the lateral distance of the basis functions.

A structure well suited for testing of different features of the method is given in Figure 3. At the right, we apply a SIW port 2 or a SIW absorber consisting of only five volume functions connected by lumped resistors with the metallic sheets.

Figure 3. Structure for testing of SIW ports and absorbers combined with a reflection-canceling slot pair. Via side length 0.17 mm and via distance p = 0.4 mm, ε<sup>r</sup> ¼ 3:0 � j0:003, and h = 0.254 mm.

The SIW ports consist of an excitation via and four via (volume) functions here only used for field (voltage) monitoring in the center of the SIW to extract the forward and backward waves in the SIW for S-parameter calculation.

with the composite Green's function:

22 Antennas and Wave Propagation

G \$H

Znk lv; lv<sup>0</sup> � �

ð

z G \$E

ZYnkð Þ¼ lv; lb

ð

ð

ð

z0

ky

ð

ð

ð

z0 G \$E

the lateral distance of the basis functions.

ky

kx

kx

F !

lb,m,n kx; ky

with ZY:

which is the sum of the Green's functions G

M,lb, comp kx; ky; zlb

¼

J,lb kx; ky; z

J,lb kx; ky; zlb; z <sup>0</sup> � �

connected by lumped resistors with the metallic sheets.

0.17 mm and via distance p = 0.4 mm, ε<sup>r</sup> ¼ 3:0 � j0:003, and h = 0.254 mm.

� � <sup>¼</sup> <sup>G</sup>

currents, we get a combined space/spectral impedance expression:

0 ; z � � \$H

parallel-plate medium) and below the metallic sheet lb (parallel-plate medium) G

� Flv, <sup>k</sup> !

> F !

\$H

M,lb kx; ky; zlb � � <sup>þ</sup> <sup>G</sup>

� � the Fourier transforms of the basis functions. For the couplings between volume

kx; ky; z <sup>0</sup> � �

with a complete analytical treatment of the space-domain integrations, and for couplings between magnetic and volume currents, we get a matrix entry without dimension, denoted

> lv, <sup>k</sup> kx; ky; z <sup>0</sup> � �

The spectral domain integrations are carried out with two different strategies depending on

A structure well suited for testing of different features of the method is given in Figure 3. At the right, we apply a SIW port 2 or a SIW absorber consisting of only five volume functions

Figure 3. Structure for testing of SIW ports and absorbers combined with a reflection-canceling slot pair. Via side length

� F !<sup>∗</sup>

� F !<sup>∗</sup>

lb,n kx; ky; zlb � �dz<sup>0</sup>

\$H

M,lb�<sup>1</sup> kx; ky; zlb � �

lv,n kx; ky; <sup>z</sup> � �dz dz<sup>0</sup>

M,lb above (radiation into a half-space or another

\$H

dkxdky (4)

dkxdky

M,lb�<sup>1</sup> and

(3)

To compute the voltages at the field monitoring vias, at first the complete coupling matrix with all via functions is computed. Afterward the matrix is compressed by deleting the matrix columns related to the monitor functions and by taking out the related line entries stored for later reuse and the voltage computation. |S11|,tl and |S12|,tl are the results for the homogeneous through line with nearly an ideal matching and transmission showing the good performance of the SIW port model and S-parameter extraction.|S11|,abs shows the broadband matching with the 5-via absorber at the right using 240 Ohm resistors connecting the vias to the ground plane. Such an absorber can be used as an effective numerical broadband-matched termination of a SIW, but it should also be possible to manufacture it using SMD resistors connecting the vias with the ground planes.

For another test the SIW is fitted with two slots of length 2.34 mm in the middle. The separation and length of the slots are chosen in such a way that we get a so-called reflectioncanceling slot pair. |S11|,ds denotes the matching with the additional reflection-canceling slot pair showing a quite good matching around 36–37 GHz.

A good test for the reliability of the numerical solutions and to get a detailed insight into the power flow of a structure is the computation of the power balance. Thus the input power at the excitation via (Figure 3) is

$$P\_{IN} = \frac{1}{2} \text{Re}\{\mathcal{U} \cdot I^\*\} \tag{5}$$

The absorbed energy within the Nabs resistors with resistance R amounts to

$$P\_{\rm abs} = \frac{1}{2} \sum\_{n=1}^{N\_{\rm abs}} R|I\_n|^2 \tag{6}$$

And the radiated space wave energy into the upper halfspace is determined with the help of the saddle point method leading to the integral

$$P\_{Rd} = \left| \frac{jk\_m \cos \theta\_r}{2\pi r} \int\_{\theta\_r = 0}^{\pi/2} \int\_{\theta\_r = 0}^{2\pi} d^{\vec{k}\_x r \sin \theta\_r \cos \varphi\_r} e^{\vec{k}\_y r \sin \theta\_r \sin \varphi\_r} \vec{G}\_M^{\Sigma, \text{FF}} \left( k\_{xr}, k\_{yr} \right) \sum\_{n=1}^{N\_s} I\_n \vec{F}\_n \left( k\_{xr}, k\_{yr} \right) r \sin \theta\_r d\varphi\_r d\theta\_r \right|^2 \tag{7}$$

with the wavenumbers kxr ¼ �km sin ϑ<sup>r</sup> cosφ<sup>r</sup> and kyr ¼ �km sin ϑ<sup>r</sup> sinφr. Here km is the wavenumber of the topmost layer m and G \$E,FF <sup>M</sup> is the far-field evaluation of the Green's function for the electric field of magnetic (surface) currents radiating into layer m.

If the structures are analyzed with infinite low dielectric and metallic losses,

$$P\_{\rm IN} = P\_{\rm Rad} + P\_{\rm Abs} \tag{8}$$

an overall integrand with an exponential decay with regard to kr. The contribution with this asymptotic representation is then added in Eq. (9) with the second integral, which can be evaluated completely analytically for all kinds of involved basis functions. In the inner area, pole rings due to surface wave and/or parallel-plate waves can occur; furthermore the inner area contains all wavenumbers related to radiating space waves. The wavenumbers in the outer areas Aout1, Aout2, and Aout3 are mainly related to evanescent waves. However, if we have only SIW structures and components like in Figure 3 without any radiating slots, we have only a contribution of a parallel-plate wave pole ring in the inner area for the coupling integrals of vias Eq. (3), whereas with a radiation by slots or other radiating elements into the upper (layered) halfspace, a branch point occurs as further singularity as well as possible surface wave poles in the integrands of the coupling integrals Eq. (2) and Eq. (4). All these

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

25

Figure 5. Top left: Typical integrand behavior in the inner integration area with pole rings. Right: First quadrant with all

Figure 6. A leaky wave antenna with slots in the cover for end-fire radiation with the same SIW as in Figure 3. Slot

integration areas. Bottom: Simultaneous integration path deformations in the inner area.

separation 0.4 mm.

must hold. The results for the SIW with the absorber and the slot pair are given in Figure 4, right. It can be seen that the radiation power is below 20 percent up to about 37 GHz (the frequency band with good matching) with only one slot pair, whereas about 80 percent is converted in the absorber. If the input power according to Eq. (5) is normalized to 100 percent, then the sum in Eq. (8) leads to the red curve in Figure 4, right. Except some small deviation in the region 40–42 GHz, the 100 percent are met with good accuracy. However, for this accuracy the number of sampling points for the numerical integration must be quite high.

For an improved computational performance, a general coupling integral reads

$$\begin{split} V\_{nm} &= \int \int \left[ I \stackrel{\leftrightarrow}{G} \left( k\_{\mathbf{x}}, k\_{\mathbf{y}}, z\_{m}, z\_{n} \right) - I \stackrel{\leftrightarrow}{G}^{Ag} \left( k\_{\mathbf{x}}, k\_{\mathbf{y}}, z\_{m}, z\_{n} \right) \right] \cdot \stackrel{\leftrightarrow}{F}\_{n0} \left( k\_{\mathbf{x}}, k\_{\mathbf{y}}, z\_{m} \right) \cdot \stackrel{\leftrightarrow}{F}\_{n0}^{\*} \left( k\_{\mathbf{x}}, k\_{\mathbf{y}}, z\_{n} \right) e^{\vec{\mathbf{x}}\_{1} \cdot \Delta \mathbf{x}\_{m}} e^{\vec{\mathbf{x}}\_{1} \cdot \Delta \mathbf{y}\_{m}} dk\_{\mathbf{x}} \, d\mathbf{k}\_{\mathbf{y}} \\ &+ \quad \int \int \stackrel{\leftrightarrow}{G} \left( k\_{\mathbf{x}}, k\_{\mathbf{y}}, z\_{m}, z\_{n} \right) \cdot \stackrel{\leftrightarrow}{F}\_{m0} \left( k\_{\mathbf{x}}, k\_{\mathbf{y}}, z\_{m} \right) \cdot \stackrel{\leftrightarrow}{F}\_{n0}^{\*} \left( k\_{\mathbf{x}}, k\_{\mathbf{y}}, z\_{n} \right) e^{\vec{\mathbf{x}}\_{1} \cdot \Delta \mathbf{x}\_{m}} e^{\vec{\mathbf{x}}\_{j} \cdot \Delta \mathbf{y}\_{m}} dk\_{\mathbf{x}} \, d\mathbf{k}\_{\mathbf{y}} \end{split} \tag{9}$$

where I G \$ is a general Green's function tensor where all analytical space-domain integrations with regard to z and z 0 are already incorporated. The F ! m,n<sup>0</sup> are the Fourier transforms of the basis functions in the origin of the coordinate system; thus, the Δxnm and Δyn,m are the lateral separations of the basis functions with regard to x and y. Figure 5 shows the different integration areas with the integrand behavior in the inner area AIn and the used simultaneous integration path deformations (for both wavenumbers kx and ky) in this area.

For basis functions with very small lateral separations or overlapping functions and selfcouplings, an asymptotic representation IG \$Asy for <sup>k</sup><sup>r</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 <sup>x</sup> þ k 2 y q ! ∞ is subtracted, leading to

Figure 4. Left: S-parameter extraction results for the SIW through line (tl), with absorber (abs) and additional double slot (ds), h = 0.254 mm. Right: power balance.

an overall integrand with an exponential decay with regard to kr. The contribution with this asymptotic representation is then added in Eq. (9) with the second integral, which can be evaluated completely analytically for all kinds of involved basis functions. In the inner area, pole rings due to surface wave and/or parallel-plate waves can occur; furthermore the inner area contains all wavenumbers related to radiating space waves. The wavenumbers in the outer areas Aout1, Aout2, and Aout3 are mainly related to evanescent waves. However, if we have only SIW structures and components like in Figure 3 without any radiating slots, we have only a contribution of a parallel-plate wave pole ring in the inner area for the coupling integrals of vias Eq. (3), whereas with a radiation by slots or other radiating elements into the upper (layered) halfspace, a branch point occurs as further singularity as well as possible surface wave poles in the integrands of the coupling integrals Eq. (2) and Eq. (4). All these

PIN ¼ PRad þ PAbs (8)

!<sup>∗</sup>

<sup>n</sup><sup>0</sup> kx; ky; zn � � e

jkyΔynm dkx dky

jkxΔxnm e

m,n<sup>0</sup> are the Fourier transforms of the

! ∞ is subtracted, leading to

jkyΔynm dkx dky

(9)

must hold. The results for the SIW with the absorber and the slot pair are given in Figure 4, right. It can be seen that the radiation power is below 20 percent up to about 37 GHz (the frequency band with good matching) with only one slot pair, whereas about 80 percent is converted in the absorber. If the input power according to Eq. (5) is normalized to 100 percent, then the sum in Eq. (8) leads to the red curve in Figure 4, right. Except some small deviation in the region 40–42 GHz, the 100 percent are met with good accuracy. However, for this accuracy

> � F !

<sup>n</sup><sup>0</sup> kx; ky; zn

basis functions in the origin of the coordinate system; thus, the Δxnm and Δyn,m are the lateral separations of the basis functions with regard to x and y. Figure 5 shows the different integration areas with the integrand behavior in the inner area AIn and the used simultaneous

For basis functions with very small lateral separations or overlapping functions and self-

Figure 4. Left: S-parameter extraction results for the SIW through line (tl), with absorber (abs) and additional double slot

\$Asy for <sup>k</sup><sup>r</sup> <sup>¼</sup>

!<sup>∗</sup>

<sup>m</sup><sup>0</sup> kx; ky:zm � � � <sup>F</sup>

� � ejkxΔxnm e

is a general Green's function tensor where all analytical space-domain integrations

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 <sup>x</sup> þ k 2 y

q

the number of sampling points for the numerical integration must be quite high. For an improved computational performance, a general coupling integral reads

are already incorporated. The F

integration path deformations (for both wavenumbers kx and ky) in this area.

\$Asy kx; ky; zm; zn

<sup>m</sup><sup>0</sup> kx; ky:zm � � � <sup>F</sup>

Vnm ¼ ð

> þ ð

where I G \$

kx

kx

ð

I G \$

24 Antennas and Wave Propagation

kx; ky; zm; zn � � � IG

0

couplings, an asymptotic representation IG

(ds), h = 0.254 mm. Right: power balance.

\$Asy kx; ky; zm; zn � � � <sup>F</sup>

� � h i

!

ky

ð

ky IG

with regard to z and z

Figure 5. Top left: Typical integrand behavior in the inner integration area with pole rings. Right: First quadrant with all integration areas. Bottom: Simultaneous integration path deformations in the inner area.

Figure 6. A leaky wave antenna with slots in the cover for end-fire radiation with the same SIW as in Figure 3. Slot separation 0.4 mm.

singularities are circumvented with the red integration path applied for both wavenumbers kx and ky(Figure 5, bottom). Since the only contribution in Eq.(3) for vias within the parallel plates comes from the parallel-plate wave pole, the number of integration sampling points in the inner area must be chosen significantly higher than typical microstrip and slot/coplanar

structures. For the structure in Figure 3 with slots and absorber, 16 sampling points must be applied for both kx and ky wavenumbers on the horizontal path to reach the power balance accuracy in Figure 4, right. On the vertical paths, two sampling points for each path are sufficient. For the numerical integration, composite Legendre-Filon quadrature techniques are

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

27

A leaky wave antenna for end-fire radiation is given in Figure 6 applying slots with constant length (l = 1.9 mm) in the middle and additional tapered slot areas at the ends for better matching. The simulated matching behavior is given in Figure 7 together with the desired

A good test for the reliability of the results is to compute the sum of the absorbed power Pabs and the radiated power Prad without dielectric losses. Thus we get for Pabs þ Prad nearly constant values around 97.5% if the integration accuracy for Eqs. (2)–(4) is sufficient. This is

For the connection of structures like in Figure 6 to a microstrip circuitry, a microstrip-SIW transition like in Figure 9 is well suited, consisting of a microstrip line with width wmic (0.33 mm) on a substrate with the height hMic (here 0.13 mm). The energy of the microstrip line is then coupled by the via (with distance d to the left wall, here 1.49 mm) through the antipad (blue, side length 0.42 mm) into the SIW with the height hSIW (0.75 mm) and width wSIW

The permittivity for both the microstrip line and the SIW is ε<sup>r</sup> ¼ 3.0. The incident (ii) and backward (ib) current waves on the microstrip line are extracted by means of the complex amplitudes of four current basis functions. The voltage waves in the SIW ui and ub are

extracted by means of four electric field monitors in the same way as in Figure 3.

Figure 9. Extraction of the S-parameter for a microstrip-SIW transition.

outlined in Figure 8 for three frequencies together with the antenna elevation patterns.

used.

(4.29 mm).

bandwidth for the MIMO radar project in [12].

3. Microstrip to SIW transitions

Figure 7. Matching behavior of the structure in Figure 6.

Figure 8. Radiation patterns and power balance of the leaky wave antenna in Figure 6.

structures. For the structure in Figure 3 with slots and absorber, 16 sampling points must be applied for both kx and ky wavenumbers on the horizontal path to reach the power balance accuracy in Figure 4, right. On the vertical paths, two sampling points for each path are sufficient. For the numerical integration, composite Legendre-Filon quadrature techniques are used.

A leaky wave antenna for end-fire radiation is given in Figure 6 applying slots with constant length (l = 1.9 mm) in the middle and additional tapered slot areas at the ends for better matching. The simulated matching behavior is given in Figure 7 together with the desired bandwidth for the MIMO radar project in [12].

A good test for the reliability of the results is to compute the sum of the absorbed power Pabs and the radiated power Prad without dielectric losses. Thus we get for Pabs þ Prad nearly constant values around 97.5% if the integration accuracy for Eqs. (2)–(4) is sufficient. This is outlined in Figure 8 for three frequencies together with the antenna elevation patterns.

## 3. Microstrip to SIW transitions

singularities are circumvented with the red integration path applied for both wavenumbers kx and ky(Figure 5, bottom). Since the only contribution in Eq.(3) for vias within the parallel plates comes from the parallel-plate wave pole, the number of integration sampling points in the inner area must be chosen significantly higher than typical microstrip and slot/coplanar

Figure 7. Matching behavior of the structure in Figure 6.

26 Antennas and Wave Propagation

Figure 8. Radiation patterns and power balance of the leaky wave antenna in Figure 6.

For the connection of structures like in Figure 6 to a microstrip circuitry, a microstrip-SIW transition like in Figure 9 is well suited, consisting of a microstrip line with width wmic (0.33 mm) on a substrate with the height hMic (here 0.13 mm). The energy of the microstrip line is then coupled by the via (with distance d to the left wall, here 1.49 mm) through the antipad (blue, side length 0.42 mm) into the SIW with the height hSIW (0.75 mm) and width wSIW (4.29 mm).

The permittivity for both the microstrip line and the SIW is ε<sup>r</sup> ¼ 3.0. The incident (ii) and backward (ib) current waves on the microstrip line are extracted by means of the complex amplitudes of four current basis functions. The voltage waves in the SIW ui and ub are extracted by means of four electric field monitors in the same way as in Figure 3.

Figure 9. Extraction of the S-parameter for a microstrip-SIW transition.

For the consistent evaluation of the wave quantities a1, b1, a2, b2, it is necessary to use the voltage-power definition for the characteristic impedance for the microstrip line and especially for the SIW. With the help of the equivalent width of a SIW:

weq <sup>¼</sup> wSIW � <sup>d</sup><sup>2</sup> circ <sup>0</sup>:<sup>95</sup> <sup>p</sup> (see, e.g. [3] and dcirc from Eq. (1)) and p the center to center distance of the vias we get for the characteristic impedance:

$$\mathcal{Z}\_{\text{L,SIV,VP}} = 2 \frac{h\_{\text{SIV}}}{w\_{eq}} \frac{\mu\_0 \omega}{\sqrt{k^2 - \left(\frac{\pi}{w\_{eq}}\right)^2}} = \frac{\left|\mathcal{U}\right|^2}{2P\_{w,a,b}} \tag{10}$$

factor of 2 due to a mistake during the derivation in [6]. For small values of hSIW , the structure in Figure 9 can be further optimized, and a broadband matching can be achieved with a microstrip circuitry as shown in the following structure. For larger values hSIW , the structure

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

29

In this case the feed via has only a length of 1.0 mm within the SIW with a height of 2.5 mm. Again the permittivity for both the microstrip line and the SIW is ε<sup>r</sup> ¼ 3.0. Such a structure is similar to a coaxial probe feed in a rectangular waveguide; thus, a start design can be made with solution methods like in [6], modified for filled waveguides. However, the influence of the antipad and optional pad structure typically has a significant influence on the overall impedance seen from the microstrip line. Therefore we started directly with numerical tests using typical values for pad and antipad size as well as for the length and the side length (here 0.43 mm) of the feed via. The other vias have a side length of 0.26 mm with a separation of 1 mm. This leads for the configuration in Figure 11 (without the matching stubs) in a short time to a design with the behavior in Figure 12, left. For the structure in Figure 11, only 311 basis functions are needed with a computation time of about 1 minute for 34 frequency points (AMD Phenom II X4 965 Quadcore processor). For the vertical discretization, six volume basis functions are used for each via of the SIW and the feed via to achieve stable

In this case, an additional pad did not show a further improvement, but with the shape of the antipad (blue), a further slight optimization could be performed, leading to a length of 0.9 mm and a width of 0.7 mm. However, with these specifications, we not yet achieve a matching below �10 dB in the desired frequency range from 30 to 40 GHz, but we get a flat curve between �7 and � 8 dB for ∣S11∣. Based on this behavior, a further matching with two additional microstrip stubs leads to the behavior in Figure 12, right, now showing a good matching from less than 30 GHz to more than 40 GHz. Figure 13 shows the matching with a SIW absorber at the right end of the SIW in Figure 11. In this case the absorber has a slightly

Besides this kind of transition, we have also specified transitions with a stepwise ridged waveguide. These structures are similar than the proposed transitions in [4], but we do not need the quite complex tapered shapes described in [4]. Our design is based on a two-stage Chebychef approach which can handle also much higher transform ratios than the designs in [6].

in Figure 11 is much better suited for broadband operation.

results.

smaller bandwidth than the transition.

Figure 11. Broadband microstrip-SIW transition for SIWs with larger thickness.

Figure 10 shows the results with the own IG-approach and with the time domain approach of CST Microwave Studio both with a very similar behavior. The green curve for ∣S11∣ in the own IG-approach is for a finer vertical discretization (three volume basis functions), whereas the other curves are computed with two volume basis functions showing already a stable solution behavior for this configuration. For ∣S22∣ we get nearly the same results as for ∣S11∣ as expected due to the very low dielectric losses considered for the simulation in contrast to the results with CST MWS. Structures like in Figure 9 with a small thickness of the SIW can be designed with the formulas, e.g., in [6]. For the input resistance, we get

$$R\_{\rm in} = 2 \cdot \frac{k\_0 \cdot h\_{\rm SIV} \cdot Z\_0}{w\_{eq} \cdot \beta\_{10}} \sin^2 \left(\beta\_{10} d\right) \sin^2 \left(\frac{\pi \alpha}{w\_{eq}}\right) \tag{11}$$

with k<sup>0</sup> and Z<sup>0</sup> the free space wave number and wave impedance, respectively, and β<sup>10</sup> the propagation constant in the filled equivalent rectangular waveguide. Typically, the feed via is centered in the SIW, i.e., x ¼ weq=2 and d is around a quarter of a guided wavelength and hSIW should be much smaller than a quarter of a guided wavelength to guarantee a nearly constant current distribution on the feed via. In contrast to the formula in [6], Eq. (11) has an additional

Figure 10. Results for the structure in Figure 9. Left: With own IG-approach. Right: With CST Microwave Studio (time domain).

factor of 2 due to a mistake during the derivation in [6]. For small values of hSIW , the structure in Figure 9 can be further optimized, and a broadband matching can be achieved with a microstrip circuitry as shown in the following structure. For larger values hSIW , the structure in Figure 11 is much better suited for broadband operation.

For the consistent evaluation of the wave quantities a1, b1, a2, b2, it is necessary to use the voltage-power definition for the characteristic impedance for the microstrip line and especially

> hSIW weq

k<sup>0</sup> hSIW Z<sup>0</sup> weq β<sup>10</sup>

<sup>0</sup>:<sup>95</sup> <sup>p</sup> (see, e.g. [3] and dcirc from Eq. (1)) and p the center to center distance of the

sin <sup>2</sup> <sup>β</sup><sup>10</sup> <sup>d</sup> � � sin <sup>2</sup> <sup>π</sup><sup>x</sup>

2Pw, a, <sup>b</sup>

weq � � (10)

(11)

μ0ω ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k <sup>2</sup> � <sup>π</sup> weq � �<sup>2</sup> <sup>r</sup> <sup>¼</sup> j j <sup>U</sup> <sup>2</sup>

Figure 10 shows the results with the own IG-approach and with the time domain approach of CST Microwave Studio both with a very similar behavior. The green curve for ∣S11∣ in the own IG-approach is for a finer vertical discretization (three volume basis functions), whereas the other curves are computed with two volume basis functions showing already a stable solution behavior for this configuration. For ∣S22∣ we get nearly the same results as for ∣S11∣ as expected due to the very low dielectric losses considered for the simulation in contrast to the results with CST MWS. Structures like in Figure 9 with a small thickness of the SIW can be designed with

with k<sup>0</sup> and Z<sup>0</sup> the free space wave number and wave impedance, respectively, and β<sup>10</sup> the propagation constant in the filled equivalent rectangular waveguide. Typically, the feed via is centered in the SIW, i.e., x ¼ weq=2 and d is around a quarter of a guided wavelength and hSIW should be much smaller than a quarter of a guided wavelength to guarantee a nearly constant current distribution on the feed via. In contrast to the formula in [6], Eq. (11) has an additional

Figure 10. Results for the structure in Figure 9. Left: With own IG-approach. Right: With CST Microwave Studio (time

for the SIW. With the help of the equivalent width of a SIW:

the formulas, e.g., in [6]. For the input resistance, we get

Rin ¼ 2 �

ZL,SIW,VP ¼ 2

weq <sup>¼</sup> wSIW � <sup>d</sup><sup>2</sup>

28 Antennas and Wave Propagation

domain).

circ

vias we get for the characteristic impedance:

In this case the feed via has only a length of 1.0 mm within the SIW with a height of 2.5 mm. Again the permittivity for both the microstrip line and the SIW is ε<sup>r</sup> ¼ 3.0. Such a structure is similar to a coaxial probe feed in a rectangular waveguide; thus, a start design can be made with solution methods like in [6], modified for filled waveguides. However, the influence of the antipad and optional pad structure typically has a significant influence on the overall impedance seen from the microstrip line. Therefore we started directly with numerical tests using typical values for pad and antipad size as well as for the length and the side length (here 0.43 mm) of the feed via. The other vias have a side length of 0.26 mm with a separation of 1 mm. This leads for the configuration in Figure 11 (without the matching stubs) in a short time to a design with the behavior in Figure 12, left. For the structure in Figure 11, only 311 basis functions are needed with a computation time of about 1 minute for 34 frequency points (AMD Phenom II X4 965 Quadcore processor). For the vertical discretization, six volume basis functions are used for each via of the SIW and the feed via to achieve stable results.

In this case, an additional pad did not show a further improvement, but with the shape of the antipad (blue), a further slight optimization could be performed, leading to a length of 0.9 mm and a width of 0.7 mm. However, with these specifications, we not yet achieve a matching below �10 dB in the desired frequency range from 30 to 40 GHz, but we get a flat curve between �7 and � 8 dB for ∣S11∣. Based on this behavior, a further matching with two additional microstrip stubs leads to the behavior in Figure 12, right, now showing a good matching from less than 30 GHz to more than 40 GHz. Figure 13 shows the matching with a SIW absorber at the right end of the SIW in Figure 11. In this case the absorber has a slightly smaller bandwidth than the transition.

Besides this kind of transition, we have also specified transitions with a stepwise ridged waveguide. These structures are similar than the proposed transitions in [4], but we do not need the quite complex tapered shapes described in [4]. Our design is based on a two-stage Chebychef approach which can handle also much higher transform ratios than the designs in [6].

Figure 11. Broadband microstrip-SIW transition for SIWs with larger thickness.

Figure 12. Results for the structure in Figure 11 with microstrip port to the left and SIW port to the right. Left: Without matching stubs. Right: With additional matching stubs.

Figure 13. Result for the structure in Figure 11 with SIW absorber to the right.

A typical design with our approach is given in Figure 14. Here, the transform ratio with the definition

$$\alpha = \frac{\text{hrw-hmic}}{\text{hrw}}$$

For the design of the transition, a quantity

discretization. Bottom: S-parameter (absolute values).

secΘ<sup>m</sup> <sup>¼</sup> cosh <sup>1</sup>

2

Figure 14. Design of a microstrip-SIW transition with two-stage ridged waveguide. Dimensions in mm.

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

31

cosh�<sup>1</sup> lnð Þ ZSIW <sup>=</sup>Zmic 2 � arip

 

 

Figure 15. Microstrip to SIW transition with two-stage ridged waveguide. Top: Back-to-back arrangement with

amounts to 7, whereas in [4], only a structure with a transform ratio of 4 was designed. hrw is the height of the standard SIW (or rectangular waveguide to the right) and hmic is the height of the microstrip line to the left.

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization… http://dx.doi.org/10.5772/intechopen.75062 31

Figure 14. Design of a microstrip-SIW transition with two-stage ridged waveguide. Dimensions in mm.

Figure 15. Microstrip to SIW transition with two-stage ridged waveguide. Top: Back-to-back arrangement with discretization. Bottom: S-parameter (absolute values).

For the design of the transition, a quantity

A typical design with our approach is given in Figure 14. Here, the transform ratio with the

Figure 13. Result for the structure in Figure 11 with SIW absorber to the right.

Figure 12. Results for the structure in Figure 11 with microstrip port to the left and SIW port to the right. Left: Without

<sup>α</sup> <sup>¼</sup> hrw-hmic hrw

amounts to 7, whereas in [4], only a structure with a transform ratio of 4 was designed. hrw is the height of the standard SIW (or rectangular waveguide to the right) and hmic is the height

definition

of the microstrip line to the left.

matching stubs. Right: With additional matching stubs.

30 Antennas and Wave Propagation

$$\sec\Theta\_m = \cosh\left(\frac{1}{2}\cosh^{-1}\left|\frac{\ln(Z\_{SIM}/Z\_{mic})}{2 \cdot a\_{rip}}\right|\right)$$

is computed, where ZSIW is the characteristic impedance of the SIW or the equivalent rectangular waveguide, here the voltage-power definition Eq. (10) should be used, and Zmic the characteristic impedance of the microstrip line. With this quantity, two reflection factors are computed with

$$
\Gamma\_0 = \frac{a\_{rip}}{2} \sec^2(\Theta\_m), \quad \Gamma\_1 = a\_{rip} \left(\sec^2(\Theta\_m) - 1\right).
$$

where arip is the tolerated intensity of the ripples in the passband, e.g., for 10 percent, αrip ¼ 0:1. With ZSIW ¼ 258 Ohm and Zmic ¼ 47 Ohm, we get secΘ<sup>m</sup> ¼ 2.18. With the reflection factors, we can compute the characteristic impedances for the ridged waveguide sections:

$$Z\_1 = Z\_{\rm mic} \cdot e^{2\Gamma\_0}, \quad Z\_2 = Z\_1 \cdot e^{2\Gamma\_1}$$

leading here to Z<sup>1</sup> ¼ 74.45 Ohm and Z<sup>2</sup> ¼ 161 Ohm. To determine the corresponding ridged waveguide dimensions, we have used CST MWS to compute the voltage from the ridge to the lower ground for a given power. This leads to the dimensions l1, w1, and h1 for length, width, and height of the first section and l2, w2, and h2 for the second section as also given in Figure 14. For the section which is connected with the microstrip line, we have chosen the same width as for the microstrip line because the differences of the characteristic impedances remain small. Of course a further adjustment of this width can be done. Figure 15, top, shows the transition in a back-to-back arrangement together with the discretization for our integral equation approach (side view shows only the first transition). In this case, we have modeled the cover of the SIW as a finite structure, whereas the bottom ground plane remains infinite.

#### 4. SIW horn antenna elements

Figure 15, bottom, shows the matching behavior of this arrangement. The reflection factor remains mainly below �15 dB; the structure may be a good alternative to the transition in Figure 11. The back-to-back structure was also built and measured successfully and was then combined with two radiating apertures shown in Figure 16. On the left the transition is combined with a simple open SIW and a short parallel-plate section. On the right an additional stepwise widening is applied. This is realized with two dielectric bars which are glued on the top and bottom side of the parallel-plate section where the metallization was removed before at these areas. As shown in the inset in Figure 16, the upper dielectric bar is metallized at the top and at the left side and analogously the lower bar to get a symmetric aperture with a height of 3.5 mm.

in Figure 17, right. The structures were manufactured and measured afterward showing the matching behavior in Figure 18 with the screenshots of the network analyzer. Similar as the simulated results, the simple structure with open SIW shows a poor matching around 5 dB up to the dip around 33.8 GHz, whereas the structure with widening shows a good matching mainly below 10 dB over the whole measured frequency range. Despite of the good matching behavior and further options to reduce the backward radiation, the stepwise widening is difficult

Figure 17. Left: Matching behavior of the structures in Figure 16. Blue: Open SIW and parallel-plate section. Green: With

Figure 16. Combinations of the transition in Figure 15 with two radiating apertures. Left: Open SIW with short parallel-

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

33

plate section. Right: With a stepwise widening at the end.

SIW horn antennas have already been studied for a longer time; typically they suffer from the substrate to the air transition which leads to larger reflections and backside radiation especially with thin substrates as we have seen with the open SIW aperture in Figure 16. In [9], a better matching and bandwidth enhancement of such kind of antennas is achieved with several parallel-plate sections in front of the origin antenna. However, the problem with the large

to fabricate, even in the case it is designed with via wholes like the feeding SIW.

widening. Right: Antenna pattern (gain) of the structure with widening. Simulations with CST MWS.

For the characterization of such structures with an end-fire radiation, we use CST Microwave Studio (time domain solver) at the moment, because it is still difficult with our integral equation framework to handle structures with finite dielectric layers to one side [5].

Figure 17 shows left the simulated matching behavior of the two structures. With the simple open SIW, the matching is still very poor as expected (blue line), whereas with the additional widening, the reflection factor remains below �10 dB for a large frequency range. However, despite of the additional widening, we still get a rather strong backward radiation as illustrated A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization… http://dx.doi.org/10.5772/intechopen.75062 33

is computed, where ZSIW is the characteristic impedance of the SIW or the equivalent rectangular waveguide, here the voltage-power definition Eq. (10) should be used, and Zmic the characteristic impedance of the microstrip line. With this quantity, two reflection factors are

ð Þ <sup>Θ</sup><sup>m</sup> , <sup>Γ</sup><sup>1</sup> <sup>¼</sup> arip sec<sup>2</sup>

where arip is the tolerated intensity of the ripples in the passband, e.g., for 10 percent, αrip ¼ 0:1. With ZSIW ¼ 258 Ohm and Zmic ¼ 47 Ohm, we get secΘ<sup>m</sup> ¼ 2.18. With the reflection factors, we

leading here to Z<sup>1</sup> ¼ 74.45 Ohm and Z<sup>2</sup> ¼ 161 Ohm. To determine the corresponding ridged waveguide dimensions, we have used CST MWS to compute the voltage from the ridge to the lower ground for a given power. This leads to the dimensions l1, w1, and h1 for length, width, and height of the first section and l2, w2, and h2 for the second section as also given in Figure 14. For the section which is connected with the microstrip line, we have chosen the same width as for the microstrip line because the differences of the characteristic impedances remain small. Of course a further adjustment of this width can be done. Figure 15, top, shows the transition in a back-to-back arrangement together with the discretization for our integral equation approach (side view shows only the first transition). In this case, we have modeled the cover of the SIW as a finite structure, whereas the bottom ground plane remains infinite.

Figure 15, bottom, shows the matching behavior of this arrangement. The reflection factor remains mainly below �15 dB; the structure may be a good alternative to the transition in Figure 11. The back-to-back structure was also built and measured successfully and was then combined with two radiating apertures shown in Figure 16. On the left the transition is combined with a simple open SIW and a short parallel-plate section. On the right an additional stepwise widening is applied. This is realized with two dielectric bars which are glued on the top and bottom side of the parallel-plate section where the metallization was removed before at these areas. As shown in the inset in Figure 16, the upper dielectric bar is metallized at the top and at the left side and analogously the lower bar to get a symmetric aperture with a height of 3.5 mm. For the characterization of such structures with an end-fire radiation, we use CST Microwave Studio (time domain solver) at the moment, because it is still difficult with our integral

equation framework to handle structures with finite dielectric layers to one side [5].

Figure 17 shows left the simulated matching behavior of the two structures. With the simple open SIW, the matching is still very poor as expected (blue line), whereas with the additional widening, the reflection factor remains below �10 dB for a large frequency range. However, despite of the additional widening, we still get a rather strong backward radiation as illustrated

<sup>2</sup>Γ<sup>0</sup> , Z<sup>2</sup> <sup>¼</sup> <sup>Z</sup><sup>1</sup> � <sup>e</sup>

ð Þ� <sup>Θ</sup><sup>m</sup> <sup>1</sup>

2Γ<sup>1</sup>

<sup>Γ</sup><sup>0</sup> <sup>¼</sup> arip

4. SIW horn antenna elements

<sup>2</sup> sec<sup>2</sup>

can compute the characteristic impedances for the ridged waveguide sections:

Z<sup>1</sup> ¼ Zmic � e

computed with

32 Antennas and Wave Propagation

Figure 16. Combinations of the transition in Figure 15 with two radiating apertures. Left: Open SIW with short parallelplate section. Right: With a stepwise widening at the end.

Figure 17. Left: Matching behavior of the structures in Figure 16. Blue: Open SIW and parallel-plate section. Green: With widening. Right: Antenna pattern (gain) of the structure with widening. Simulations with CST MWS.

in Figure 17, right. The structures were manufactured and measured afterward showing the matching behavior in Figure 18 with the screenshots of the network analyzer. Similar as the simulated results, the simple structure with open SIW shows a poor matching around 5 dB up to the dip around 33.8 GHz, whereas the structure with widening shows a good matching mainly below 10 dB over the whole measured frequency range. Despite of the good matching behavior and further options to reduce the backward radiation, the stepwise widening is difficult to fabricate, even in the case it is designed with via wholes like the feeding SIW.

SIW horn antennas have already been studied for a longer time; typically they suffer from the substrate to the air transition which leads to larger reflections and backside radiation especially with thin substrates as we have seen with the open SIW aperture in Figure 16. In [9], a better matching and bandwidth enhancement of such kind of antennas is achieved with several parallel-plate sections in front of the origin antenna. However, the problem with the large

Figure 18. Measured matching behavior of the structures in Figure 16. Left: Open SIW and parallel-plate section. Right: With widening.

35 GHz. At the right the matching behavior of the structure with H10wave excitation and variation of xslot is illustrated. With a desired center frequency of 35 GHz and a bandwidth with a range of 30 to 40 Ghz at least, a value of 1.95 mm for xslot is best suited in this context. The dependence of the matching behavior with regard to lrod and the slot width is much lower; in this case, a slot width of 0.2 mm was used. Figure 20 shows the azimuth diagrams for three values of xslot together with the forward-to-backward ratio (FTBR). It can be observed that the best value is reached for xslot = 1.95 mm, which also leads to the best matching behavior.

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

35

Figure 21. Complete antenna element in SIW technology. Top: Representation with hidden components. Bottom: With

transparent substrates. Dimensions in mm.

Figure 20. Azimuth diagrams at 35 GHz and FBTR for three values of xslot.

Figure 19. Left: Rectangular waveguide with ringslot and dielectric rod. Right: Matching behavior of the left structure in dependence of xslot (in mm) and lrod fixed to 2.5 mm (CST MWS).

backside radiation is only slightly reduced or not at all; typically, a good matching is achieved but simultaneously with a large backside radiation and vice versa. Some further improvements are made in [10] using transitions with a saw-tooth geometry. But for a good performance, the aperture width must be typically larger than one free space wavelength; thus, an application within a typical phased array arrangement requiring an element distance around λ0=2 is not possible.

In contrast to the parallel-plate configurations in [9, 10] or the widening in Figure 16, right, for a better matching of SIW horn antennas, we use in a further study a (rectangular) ringslot to improve the matching behavior and bandwidth together with a reduction of the backside radiation. Figure 19 shows the structure with the rectangular ringslot located with the distance xslot apart from the right metallic edge of the rectangular waveguide and a dielectric extension with the length lrod (fixed to 2.5 mm). The width of the waveguide and dielectric is set to 4.0 mm, the height to 2.0 mm. This means that the width is only slightly higher than λ0=2 at A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization… http://dx.doi.org/10.5772/intechopen.75062 35

Figure 20. Azimuth diagrams at 35 GHz and FBTR for three values of xslot.

backside radiation is only slightly reduced or not at all; typically, a good matching is achieved but simultaneously with a large backside radiation and vice versa. Some further improvements are made in [10] using transitions with a saw-tooth geometry. But for a good performance, the aperture width must be typically larger than one free space wavelength; thus, an application within a typical phased array arrangement requiring an element distance around λ0=2 is not

Figure 19. Left: Rectangular waveguide with ringslot and dielectric rod. Right: Matching behavior of the left structure in

dependence of xslot (in mm) and lrod fixed to 2.5 mm (CST MWS).

Figure 18. Measured matching behavior of the structures in Figure 16. Left: Open SIW and parallel-plate section. Right:

In contrast to the parallel-plate configurations in [9, 10] or the widening in Figure 16, right, for a better matching of SIW horn antennas, we use in a further study a (rectangular) ringslot to improve the matching behavior and bandwidth together with a reduction of the backside radiation. Figure 19 shows the structure with the rectangular ringslot located with the distance xslot apart from the right metallic edge of the rectangular waveguide and a dielectric extension with the length lrod (fixed to 2.5 mm). The width of the waveguide and dielectric is set to 4.0 mm, the height to 2.0 mm. This means that the width is only slightly higher than λ0=2 at

possible.

With widening.

34 Antennas and Wave Propagation

35 GHz. At the right the matching behavior of the structure with H10wave excitation and variation of xslot is illustrated. With a desired center frequency of 35 GHz and a bandwidth with a range of 30 to 40 Ghz at least, a value of 1.95 mm for xslot is best suited in this context.

The dependence of the matching behavior with regard to lrod and the slot width is much lower; in this case, a slot width of 0.2 mm was used. Figure 20 shows the azimuth diagrams for three values of xslot together with the forward-to-backward ratio (FTBR). It can be observed that the best value is reached for xslot = 1.95 mm, which also leads to the best matching behavior.

Figure 21. Complete antenna element in SIW technology. Top: Representation with hidden components. Bottom: With transparent substrates. Dimensions in mm.

The structure in Figure 19 has then been transformed into SIW technology and combined with the microstrip to SIW transition of Figure 11 with the substrate thickness 2.5 mm. This leads to the structure in Figure 21. The slot width in the lower and upper metallization amounts to 0.3 mm; the via separation near the slot was reduced to 0.72 mm. In the region of the feeding via, the SIW has a width of 3.5 mm as in Figure 11 (center to center of the vias) and is then increased by a smooth taper to 4.1 mm.

The vias close to the slot are modeled with a round cross section, but the differences compared with a quadratic cross section are negligible.

Figure 22 shows the matching behavior simulated with CST. We can see that the reflection factor is even below 15 dB in the range from 30 to 40 GHz.

Finally, Figure 18 shows the antenna patterns derived with CST MWS. Especially by hands of the 3D pattern, we can observe that the overall backside radiation is very low. A similar antenna element like in Figure 21 was already built, but in the microstrip-to-SIW transition, the feeding via is connected with the bottom ground plane like in Figure 9 leading therefore to a (desired) smaller bandwidth of about 2 GHz around the center frequency 35 GHz well suited for the project in [12] mentioned in context with the leaky wave antenna in Figures 6 and 7. The fabricated elements show a similar antenna pattern than in Figure 23 and are already tested in a phased array arrangement.

5. Conclusions

Author details

Thomas Vaupel

Germany

This chapter comprises the application of an integral equation framework to the analysis and design of substrate integrated waveguide (SIW) components and antennas. For SIW circuits, microstrip to SIW transitions, and slot antennas, only the vias must be considered for the discretization of the SIWs in many cases, whereas radiating slots or antipads are modeled with magnetic surface currents. The computational performance and accuracy are significantly improved by a subdivision of the Cartesian wavenumber plane, integration path deformations, an asymptotic subtraction technique, as well as extended quadrature methods. The introduction of special SIW ports, field monitoring, and compact SIW absorbers allows an effective S-parameter extraction and power balance control also in combination with microstrip ports. For SIWs with larger thickness, broadband microstrip-to-SIW transitions have been designed based on a feed via with pad/antipad combination and a two-stage ridged-SIW structure. The latter was combined with an end-fire horn antenna based on a stepwise widening and was successfully built and measured. But the best radiation properties could be obtained with a combination of a SIW with a rectangular ringslot and a dielectric rod showing both broadband behavior and a small backward radiation. This structure is already tested within different array arrangements for a MIMO radar using a transition with feed via connected to the lower ground and smaller bandwidth, whereas a

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization…

http://dx.doi.org/10.5772/intechopen.75062

37

Figure 23. Azimuth, elevation, and 3D antenna pattern of the antenna element in Figure 21 at 35 GHz.

broadband version is used in near future for an airborne synthetic aperture radar.

Fraunhofer Institute for High Frequency Physics and Radar Techniques FHR, Wachtberg,

Address all correspondence to: thomas.vaupel@fhr.fraunhofer.de

Figure 22. Left: Matching behavior of the complete antenna element in Figure 21 (CST MWS). Right: Fabricated element (here for the project in [12]), top view with microstrip line and matching stubs and bottom view with 1 Euro coin.

A Combined Electric/Magnetic Field Surface Volume Integral Equation Approach for the Fast Characterization… http://dx.doi.org/10.5772/intechopen.75062 37

Figure 23. Azimuth, elevation, and 3D antenna pattern of the antenna element in Figure 21 at 35 GHz.

### 5. Conclusions

The structure in Figure 19 has then been transformed into SIW technology and combined with the microstrip to SIW transition of Figure 11 with the substrate thickness 2.5 mm. This leads to the structure in Figure 21. The slot width in the lower and upper metallization amounts to 0.3 mm; the via separation near the slot was reduced to 0.72 mm. In the region of the feeding via, the SIW has a width of 3.5 mm as in Figure 11 (center to center of the vias) and is then

The vias close to the slot are modeled with a round cross section, but the differences compared

Figure 22 shows the matching behavior simulated with CST. We can see that the reflection

Finally, Figure 18 shows the antenna patterns derived with CST MWS. Especially by hands of the 3D pattern, we can observe that the overall backside radiation is very low. A similar antenna element like in Figure 21 was already built, but in the microstrip-to-SIW transition, the feeding via is connected with the bottom ground plane like in Figure 9 leading therefore to a (desired) smaller bandwidth of about 2 GHz around the center frequency 35 GHz well suited for the project in [12] mentioned in context with the leaky wave antenna in Figures 6 and 7. The fabricated elements show a similar antenna pattern than in Figure 23 and are already

Figure 22. Left: Matching behavior of the complete antenna element in Figure 21 (CST MWS). Right: Fabricated element (here for the project in [12]), top view with microstrip line and matching stubs and bottom view with 1 Euro coin.

increased by a smooth taper to 4.1 mm.

36 Antennas and Wave Propagation

tested in a phased array arrangement.

with a quadratic cross section are negligible.

factor is even below 15 dB in the range from 30 to 40 GHz.

This chapter comprises the application of an integral equation framework to the analysis and design of substrate integrated waveguide (SIW) components and antennas. For SIW circuits, microstrip to SIW transitions, and slot antennas, only the vias must be considered for the discretization of the SIWs in many cases, whereas radiating slots or antipads are modeled with magnetic surface currents. The computational performance and accuracy are significantly improved by a subdivision of the Cartesian wavenumber plane, integration path deformations, an asymptotic subtraction technique, as well as extended quadrature methods. The introduction of special SIW ports, field monitoring, and compact SIW absorbers allows an effective S-parameter extraction and power balance control also in combination with microstrip ports. For SIWs with larger thickness, broadband microstrip-to-SIW transitions have been designed based on a feed via with pad/antipad combination and a two-stage ridged-SIW structure. The latter was combined with an end-fire horn antenna based on a stepwise widening and was successfully built and measured. But the best radiation properties could be obtained with a combination of a SIW with a rectangular ringslot and a dielectric rod showing both broadband behavior and a small backward radiation. This structure is already tested within different array arrangements for a MIMO radar using a transition with feed via connected to the lower ground and smaller bandwidth, whereas a broadband version is used in near future for an airborne synthetic aperture radar.

## Author details

Thomas Vaupel

Address all correspondence to: thomas.vaupel@fhr.fraunhofer.de

Fraunhofer Institute for High Frequency Physics and Radar Techniques FHR, Wachtberg, Germany

## References

[1] Buchta M, Heinrich W. On the equivalence between cylindrical and rectangular via-holes in electromagnetic modeling. In: Proceedings of the 37th EuMC. Oct. 2007:142-145

**Chapter 3**

**Provisional chapter**

**Time-Domain Analysis of Modified Vivaldi Antennas**

In the ultra-wideband (UWB) application frequency domain parameters such as gain, group delay isn't sufficient to demonstrate the performance of the antenna. Besides frequency domain analysis, a time-domain analysis is required to characterize the transient behavior of UWB antennas for pulsed operations since pulse distortion of the UWB antenna reduces the system performance and decreases the signal to noise ratio (SNR) of the UWB communication system. Vivaldi antenna is a widely used UWB antenna, especially in microwave imaging applications. Performance of Vivaldi antennas is enhanced by adding corrugation on the edge of exponential flaring and/or grating elements on the slot area. From the measured scattering parameters of modified Vivaldi antennas, pulse preserving capabilities are observed. Pulse width extension and fidelity factor parameters are used to define the similarity between the transmitted and received pulse. The analysis is performed with angular dependence with respect to the signal transmitted at the main

**Keywords:** UWB antenna, Vivaldi antenna, time domain, pulse distortion, fidelity

Ultra-wideband (UWB) systems have been used in various applications that range from deep space investigation to commercial telecommunication links and radars with high spatial resolutions [1–3]. Due to its low complexity, small physical size, low manufacturing cost, low interference and high time-domain resolution, it is widely used in communication systems,

In 2012, federal communications commission (FCC) has allocated 7.5 GHz-wide frequency band that ranges from 3.1 to 10.6 GHz for UWB applications [4]. In UWB systems, antennas

**Time-Domain Analysis of Modified Vivaldi Antennas**

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

DOI: 10.5772/intechopen.74945

Sultan Aldırmaz Çolak and Nurhan Türker Tokan

Sultan Aldırmaz Çolak and Nurhan Türker Tokan

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74945

**Abstract**

beam direction.

**1. Introduction**

analysis, corrugation

microwave imaging, remote sensing and radar.


#### **Time-Domain Analysis of Modified Vivaldi Antennas Time-Domain Analysis of Modified Vivaldi Antennas**

DOI: 10.5772/intechopen.74945

Sultan Aldırmaz Çolak and Nurhan Türker Tokan Sultan Aldırmaz Çolak and Nurhan Türker Tokan

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74945

#### **Abstract**

References

38 Antennas and Wave Propagation

Sep-Oct. 2012;3(5):36-40

ISBN: 978-953-307-083-4

April 2013;61(4):1923-1930

[1] Buchta M, Heinrich W. On the equivalence between cylindrical and rectangular via-holes in electromagnetic modeling. In: Proceedings of the 37th EuMC. Oct. 2007:142-145

[2] Caballero ED, Esteban H, Belenguer A, et al. Efficient analysis of substrate integrated waveguide devices using hybride mode-matching between cylindrical and guided modes.

[3] Kumar H, Jadhav R, Ranade S. A review on substrate integrated waveguide and its microstrip interconnect. IOSR Journal of Electronics and Communication Engineering.

[4] Ding Y, Wu K. Substrate integrated waveguide-to-microstrip transition in multilayer substrate. IEEE Transactions on Microwave Theory and Techniques. Dec. 2007;55(12):2839-2844

[5] Vaupel T. Design of end-fire substrate integrated waveguide antenna elements using in-house planar-3D integral equation frameworks and commercial full-3D methods. In: 8th European

[6] Yau CK, Huang TY, Shen TM, et al. Design and modelling of microstrip line to substrate integrated waveguide transitions. In: Zhurbenko V, editor. Passive Microwave Components and Antennas. InTechOpen. April 2010:225-246. Chapter 11. DOI: 10.5772/9418

[7] Vaupel T. A MFIE/Volume integral equation approach with minimum discretization effort for substrate integrated waveguide structures and leaky wave/slot antennas. IEEE

[8] Liu J, Jackson D, Long Y. Substrate integrated waveguide (SIW) leaky-wave antenna with transverse slots. IEEE Transaction on Antennas and Propagation. Jan 2012;60(1):20-29 [9] Esquis-Morote M, Fuchs B, Zürcher J-F, Mosig JR. A printed transition for matching improvement of SIW horn antennas. IEEE Transactions on Antennas and Propagation.

[10] Esquis-Morote M, Fuchs B, Zürcher J-F, Mosig JR. Novel thin and compact H-Plane SIW horn antenna. IEEE Transactions on Antennas and Propagation. June 2013;61(6):2911-2920

[11] Vaupel T, Eibert TF, Hansen V. Spectral domain analysis of large (M)MIC-structures using novel quadrature methods. International Journal of Numerical Modelling: Electronic Net-

[12] Panhuber R, Klenke R, Biallawons O, Klare J. System concept for the Imaging MIMO Radar of the radar warning and information System RAWIS. In: Proceedings of EUSAR 2016: 11th

European Conference on Synthetic Aperture Radar. Hamburg, Germany. 2016:1-4

APS Symposium. Fajardo, Puerto Rico; 2016:1327-1328

works, Devices and Fields. Jan-Feb. 2005;18:23-38

Conference on Antennas and Propagation. Netherlands: The Hague; 2014:2512-2515

IEEE Transactions on Microwave Theory and Techniques. Feb. 2012;60(2):232-243

In the ultra-wideband (UWB) application frequency domain parameters such as gain, group delay isn't sufficient to demonstrate the performance of the antenna. Besides frequency domain analysis, a time-domain analysis is required to characterize the transient behavior of UWB antennas for pulsed operations since pulse distortion of the UWB antenna reduces the system performance and decreases the signal to noise ratio (SNR) of the UWB communication system. Vivaldi antenna is a widely used UWB antenna, especially in microwave imaging applications. Performance of Vivaldi antennas is enhanced by adding corrugation on the edge of exponential flaring and/or grating elements on the slot area. From the measured scattering parameters of modified Vivaldi antennas, pulse preserving capabilities are observed. Pulse width extension and fidelity factor parameters are used to define the similarity between the transmitted and received pulse. The analysis is performed with angular dependence with respect to the signal transmitted at the main beam direction.

**Keywords:** UWB antenna, Vivaldi antenna, time domain, pulse distortion, fidelity analysis, corrugation

## **1. Introduction**

Ultra-wideband (UWB) systems have been used in various applications that range from deep space investigation to commercial telecommunication links and radars with high spatial resolutions [1–3]. Due to its low complexity, small physical size, low manufacturing cost, low interference and high time-domain resolution, it is widely used in communication systems, microwave imaging, remote sensing and radar.

In 2012, federal communications commission (FCC) has allocated 7.5 GHz-wide frequency band that ranges from 3.1 to 10.6 GHz for UWB applications [4]. In UWB systems, antennas

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

can be considered as the key component that affects the system performance. The performance and quality of UWB antennas are mostly analyzed in the frequency domain. For narrow-band systems, traditional frequency domain parameters (such as gain, group delay time, etc.) are enough to assess the performance of the antenna. However, in the ultra-wideband applications, these parameters are not sufficient to demonstrate the performance of the antenna. Besides frequency domain analysis, a time-domain analysis is required to characterize the transient behavior of UWB antennas for pulsed operations [5]. The antenna's transient response is the quantity for the characterization of the signal distortion and correlation of the radiated pulse in the time domain [6–7]. The time-dispersion characteristic of the antenna is one of the most important factors that should be considered in ultra-wideband applications since antennas with smaller time dispersion will have a better energy-focusing property and anti-interference performance. Antenna with larger dispersion leads to a less stable phase center and increases the interference in UWB communications based on pulse radio [8]. Consequently, this will result in time spreading of pulses and make signal transmission less predictable and less reliable in sensor system. Thus, it is necessary to investigate the antenna and radio propagation performance. This analysis should be performed not only in the main beam direction of the antenna but also in an arbitrary angular direction. By calculating the correlation between the transmitted signal and radiated signal in an arbitrary angular direction, the angular region where the pulse is preserved can be determined.

Pulse-preserving capabilities of modified Vivaldi antennas given in **Figure 1** are measured in terms of two parameters: the pulse width extension and the fidelity factor. In addition, the results are compared with that of the standard Vivaldi antennas. Their time-domain performance is quantified by their respective standard deviations from the width of the ideal signal that has constant value. Standard Fourier transform relationship is used to recover time-domain waveforms. Although this chapter concentrates on Vivaldi structures operating

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945 41

This chapter is organized as follows: In the next Section, time-domain analysis and the procedure for the analysis of the angular distortion of radiated pulses are presented. In Section 3, the modified Vivaldi antennas are introduced and their frequency-domain performance is demonstrated. Measurement setup and time-domain analysis results of the antennas are

Despite the wide frequency of the radiation, the transmitted UWB waveform is dispersive. This is one major difference from narrow-band circumstances. When the signal is transmitted through a UWB antenna, the antenna output signal contains the input signal and its derivatives with varying delays, caused mainly by the resonances in the radiator structure. Frequency-dependent variations of antenna characteristics and reflection coefficients, and RF circuitry, are considered as the sources of waveform dispersion [18]. When *S*21 parameter of the link in the frequency domain presents linear phase variation which results as non-dispersive radiation and constant amplitude which results as no amplitude distortion, the time-domain

Impulse response of a link, *S*21(*t*, *<sup>θ</sup>*, *<sup>ϕ</sup>*), can be derived by taking the inverse Fourier transform

in 3.1–10.6 GHz, the analysis is also applicable to any UWB antenna.

**Figure 1.** Modified Vivaldi antennas with (a) corrugation and (b) corrugation and strip.

given in Section 4. Section 5 concludes the whole chapter.

**2. Time-domain analysis**

pulse is not distorted.

of *S*21(*ω*, *<sup>θ</sup>*, *<sup>ϕ</sup>*):

In order to radiate a short pulse with low distortion, the antenna needs to efficiently operate over a broad bandwidth and be non-dispersive, with a stable phase center at all frequencies. One of the most widely employed solutions for broadband planar, directive antenna is tapered slot antenna. With its relatively small physical size, low cost, easy integration with the circuit board and almost stable radiation properties within its band, exponentially tapered slot antenna, which is also known as Vivaldi antenna, became a good candidate for UWB communication, radar and microwave imaging applications [9–10]. Recent works have aimed to increase its radiation performance by physically modifying its structure. A parasitic elliptical patch is inserted in the slot aperture for radiation stability and directivity improvement at high frequencies [11]. In [12], to reduce side- and back-lobe levels, extend the bandwidth and increase the main lobe gain, an exponential slot edge is added to the structure. In [13], a double slot structure is proposed in the Vivaldi antenna, to enhance the directivity and radiation performance, whereas double antipodal structure having corrugated edges and the semicircle director is presented in [14] for the same purpose. Vivaldi antenna is modified by incorporating corrugations on the edges of the exponential tapered slots and periodic grating elements consisting of metallic strips on the slot area in [15] to increase gain, especially at the lower end of the frequency band.

Time-domain characteristics of standard Vivaldi antennas are investigated and proved to be weakly dispersive in [16]. In [17], the time-domain radiation properties of the Vivaldi antenna are analyzed with angular dependence with respect to the signal transmitted at the main beam direction. In literature, time-domain characteristics of the modified Vivaldi antennas are not considered so far. With this work, effects of the physical modification on the Vivaldi structure will be observed in the time domain. If the modifications made to improve frequency-domain parameters would benefit time-domain parameters, they will be investigated, and potentials of these antennas to be used as basic element for non-distorted radiated link are discussed.

**Figure 1.** Modified Vivaldi antennas with (a) corrugation and (b) corrugation and strip.

Pulse-preserving capabilities of modified Vivaldi antennas given in **Figure 1** are measured in terms of two parameters: the pulse width extension and the fidelity factor. In addition, the results are compared with that of the standard Vivaldi antennas. Their time-domain performance is quantified by their respective standard deviations from the width of the ideal signal that has constant value. Standard Fourier transform relationship is used to recover time-domain waveforms. Although this chapter concentrates on Vivaldi structures operating in 3.1–10.6 GHz, the analysis is also applicable to any UWB antenna.

This chapter is organized as follows: In the next Section, time-domain analysis and the procedure for the analysis of the angular distortion of radiated pulses are presented. In Section 3, the modified Vivaldi antennas are introduced and their frequency-domain performance is demonstrated. Measurement setup and time-domain analysis results of the antennas are given in Section 4. Section 5 concludes the whole chapter.

## **2. Time-domain analysis**

can be considered as the key component that affects the system performance. The performance and quality of UWB antennas are mostly analyzed in the frequency domain. For narrow-band systems, traditional frequency domain parameters (such as gain, group delay time, etc.) are enough to assess the performance of the antenna. However, in the ultra-wideband applications, these parameters are not sufficient to demonstrate the performance of the antenna. Besides frequency domain analysis, a time-domain analysis is required to characterize the transient behavior of UWB antennas for pulsed operations [5]. The antenna's transient response is the quantity for the characterization of the signal distortion and correlation of the radiated pulse in the time domain [6–7]. The time-dispersion characteristic of the antenna is one of the most important factors that should be considered in ultra-wideband applications since antennas with smaller time dispersion will have a better energy-focusing property and anti-interference performance. Antenna with larger dispersion leads to a less stable phase center and increases the interference in UWB communications based on pulse radio [8]. Consequently, this will result in time spreading of pulses and make signal transmission less predictable and less reliable in sensor system. Thus, it is necessary to investigate the antenna and radio propagation performance. This analysis should be performed not only in the main beam direction of the antenna but also in an arbitrary angular direction. By calculating the correlation between the transmitted signal and radiated signal in an arbitrary

40 Antennas and Wave Propagation

angular direction, the angular region where the pulse is preserved can be determined.

In order to radiate a short pulse with low distortion, the antenna needs to efficiently operate over a broad bandwidth and be non-dispersive, with a stable phase center at all frequencies. One of the most widely employed solutions for broadband planar, directive antenna is tapered slot antenna. With its relatively small physical size, low cost, easy integration with the circuit board and almost stable radiation properties within its band, exponentially tapered slot antenna, which is also known as Vivaldi antenna, became a good candidate for UWB communication, radar and microwave imaging applications [9–10]. Recent works have aimed to increase its radiation performance by physically modifying its structure. A parasitic elliptical patch is inserted in the slot aperture for radiation stability and directivity improvement at high frequencies [11]. In [12], to reduce side- and back-lobe levels, extend the bandwidth and increase the main lobe gain, an exponential slot edge is added to the structure. In [13], a double slot structure is proposed in the Vivaldi antenna, to enhance the directivity and radiation performance, whereas double antipodal structure having corrugated edges and the semicircle director is presented in [14] for the same purpose. Vivaldi antenna is modified by incorporating corrugations on the edges of the exponential tapered slots and periodic grating elements consisting of metallic strips on the slot area in [15] to increase gain, especially at the lower end of the frequency band.

Time-domain characteristics of standard Vivaldi antennas are investigated and proved to be weakly dispersive in [16]. In [17], the time-domain radiation properties of the Vivaldi antenna are analyzed with angular dependence with respect to the signal transmitted at the main beam direction. In literature, time-domain characteristics of the modified Vivaldi antennas are not considered so far. With this work, effects of the physical modification on the Vivaldi structure will be observed in the time domain. If the modifications made to improve frequency-domain parameters would benefit time-domain parameters, they will be investigated, and potentials of these antennas to be used as basic element for non-distorted radiated link are discussed.

Despite the wide frequency of the radiation, the transmitted UWB waveform is dispersive. This is one major difference from narrow-band circumstances. When the signal is transmitted through a UWB antenna, the antenna output signal contains the input signal and its derivatives with varying delays, caused mainly by the resonances in the radiator structure. Frequency-dependent variations of antenna characteristics and reflection coefficients, and RF circuitry, are considered as the sources of waveform dispersion [18]. When *S*21 parameter of the link in the frequency domain presents linear phase variation which results as non-dispersive radiation and constant amplitude which results as no amplitude distortion, the time-domain pulse is not distorted.

Impulse response of a link, *S*21(*t*, *<sup>θ</sup>*, *<sup>ϕ</sup>*), can be derived by taking the inverse Fourier transform of *S*21(*ω*, *<sup>θ</sup>*, *<sup>ϕ</sup>*):

$$\mathcal{S}\_{\text{2}}(t,\theta,\varphi) = \text{IFT}\left\{ \mathcal{S}\_{\text{2}}(\omega,\theta,\varphi) \right\} \tag{1}$$

*FF* = *max<sup>τ</sup>*

about the amplitudes of signals.

**3.1. Antenna design**

**3. Modified Vivaldi antennas**

The dielectric constant of the dielectric material is *<sup>ε</sup><sup>r</sup>* <sup>=</sup> 2.33.

∫ −∞ +∞

\_\_\_\_\_\_\_\_\_\_\_\_ <sup>∫</sup> −∞ +∞ <sup>|</sup>*Sref*(*t*) | 2 *dt* <sup>√</sup>

where *Sref*(*t*) and *S*21(*t*) are the transmitted and received signals, respectively. If the transmitted and received signals are exactly same, *FF* coefficient has its maximum of 1. When *FF* coefficient is 1, input signal isn't distorted by the antenna. Fidelity depends on the spatial radiation characteristics of the antenna. Thus, angular variation of the *FF* coefficient should also be observed. Because of the normalization procedure, fidelity factor cannot provide information

The Vivaldi antenna is one of the classical ultra-wideband antennas with many applications [8]. It is a traveling-wave, end-fire antenna and due to its completely planar structure, it can be easily integrated in UWB sensor circuit. It has almost symmetric radiation patterns in the E and H planes. Theoretically, with its exponentially tapered slot, the Vivaldi antenna has an unlimited range of operating frequencies. However, in practice, it is constrained by the physical dimensions such as taper dimensions, the slot line width and transition from the feed line. The structure of the standard Vivaldi antenna together with its dimensions is shown in **Figure 2a**. The proposed Vivaldi antenna consists of a microstrip feed line, microstrip line to slot line transition and the radiating structure. It is designed to operate efficiently as the transmitter and receiver in the unlicensed band of 3.1–10.6 GHz (7.5 GHz bandwidth). The slot curve of the Vivaldi antenna is the exponential function, which is expressed as *S*(*z*) <sup>=</sup> (*Wslot*/2) *<sup>e</sup> az* where *<sup>a</sup>* <sup>=</sup> 0.165 and *Wslot* <sup>=</sup> 0.25 mm. A quarter wavelength open circuit stub is used for wideband matching. The aperture coupling is optimized for the frequency range from 3.1 to 10.6 GHz. The size of the Vivaldi antenna is 50 × 50 mm. Its dimensions are given in **Figure 2a**. Rogers RT/ Duroid 5870 with 0.51-mm dielectric thickness and 17.5-um copper is chosen for the design.

One of the bottlenecks of the conventional Vivaldi antenna is its relatively low directivity, especially at lower frequencies of the band. The lower frequency response of Vivaldi antennas with satisfactory impedance match and effective radiation is usually improved by increasing the aperture size. Another solution is introducing variable length slots to effectively increase the aperture of the antenna [15, 20]. It is shown that by incorporating a corrugated profile on the sides of exponential flaring, more suitable characteristics, especially for microwave imaging applications (i.e., higher gain, broader bandwidth), can be obtained compared to standard Vivaldi designs [21].

Performance of these antennas is widely discussed in the frequency domain. Time-domain analysis of these antennas is also needed since these antennas are considered as a good choice for microwave imaging applications [21]. With this aim, Vivaldi antenna with corrugations is designed to operate at UWB frequencies. It has the same size and uses the same material as

√

*Sref*(*t*) *S*21(*t* − *τ*) *dt* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\_\_\_\_\_\_\_\_\_\_\_\_ <sup>∫</sup> −∞ +∞ |*S*21(*t*) | 2 *dt*

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945

(5)

43

where the angle *θ* starts from the positive *z*-axis and the angle *φ* starts from the positive *x*-axis. *ω* refers to angular frequency, whereas *t* denotes time. Pulse distortion in the time domain can be observed from the difference between the received and transmitted UWB pulse in shape.

#### **2.1. Time-domain parameters**

The main parameters for the evaluation of the pulse characteristics in the time domain are obtained from the transient response of the antenna [5]. The peak value of the antenna impulse response, *P*(*θ*, *ϕ*), is expressed in Eq. (2):

$$P(\theta,\varphi) = \max \left| \mathbb{S}\_{\text{21}}(t,\theta,\varphi) \right| \tag{2}$$

The angular dependency of *P* is the result of angular-dependent impulse response. Most energy is contained around the peak of the impulse response. The higher values of *P*(*θ*, *φ*) demonstrates lower loss of the link. The pulse width is the width of the time window that contains a certain percentage of the total energy. Half-power width, *τ*, is a parameter used to define the broadening of the signal:

$$\mathbf{T} = \left. t\_2 \right|\_{\left[ S\_{\mathbf{x}}(t, \theta, \rho) \right] \neq 0.7 \ P\_{\mathbf{s}, \mathbf{s} \text{max}}(\theta, \rho)} - t\_1 \Big|\_{\left. t\_1 \circ \mathbf{e}\_p \right| \left[ S\_{\mathbf{x}}(t, \theta, \rho) \right] \neq 0.7 \ P\_{\mathbf{s}, \mathbf{s} \text{max}}(\theta, \rho)} \tag{3}$$

where *Pnor*,*max* is the maximum value of the normalized version of the antenna impulse response. *t* 2 and *t* 1 are the instants when half power width occurs, after and before the maximum, respectively. Time difference between the half power width of the received signal and the reference signal describes the broadening of the transmitted pulse. Thus, when there is no distortion, *τ* of the received pulse is equal to the *τ* of the transmitted pulse. This is the ideal case. However, mostly the link is distorted and the widening affects the communication quality. Ringing duration parameter, *τ<sup>r</sup>* given in Eq. (4), defines the oscillations in the antenna impulse response:

$$\pi\_r = \left. t\_r \right|\_{\left[ \mathcal{S}\_x(t, \theta, \rho) \right] \ast \mathcal{B}^q(\theta, \rho)} - t\_p \Big|\_{\left. t\_r \circ t\_r \right|\_{\left. \mathcal{S}\_x(t, \theta, \rho) \right| \ast \mathcal{B}^q(\theta, \rho)}} \tag{4}$$

*R* is a coefficient that is used to define the instant of the ringing. *<sup>t</sup> <sup>P</sup>* and *<sup>t</sup> <sup>r</sup>* are the instants when the pulse has its maximum and first ringing, respectively. Although these parameters are widely used to quantify the time-domain signal, correlation between the transmitted and received pulse should be observed as well. Besides, due to the angular variation of the transmitted signal, crosscorrelation between the transmitted and received signals should be investigated and quantified not only at the main beam direction but also with angular dependence as given in [19].

#### **2.2. Fidelity analysis**

The correlation coefficient between the received pulse and transmitted pulse demonstrates the amount of pulse distortion which the antenna induced. The fidelity factor, *FF,* is a parameter used to quantify the similarity between transmitted and received signal [16]:

$$\begin{aligned} \text{http://t.org/t/dt.do/dorg/10.5772/ntte0ppen.74945} \\\\ FF &= \max\_{\tau} \frac{\stackrel{\textstyle \int}{\longrightarrow} \stackrel{\textstyle S\_{\eta\eta}}{\longrightarrow} (t) \stackrel{\textstyle S\_{\eta}}{\longrightarrow} (t-\tau) \stackrel{\textstyle t}{dt}}{\stackrel{\textstyle \int}{\longrightarrow} \stackrel{\textstyle S\_{\eta\eta}}{\longrightarrow} (t) \stackrel{\textstyle \int}{\longrightarrow} \stackrel{\textstyle S\_{\eta 1}}{\longrightarrow} (t-\tau)^{2}} \end{aligned} \tag{5}$$

where *Sref*(*t*) and *S*21(*t*) are the transmitted and received signals, respectively. If the transmitted and received signals are exactly same, *FF* coefficient has its maximum of 1. When *FF* coefficient is 1, input signal isn't distorted by the antenna. Fidelity depends on the spatial radiation characteristics of the antenna. Thus, angular variation of the *FF* coefficient should also be observed. Because of the normalization procedure, fidelity factor cannot provide information about the amplitudes of signals.

## **3. Modified Vivaldi antennas**

#### **3.1. Antenna design**

*S*21(*t*, *θ*, *ϕ*) = *IFT* {*S*21(*ω*, *θ*, *ϕ*)} (1)

where the angle *θ* starts from the positive *z*-axis and the angle *φ* starts from the positive *x*-axis. *ω* refers to angular frequency, whereas *t* denotes time. Pulse distortion in the time domain can be observed from the difference between the received and transmitted UWB pulse in shape.

The main parameters for the evaluation of the pulse characteristics in the time domain are obtained from the transient response of the antenna [5]. The peak value of the antenna impulse

*P*(*θ*, *ϕ*) = *max* |*S*21(*t*, *θ*, *ϕ*)| (2)

The angular dependency of *P* is the result of angular-dependent impulse response. Most energy is contained around the peak of the impulse response. The higher values of *P*(*θ*, *φ*) demonstrates lower loss of the link. The pulse width is the width of the time window that contains a certain percentage of the total energy. Half-power width, *τ*, is a parameter used to

> − *t* 1|*t* 1 <*t* 2 ,

where *Pnor*,*max* is the maximum value of the normalized version of the antenna impulse response.

 are the instants when half power width occurs, after and before the maximum, respectively. Time difference between the half power width of the received signal and the reference signal describes the broadening of the transmitted pulse. Thus, when there is no distortion, *τ* of the received pulse is equal to the *τ* of the transmitted pulse. This is the ideal case. However, mostly the link is distorted and the widening affects the communication quality. Ringing dura-

> − *t P*|*t P*<*t r*

the pulse has its maximum and first ringing, respectively. Although these parameters are widely used to quantify the time-domain signal, correlation between the transmitted and received pulse should be observed as well. Besides, due to the angular variation of the transmitted signal, crosscorrelation between the transmitted and received signals should be investigated and quantified

The correlation coefficient between the received pulse and transmitted pulse demonstrates the amount of pulse distortion which the antenna induced. The fidelity factor, *FF,* is a param-

given in Eq. (4), defines the oscillations in the antenna impulse response:

<sup>|</sup>*S*21(*t*,*θ*,*ϕ*)|=0.7 *Pnor*,*max*(*θ*,*ϕ*) (3)

, <sup>|</sup>*S*21(*t*,*θ*,*ϕ*)|=*P*(*θ*,*ϕ*) (4)

are the instants when

and *<sup>t</sup> <sup>r</sup>*

2||*<sup>S</sup>*21(*t*,*θ*,*ϕ*)|=0.7 *Pnor*,*max*(*θ*,*ϕ*)

*<sup>r</sup>*||*<sup>S</sup>*21(*t*,*θ*,*φ*)|=*RP*(*θ*,*ϕ*)

not only at the main beam direction but also with angular dependence as given in [19].

eter used to quantify the similarity between transmitted and received signal [16]:

*R* is a coefficient that is used to define the instant of the ringing. *<sup>t</sup> <sup>P</sup>*

**2.1. Time-domain parameters**

42 Antennas and Wave Propagation

response, *P*(*θ*, *ϕ*), is expressed in Eq. (2):

define the broadening of the signal:

*τ* = *t*

*τ<sup>r</sup>* = *t*

*t* 2 and *t* 1

tion parameter, *τ<sup>r</sup>*

**2.2. Fidelity analysis**

The Vivaldi antenna is one of the classical ultra-wideband antennas with many applications [8]. It is a traveling-wave, end-fire antenna and due to its completely planar structure, it can be easily integrated in UWB sensor circuit. It has almost symmetric radiation patterns in the E and H planes. Theoretically, with its exponentially tapered slot, the Vivaldi antenna has an unlimited range of operating frequencies. However, in practice, it is constrained by the physical dimensions such as taper dimensions, the slot line width and transition from the feed line.

The structure of the standard Vivaldi antenna together with its dimensions is shown in **Figure 2a**. The proposed Vivaldi antenna consists of a microstrip feed line, microstrip line to slot line transition and the radiating structure. It is designed to operate efficiently as the transmitter and receiver in the unlicensed band of 3.1–10.6 GHz (7.5 GHz bandwidth). The slot curve of the Vivaldi antenna is the exponential function, which is expressed as *S*(*z*) <sup>=</sup> (*Wslot*/2) *<sup>e</sup> az* where *<sup>a</sup>* <sup>=</sup> 0.165 and *Wslot* <sup>=</sup> 0.25 mm. A quarter wavelength open circuit stub is used for wideband matching. The aperture coupling is optimized for the frequency range from 3.1 to 10.6 GHz. The size of the Vivaldi antenna is 50 × 50 mm. Its dimensions are given in **Figure 2a**. Rogers RT/ Duroid 5870 with 0.51-mm dielectric thickness and 17.5-um copper is chosen for the design. The dielectric constant of the dielectric material is *<sup>ε</sup><sup>r</sup>* <sup>=</sup> 2.33.

One of the bottlenecks of the conventional Vivaldi antenna is its relatively low directivity, especially at lower frequencies of the band. The lower frequency response of Vivaldi antennas with satisfactory impedance match and effective radiation is usually improved by increasing the aperture size. Another solution is introducing variable length slots to effectively increase the aperture of the antenna [15, 20]. It is shown that by incorporating a corrugated profile on the sides of exponential flaring, more suitable characteristics, especially for microwave imaging applications (i.e., higher gain, broader bandwidth), can be obtained compared to standard Vivaldi designs [21].

Performance of these antennas is widely discussed in the frequency domain. Time-domain analysis of these antennas is also needed since these antennas are considered as a good choice for microwave imaging applications [21]. With this aim, Vivaldi antenna with corrugations is designed to operate at UWB frequencies. It has the same size and uses the same material as the standard Vivaldi. The dimensions of the Vivaldi with corrugations are given in **Figure 2b**. The edge of Vivaldi is symmetrically corrugated by slots along the y-axis. The corrugations are rectangular slots with varying lengths. Design parameters of the corrugations are the distance between slots, the width of slots and the length of slots. The width of slots and distance between the rectangular slots of corrugation remain same. The length of the slots decreases gradually toward the flaring. Simulations proved that increasing the number of slots improves the radiation characteristics of the designed antenna by triggering extra resonances and modifying the direction of the current on the edges. The corrugations on the edges of the flaring act like a resistive loading. These corrugations are useful to concentrate the wave toward the slot area and contribute to the end-fire radiation patterns. The design parameters of the corrugation are optimized as 1 mm, 1 and 20–14.5 mm, respectively.

located to the flaring are the distance between strips, the width of strips and the length of strips. They are optimized as 3, 0.3 and 8 mm, respectively. All of the three Vivaldi designs use the same exponential tapering and balun. To match the antenna over a wide frequency band, a microstrip line to slot line transition and feed balun is designed as shown in **Figure 2d**. The selected reference axis system is also presented. The overall size of the antenna is not affected by the techniques used to increase the gain and improve the radiation patterns of the antenna

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945 45

To demonstrate the pulse distortion properties of the modified Vivaldi antennas, the prototypes have been manufactured with printed circuit board technology. The prototypes are shown in **Figures 1** and **3** (Vivaldi with corrugation and Vivaldi with corrugation and strip in **Figure 1**, standard Vivaldi in **Figure 3**). The scattering parameters of the antenna are measured using an Agilent vector network analyzer. The reflection behavior of each antenna has been investigated in terms of *S*11. The measured return loss variation of the antennas is given in **Figure 4**. Simulations performed with a commercial finite integration technique-based software package computer simulation technology (CST) microwave studio, not reported for brevity, are in excel-

Simulated gain variations of the antennas are given in **Figure 5**. The realized gain of the modified antennas improves significantly throughout the frequency band compared to standard Vivaldi. Existence of the corrugations and grating elements maximizes the radiation in the bore sight direction. With the corrugations added, at the lower frequencies of the band, both of the modified Vivaldi antennas have higher gain compared to standard Vivaldi antenna. Moreover, Vivaldi with corrugation and strip has a 0.2 dB more gain than Vivaldi with corrugation at the whole frequency band. With these results, the positive effect of the existence of corrugation and metallic strips is observed in the frequency domain. However, since the antennas are aimed to be used for UWB applications, their time-domain performance should

in bore sight direction; therefore, the overall size of the antenna remains compact.

**3.2. Antenna performance**

also be investigated.

lent agreement with the measurement results.

**Figure 3.** Fabricated Vivaldi antenna (a) top view; (b) back view.

Besides adding corrugations on the edges of the flaring, adding grating elements on the slot area in the direction of the antenna axis is another technique to enhance the gain of the antenna. These elements work as directive elements and contribute to the radiation in the endfire direction. With the combination of both the corrugations and grating elements, the gain of the antenna increases significantly in the end-fire direction [15].

The third design for the Vivaldi antennae is achieved by adding three metallic strips on the slot area as demonstrated in **Figure 2c**. Design parameters of the grating elements that are

**Figure 2.** UWB Vivaldi antennas with its dimensions: (a) top view of Vivaldi; (b) top view of Vivaldi with corrugation; (c) top view of Vivaldi with corrugation and strip; (d) bottom view.

located to the flaring are the distance between strips, the width of strips and the length of strips. They are optimized as 3, 0.3 and 8 mm, respectively. All of the three Vivaldi designs use the same exponential tapering and balun. To match the antenna over a wide frequency band, a microstrip line to slot line transition and feed balun is designed as shown in **Figure 2d**. The selected reference axis system is also presented. The overall size of the antenna is not affected by the techniques used to increase the gain and improve the radiation patterns of the antenna in bore sight direction; therefore, the overall size of the antenna remains compact.

#### **3.2. Antenna performance**

the standard Vivaldi. The dimensions of the Vivaldi with corrugations are given in **Figure 2b**. The edge of Vivaldi is symmetrically corrugated by slots along the y-axis. The corrugations are rectangular slots with varying lengths. Design parameters of the corrugations are the distance between slots, the width of slots and the length of slots. The width of slots and distance between the rectangular slots of corrugation remain same. The length of the slots decreases gradually toward the flaring. Simulations proved that increasing the number of slots improves the radiation characteristics of the designed antenna by triggering extra resonances and modifying the direction of the current on the edges. The corrugations on the edges of the flaring act like a resistive loading. These corrugations are useful to concentrate the wave toward the slot area and contribute to the end-fire radiation patterns. The design parameters of the corruga-

Besides adding corrugations on the edges of the flaring, adding grating elements on the slot area in the direction of the antenna axis is another technique to enhance the gain of the antenna. These elements work as directive elements and contribute to the radiation in the endfire direction. With the combination of both the corrugations and grating elements, the gain of

The third design for the Vivaldi antennae is achieved by adding three metallic strips on the slot area as demonstrated in **Figure 2c**. Design parameters of the grating elements that are

**Figure 2.** UWB Vivaldi antennas with its dimensions: (a) top view of Vivaldi; (b) top view of Vivaldi with corrugation;

(c) top view of Vivaldi with corrugation and strip; (d) bottom view.

tion are optimized as 1 mm, 1 and 20–14.5 mm, respectively.

44 Antennas and Wave Propagation

the antenna increases significantly in the end-fire direction [15].

To demonstrate the pulse distortion properties of the modified Vivaldi antennas, the prototypes have been manufactured with printed circuit board technology. The prototypes are shown in **Figures 1** and **3** (Vivaldi with corrugation and Vivaldi with corrugation and strip in **Figure 1**, standard Vivaldi in **Figure 3**). The scattering parameters of the antenna are measured using an Agilent vector network analyzer. The reflection behavior of each antenna has been investigated in terms of *S*11. The measured return loss variation of the antennas is given in **Figure 4**. Simulations performed with a commercial finite integration technique-based software package computer simulation technology (CST) microwave studio, not reported for brevity, are in excellent agreement with the measurement results.

Simulated gain variations of the antennas are given in **Figure 5**. The realized gain of the modified antennas improves significantly throughout the frequency band compared to standard Vivaldi. Existence of the corrugations and grating elements maximizes the radiation in the bore sight direction. With the corrugations added, at the lower frequencies of the band, both of the modified Vivaldi antennas have higher gain compared to standard Vivaldi antenna. Moreover, Vivaldi with corrugation and strip has a 0.2 dB more gain than Vivaldi with corrugation at the whole frequency band. With these results, the positive effect of the existence of corrugation and metallic strips is observed in the frequency domain. However, since the antennas are aimed to be used for UWB applications, their time-domain performance should also be investigated.

**Figure 3.** Fabricated Vivaldi antenna (a) top view; (b) back view.

The antennas were placed at about 20 cm of distance. In **Figure 7**, measurement setup is shown

The procedure for the measurement of *S*21(*ω*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) with angular variation can be summarized as follows: The measurements are performed by shifting one of the antennas in the range

Afterward, the impulse response of the link with angular variation is derived by means of

The link between the transmitting and receiving antennas can be characterized in terms of its

*H*(*ω*) = *URX*(*ω*)/*UTX*(*ω*) (6)

) and H-planes (*ϕ* = 0°

are plotted as the func-

47

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945

) and measuring *S*21(*ω*, *<sup>θ</sup>*, *<sup>ϕ</sup>*).

for E and H planes. In **Figure 8**, the amplitudes of *S*21 parameters at *<sup>θ</sup>* <sup>=</sup> <sup>0</sup>°

steps in E- (*ϕ* = 90°

inverse Fourier transform of the measured *S*21 as shown in the next section.

**Figure 7.** Measurement setup for the characterization of the antenna link (a) E-plane; (b) H-plane.

tion of the frequency in the range 0–12 GHz.

with 5<sup>o</sup>

**Figure 6.** Demonstration of the measurement setup.

**4.2. Time-domain analysis**

complex transfer function:

*4.2.1. Pulse comparison*

of −90° ≤ θ ≤ 90°

**Figure 4.** Measured return loss of Vivaldi antennas.

**Figure 5.** Simulated gain variations of Vivaldi antennas.

## **4. Time-domain analysis of modified Vivaldi antennas**

#### **4.1. Measurement setup**

Time-domain analysis of modified Vivaldi antennas is performed and compared with that of standard Vivaldi antenna. A link composed of two identical Vivaldi antennas has been experimentally characterized. The measurements were performed with the same setup. The transmit-receive antenna link measurement setup demonstration for E-plane is shown in **Figure 6**. The antennas were placed at about 20 cm of distance. In **Figure 7**, measurement setup is shown for E and H planes. In **Figure 8**, the amplitudes of *S*21 parameters at *<sup>θ</sup>* <sup>=</sup> <sup>0</sup>° are plotted as the function of the frequency in the range 0–12 GHz.

The procedure for the measurement of *S*21(*ω*, *<sup>θ</sup>*, *<sup>ϕ</sup>*) with angular variation can be summarized as follows: The measurements are performed by shifting one of the antennas in the range of −90° ≤ θ ≤ 90° with 5<sup>o</sup> steps in E- (*ϕ* = 90° ) and H-planes (*ϕ* = 0° ) and measuring *S*21(*ω*, *<sup>θ</sup>*, *<sup>ϕ</sup>*). Afterward, the impulse response of the link with angular variation is derived by means of inverse Fourier transform of the measured *S*21 as shown in the next section.

#### **4.2. Time-domain analysis**

#### *4.2.1. Pulse comparison*

The link between the transmitting and receiving antennas can be characterized in terms of its complex transfer function:

$$H(\omega) = \text{\textquotedblleft}\mathcal{U}\_{\text{\textquotedblleft}\lambda}(\omega) / \mathcal{U}\_{\text{\textquotedblleft}\lambda}(\omega)\tag{6}$$

**Figure 6.** Demonstration of the measurement setup.

**Figure 5.** Simulated gain variations of Vivaldi antennas.

**Figure 4.** Measured return loss of Vivaldi antennas.

46 Antennas and Wave Propagation

**4.1. Measurement setup**

**4. Time-domain analysis of modified Vivaldi antennas**

Time-domain analysis of modified Vivaldi antennas is performed and compared with that of standard Vivaldi antenna. A link composed of two identical Vivaldi antennas has been experimentally characterized. The measurements were performed with the same setup. The transmit-receive antenna link measurement setup demonstration for E-plane is shown in **Figure 6**.

**Figure 7.** Measurement setup for the characterization of the antenna link (a) E-plane; (b) H-plane.

**Figure 8.** Measured insertion loss of the antennas.

where *URX*(*ω*) and *UTX*(*ω*) are the spectra of the received and transmitted voltages, respectively. The coupling parameter between the antennas is related with the complex transfer function of the antennas as [22]:

$$S\_{21}(\omega) = \left. H\_{\text{rx}}(\omega) \right| H\_{\text{RX}}(\omega) \frac{j\omega}{2\pi r c} e^{-j\omega c/c} \tag{7}$$

port is 50 cm. The Teflon coaxial cables for the transmitter/receiver antennas correspond to 4.83 ns of propagation time. The signal path in the microstrip antenna is 65.2 mm. With an effective dielectric constant of 1.97 at 5 GHz, the propagation time inside transmitter/ receiver antennas are calculated approximately as 0.61 ns. A total of 20 cm free space propagation corresponds to 0.666 ns of propagation time. When the 2 cm adapters used at the network analyzer ports are added, total propagation time of the signal can be calculated as approximately 6.3 ns. This is observed from the time-domain representations obtained from the measurement results as well. The main peak of the signal arrives to the receiver after

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945 49

The half power width of the reference signal is 0.119 ns. When the antenna is at bore sight, half power width of the pulse for standard Vivaldi is measured as 0.011 ns wider than that of the reference signal. Similarly, the pulse is 0.076 and 0.03 ns wider for Vivaldi with

**Figure 9.** Comparison between the impulse response and an ideal delayed pulse in E-plane (a) *θ* = 0 °; (b) *θ* = 10 °; (c)

approximately 6.3 ns.

*θ* = 30 °; (d) *θ* = 45 °.

where *HTX*(*ω*) and *HRX*(*ω*) are the transfer functions of transmit and receive antenna. *r* is the distance between the antennas. The impulse response of the links is derived over 7.5 GHz bandwidth, from 3.1 to 10.6 GHz, by means of the inverse fourier transform (IFT) of the measured *<sup>S</sup>*21(*ω*). In the application of time-domain analysis, the reference signal *Sref*(*t*) will be a sinc pulse associated with the mentioned 7.5 GHz band, which can be expressed as:

$$S\_{\eta\eta}(t) = \text{IFT}\{S\_{\eta\eta}(\omega)\} = \frac{1}{2\pi} \int\_{\omega\_1}^{\omega\_j} e^{-j\omega t} d\alpha \tag{8}$$

where *ω*<sup>1</sup> <sup>=</sup> <sup>2</sup>*π<sup>f</sup>* 1 and *ω*<sup>2</sup> <sup>=</sup> <sup>2</sup>*π<sup>f</sup>* 2 . In **Figure 9a–d**, a comparison between the response of the antenna link in time domain and the reference signal delayed to the present maximum in correspondence of the main peak of the link's impulse response is shown for *θ* = 0, 10, 30, 45° in the E-plane (*ϕ* = 90° ). The green dash-dot line is the ideal delayed pulse obtained by Eq. (8). Blue solid, red-dashed and black dash-dot lines belong to impulse response of the Vivaldi, Vivaldi with corrugation and Vivaldi with corrugation and strip, respectively. For a rigorous comparison, path loss effect is removed by scaling the received pulse amplitude to the transmitted amplitude. Thus, in the time-domain representations, the amplitude of the *S*21 parameter measured with the setup in **Figure 7** is normalized to its maximum. The length of the coaxial cables used between the connector of the antenna and network analyzer

port is 50 cm. The Teflon coaxial cables for the transmitter/receiver antennas correspond to 4.83 ns of propagation time. The signal path in the microstrip antenna is 65.2 mm. With an effective dielectric constant of 1.97 at 5 GHz, the propagation time inside transmitter/ receiver antennas are calculated approximately as 0.61 ns. A total of 20 cm free space propagation corresponds to 0.666 ns of propagation time. When the 2 cm adapters used at the network analyzer ports are added, total propagation time of the signal can be calculated as approximately 6.3 ns. This is observed from the time-domain representations obtained from the measurement results as well. The main peak of the signal arrives to the receiver after approximately 6.3 ns.

The half power width of the reference signal is 0.119 ns. When the antenna is at bore sight, half power width of the pulse for standard Vivaldi is measured as 0.011 ns wider than that of the reference signal. Similarly, the pulse is 0.076 and 0.03 ns wider for Vivaldi with

where *URX*(*ω*) and *UTX*(*ω*) are the spectra of the received and transmitted voltages, respectively. The coupling parameter between the antennas is related with the complex transfer function

where *HTX*(*ω*) and *HRX*(*ω*) are the transfer functions of transmit and receive antenna. *r* is the distance between the antennas. The impulse response of the links is derived over 7.5 GHz bandwidth, from 3.1 to 10.6 GHz, by means of the inverse fourier transform (IFT) of the measured *<sup>S</sup>*21(*ω*). In the application of time-domain analysis, the reference signal *Sref*(*t*) will be a sinc pulse

> } <sup>=</sup> \_\_\_1 <sup>2</sup>*<sup>π</sup>* ∫ *ω*1 *ω*2

antenna link in time domain and the reference signal delayed to the present maximum in correspondence of the main peak of the link's impulse response is shown for *θ* = 0, 10, 30,

Eq. (8). Blue solid, red-dashed and black dash-dot lines belong to impulse response of the Vivaldi, Vivaldi with corrugation and Vivaldi with corrugation and strip, respectively. For a rigorous comparison, path loss effect is removed by scaling the received pulse amplitude to the transmitted amplitude. Thus, in the time-domain representations, the amplitude of the *S*21 parameter measured with the setup in **Figure 7** is normalized to its maximum. The length of the coaxial cables used between the connector of the antenna and network analyzer

. In **Figure 9a–d**, a comparison between the response of the

). The green dash-dot line is the ideal delayed pulse obtained by

<sup>2</sup>*rc <sup>e</sup>* <sup>−</sup>*jr*/*<sup>c</sup>* (7)

*e* <sup>−</sup>*j<sup>t</sup> d* (8)

of the antennas as [22]:

48 Antennas and Wave Propagation

**Figure 8.** Measured insertion loss of the antennas.

where *ω*<sup>1</sup> <sup>=</sup> <sup>2</sup>*π<sup>f</sup>*

45°

1

in the E-plane (*ϕ* = 90°

*<sup>S</sup>*21(*ω*) <sup>=</sup> *HTX*(*ω*) *HRX*(*ω*) *<sup>j</sup>*\_\_\_\_

*Sref*(*t*) <sup>=</sup> *IFT*{*Sref*(*ω*)

2

and *ω*<sup>2</sup> <sup>=</sup> <sup>2</sup>*π<sup>f</sup>*

associated with the mentioned 7.5 GHz band, which can be expressed as:

**Figure 9.** Comparison between the impulse response and an ideal delayed pulse in E-plane (a) *θ* = 0 °; (b) *θ* = 10 °; (c) *θ* = 30 °; (d) *θ* = 45 °.

corrugation and Vivaldi with corrugation and strip, respectively. The pulse is visible in the inset (**Figure 9a**). The pulses widen for larger values of *θ*. This is observed in **Figure 9b–d**. When *θ* = 0 ° in E-field, the shape of the transmitted pulse is close to that of standard Vivaldi and Vivaldi with corrugation and strip. As theta gets larger, the pulses widen. This is observed at *θ* = 10 ° given in **Figure 9b**. At 10 ° the main beam of Vivaldi with corrugation and strip pulse is similar to the main beam of reference pulse but secondary pulses are generated. A very similar case occurs at *θ* = 30 ° . At *θ* = 45 ° , secondary pulse of the standard Vivaldi is also generated.

better than the other Vivaldi antennas. Although only the measurement results are shared in this communication, simulations performed with CST, not reported for brevity, are in very

Based on the comparison between the impulse response of link and ideal delayed signal, one can clearly establish the presence of pulse widening. To quantify the amount of widening, pulse analysis with respect to *θ* in E- and H-planes are performed using the definition given in Eq. (3). In **Figure 11(a)** and **(b)**, half power width of the impulse response, *τ* is demonstrated in E- and H-planes, respectively. The green dotted line shows the width of the reference pulse

Secondary pulse signal that has its maximum reach to half power of the main beam is gener-

results, it can be concluded that the width of the pulse that belongs to Vivaldi with corrugation widens more than the standard and corrugation and strip Vivaldi. This is valid both in E- and in H-planes. This result is even more obvious when the pulse width is compared to that of the reference signal. In **Figure 12**, pulse extension ratio of the measured pulse is given.

The pulse extension ratio of Vivaldi with corrugation is below 65% in −60° ≤ *θ* ≤ 60°

above 100% afterward. Pulse width ratio of Vivaldi and Vivaldi with corrugation and strip is more stable than Vivaldi with corrugation. Based on pulse analysis results, one can accept the standard Vivaldi to have the best pulse distortion performance in the time

in H-plane. As a result, pulse width is observed

\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>τ</sup>reference pulse*

in H-plane. Based on the pulse width

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945

(9)

51

. It goes

in E-plane and 70°

*Pulse Extension Ratio* <sup>=</sup> *<sup>τ</sup>pulse* <sup>−</sup> *<sup>τ</sup>reference pulse*

**Figure 11.** Half power width of the measured pulse (a) E-plane; (b) H-plane.

for E-plane and −70° ≤ *θ* ≤ 70°

good agreement with the measured results.

It is calculated by the following expression:

*4.2.2. Pulse analysis*

which is equal to 0.119 ns.

ated by the link after 65°

between −65° ≤ *θ* ≤ 65°

Similarly, a comparison in H-plane (*ϕ* = 0° ) between the time-domain response of the antenna link that consists of standard Vivaldi and modified Vivaldi antennas and the reference signal, delayed by 6.3 ns to present the maximum in correspondence of the main peak of the link impulse response, is shown in **Figure 10**. In H-plane, pulse characteristics are different than the E-plane. Clearly, the pulse-preserving capability of Vivaldi with corrugation and strip is

**Figure 10.** Comparison between the impulse response and an ideal delayed pulse in H-plane (a) *θ* = 0 °; (b) *θ* = 10 °; (c) *θ* = 30 °; (d) *θ* = 45 °.

better than the other Vivaldi antennas. Although only the measurement results are shared in this communication, simulations performed with CST, not reported for brevity, are in very good agreement with the measured results.

#### *4.2.2. Pulse analysis*

Based on the comparison between the impulse response of link and ideal delayed signal, one can clearly establish the presence of pulse widening. To quantify the amount of widening, pulse analysis with respect to *θ* in E- and H-planes are performed using the definition given in Eq. (3). In **Figure 11(a)** and **(b)**, half power width of the impulse response, *τ* is demonstrated in E- and H-planes, respectively. The green dotted line shows the width of the reference pulse which is equal to 0.119 ns.

Secondary pulse signal that has its maximum reach to half power of the main beam is generated by the link after 65° in E-plane and 70° in H-plane. As a result, pulse width is observed between −65° ≤ *θ* ≤ 65° for E-plane and −70° ≤ *θ* ≤ 70° in H-plane. Based on the pulse width results, it can be concluded that the width of the pulse that belongs to Vivaldi with corrugation widens more than the standard and corrugation and strip Vivaldi. This is valid both in E- and in H-planes. This result is even more obvious when the pulse width is compared to that of the reference signal. In **Figure 12**, pulse extension ratio of the measured pulse is given. It is calculated by the following expression:

$$\text{PulseExtension Ratio} = \frac{\tau\_{\text{pulse}} - \tau\_{\text{refram} \text{pulse}}}{\tau\_{\text{refram} \text{pulse}}} \tag{9}$$

The pulse extension ratio of Vivaldi with corrugation is below 65% in −60° ≤ *θ* ≤ 60° . It goes above 100% afterward. Pulse width ratio of Vivaldi and Vivaldi with corrugation and strip is more stable than Vivaldi with corrugation. Based on pulse analysis results, one can accept the standard Vivaldi to have the best pulse distortion performance in the time

**Figure 11.** Half power width of the measured pulse (a) E-plane; (b) H-plane.

**Figure 10.** Comparison between the impulse response and an ideal delayed pulse in H-plane (a) *θ* = 0 °; (b) *θ* = 10 °;

corrugation and Vivaldi with corrugation and strip, respectively. The pulse is visible in the inset (**Figure 9a**). The pulses widen for larger values of *θ*. This is observed in **Figure 9b–d**.

and Vivaldi with corrugation and strip. As theta gets larger, the pulses widen. This is

and strip pulse is similar to the main beam of reference pulse but secondary pulses are

link that consists of standard Vivaldi and modified Vivaldi antennas and the reference signal, delayed by 6.3 ns to present the maximum in correspondence of the main peak of the link impulse response, is shown in **Figure 10**. In H-plane, pulse characteristics are different than the E-plane. Clearly, the pulse-preserving capability of Vivaldi with corrugation and strip is

given in **Figure 9b**. At 10 °

generated. A very similar case occurs at *θ* = 30 °

Similarly, a comparison in H-plane (*ϕ* = 0°

in E-field, the shape of the transmitted pulse is close to that of standard Vivaldi

. At *θ* = 45 °

the main beam of Vivaldi with corrugation

) between the time-domain response of the antenna

, secondary pulse of the standard

(c) *θ* = 30 °; (d) *θ* = 45 °.

When *θ* = 0 °

observed at *θ* = 10 °

50 Antennas and Wave Propagation

Vivaldi is also generated.

domain. Although, these results give an idea about the pulse distortion introduced by the Vivaldi antennas, to more rigorously quantify the pulse, distortion fidelity analysis should be performed.

for −45° ≤ *θ* ≤ 45°

**5. Conclusion**

band instruments.

**Acknowledgements**

Number: FBA-2017-3071).

. It has a lower value for the angles greater than 45°

better pulse-preserving properties in the time domain.

corrugation and strip has clearly a higher fidelity value than the others. Although standard Vivaldi was considered to have better impulse response in terms of pulse widening, fidelity analysis results represent Vivaldi with corrugation and strip to have the highest similarity between the transmitted and received signal. A good antenna performance requires simultaneously both a high fidelity and small pulse extension ratio of the impulse response. Thus, the modified version of Vivaldi antenna having corrugation and strip is a good candidate for UWB applications with its higher gain and wider bandwidth in the frequency domain and

In this chapter, the analysis procedure for the pulse-preserving properties of impulse-radiating antennas is defined. The analysis procedure is applied to a widely used UWB antenna, namely Vivaldi antenna. Vivaldi antennas are popular in UWB applications due to its complete planar structure which enables it to be easily integrated to UWB sensor circuit. However, Vivaldi antenna has relatively low directivity, especially at lower frequencies of the band. The lower frequency response of Vivaldi antennas may be improved by increasing the aperture size of the antenna. In the cases where physical size matters slots are added on the edges of exponential flaring to effectively increase the aperture of the antenna. The corrugated profile results in more suitable characteristics (i.e., higher gain, higher directivity, broader bandwidth). Besides adding slots on the edges of the flaring, adding grating elements on the slot area in the direction of the antenna axis is another technique to enhance the gain of the antenna. With the combination of both the corrugations and grating elements, the gain of the antenna increases significantly in the end-fire direction. Although these modified Vivaldi antennas are used in many UWB applications, their time-domain performance is not observed. With this contribution, pulse-preserving capabilities of modified Vivaldi antennas based on measurements are observed. Two parameters are used to quantify the capability of the antenna. First one is the pulse width extension that defines the broadening of the signal at its half power. Since most of the energy is stored around the peak of the pulse, this parameter is useful to demonstrate the pulse-preserving capability of the antenna but not sufficient. The second parameter is the fidelity factor that measures the correlation between the transmitted and received pulse. The performance of the modified Vivaldi antennas is also analyzed in different angular directions with respect to the main beam. Among the Vivaldi structures observed, Vivaldi antenna with corrugation and strip is proved to be potentially suited for both pulsed and harmonic broad-

This work was supported by Research Fund of the Yıldız Technical University (Project

. In H-plane, Vivaldi with

53

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945

#### *4.2.3. Fidelity analysis*

Most of the energy carried by the pulse is stored around the peak of the impulse. The correlation coefficient between the received pulse and transmitted pulse quantifies the similarity between transmitted and received signal. For the 3.1–10.6 GHz band, the fidelity factor of the link between two identical antennas is shown in **Figure 13** for E- and H-planes. The fidelity variation obtained from the measured *S21* has unexpected pits at some angles. This may be due to the structure of the antenna profile. The fidelity values in E-plane are mostly close to 0.9 in

**Figure 12.** Pulse extension ratio of the measured pulse in (a) E-plane; (b) H-plane.

**Figure 13.** Fidelity factor variations of the pulses (a) E-plane; (b) H-plane.

for −45° ≤ *θ* ≤ 45° . It has a lower value for the angles greater than 45° . In H-plane, Vivaldi with corrugation and strip has clearly a higher fidelity value than the others. Although standard Vivaldi was considered to have better impulse response in terms of pulse widening, fidelity analysis results represent Vivaldi with corrugation and strip to have the highest similarity between the transmitted and received signal. A good antenna performance requires simultaneously both a high fidelity and small pulse extension ratio of the impulse response. Thus, the modified version of Vivaldi antenna having corrugation and strip is a good candidate for UWB applications with its higher gain and wider bandwidth in the frequency domain and better pulse-preserving properties in the time domain.

## **5. Conclusion**

In this chapter, the analysis procedure for the pulse-preserving properties of impulse-radiating antennas is defined. The analysis procedure is applied to a widely used UWB antenna, namely Vivaldi antenna. Vivaldi antennas are popular in UWB applications due to its complete planar structure which enables it to be easily integrated to UWB sensor circuit. However, Vivaldi antenna has relatively low directivity, especially at lower frequencies of the band. The lower frequency response of Vivaldi antennas may be improved by increasing the aperture size of the antenna. In the cases where physical size matters slots are added on the edges of exponential flaring to effectively increase the aperture of the antenna. The corrugated profile results in more suitable characteristics (i.e., higher gain, higher directivity, broader bandwidth). Besides adding slots on the edges of the flaring, adding grating elements on the slot area in the direction of the antenna axis is another technique to enhance the gain of the antenna. With the combination of both the corrugations and grating elements, the gain of the antenna increases significantly in the end-fire direction. Although these modified Vivaldi antennas are used in many UWB applications, their time-domain performance is not observed. With this contribution, pulse-preserving capabilities of modified Vivaldi antennas based on measurements are observed. Two parameters are used to quantify the capability of the antenna. First one is the pulse width extension that defines the broadening of the signal at its half power. Since most of the energy is stored around the peak of the pulse, this parameter is useful to demonstrate the pulse-preserving capability of the antenna but not sufficient. The second parameter is the fidelity factor that measures the correlation between the transmitted and received pulse. The performance of the modified Vivaldi antennas is also analyzed in different angular directions with respect to the main beam. Among the Vivaldi structures observed, Vivaldi antenna with corrugation and strip is proved to be potentially suited for both pulsed and harmonic broadband instruments.

## **Acknowledgements**

**Figure 13.** Fidelity factor variations of the pulses (a) E-plane; (b) H-plane.

**Figure 12.** Pulse extension ratio of the measured pulse in (a) E-plane; (b) H-plane.

domain. Although, these results give an idea about the pulse distortion introduced by the Vivaldi antennas, to more rigorously quantify the pulse, distortion fidelity analysis should

Most of the energy carried by the pulse is stored around the peak of the impulse. The correlation coefficient between the received pulse and transmitted pulse quantifies the similarity between transmitted and received signal. For the 3.1–10.6 GHz band, the fidelity factor of the link between two identical antennas is shown in **Figure 13** for E- and H-planes. The fidelity variation obtained from the measured *S21* has unexpected pits at some angles. This may be due to the structure of the antenna profile. The fidelity values in E-plane are mostly close to 0.9 in

be performed.

*4.2.3. Fidelity analysis*

52 Antennas and Wave Propagation

This work was supported by Research Fund of the Yıldız Technical University (Project Number: FBA-2017-3071).

## **Author details**

Sultan Aldırmaz Çolak1 and Nurhan Türker Tokan<sup>2</sup> \*

\*Address all correspondence to: nurhanturker@gmail.com

1 Department of Electronics and Communications Engineering, Kocaeli University, Kocaeli, Turkey

[11] Nassar IT, Weller TM. A novel method for improving antipodal Vivaldi antenna performance. IEEE Transactions on Antennas and Propagation. 2015;**63**:3321-3324

Time-Domain Analysis of Modified Vivaldi Antennas http://dx.doi.org/10.5772/intechopen.74945 55

[12] Oliveira AMD, Perotoni MB, Kofuji ST, Justo JF. A palm tree antipodal Vivaldi antenna with exponential slot edge for improved radiation pattern. IEEE Antennas and Wireless

[13] Wang YW, Wang GM, Zong BF. Directivity improvement of Vivaldi antenna using double-slot structure. IEEE Antennas and Wireless Propagation Letters. 2013;**12**:1380-1383.

[14] Zhang Y, ChaoWang EL, Guo G. Radiation enhanced Vivaldi antenna with doubleantipodal structure. IEEE Antennas and Wireless Propagation Letters. 2017;**16**:561-564.

[15] Pandey GK, Singh HS, Bharti PK, Pandey A, Meshram MK. High gain Vivaldi antenna for radar and microwave imaging applications. International Journal of Signal Processing

[16] Mehdipour A, Mohammadpour-Aghdam K, Faraji-Dana R. Complete dispersion analysis of Vivaldi antenna for ultra wide band applications. Progress in Electromagnetics

[17] Pancera E. Strategies for time domain characterization of UWB components and systems [Thesis] Universität Karlsruhe (TH) Fakultät für Elektrotechnik und Informationstechnik,

[18] Do-Hoon K. Effect of antenna gain and group delay variations on pulse-preserving capabilities of ultra wideband antennas. IEEE Transactions on Antennas and Propagation.

[19] Tokan NT, Neto A, Tokan F, Cavallo D. Comparative study on pulse distortion and phase aberration of directive ultra-wide band antennas. IET Microwaves, Antennas and

[20] Gopikrishnan G, Akhterand Z, Jaleel Akhtar M. A novel corrugated four slot Vivaldi antenna loaded with metamaterial cells for microwave imaging. In: Proceeding of the

[21] Abbak M, Akıncı MN, Çayören M, Akduman L. Experimental microwave imaging with a novel corrugated Vivaldi antenna. IEEE Transactions on Antennas and Propagation.

[22] Neto A. UWB, non dispersive radiation from the planarly fed leaky lens antenna. Part 1: Theory and design. IEEE Transactions on Antennas and Propagation. 2010;**58**:2238-2247

Propagation. 2013;**7**(12):1021-1026. DOI: 10.1049/iet-map.2013.0032

Asia-Pacific Microwave Conference (APMC), New Delhi; 2016. pp. 1-4

Propagation Letters. 2015;**14**:1334-1337. DOI: 10.1109/LAWP.2015.2404875

DOI: 10.1109/LAWP.2013.2285182

DOI: 10.1109/LAWP.2016.2588882

Germany; 2009

Systems. 2015;**3**:35-39. DOI: 10.12720/ijsps.3.1.35-39

Research. 2007;**77**:85-96. DOI: 10.2528/PIER07072904

2006;**54**(8):2208-2215. DOI: 10.1109/TAP.2006.879189

2017;**65**:3302-3307. DOI: 10.1109/TAP.2017.2670228

2 Department of Electronics and Communications Engineering, Yildiz Technical University, Istanbul, Turkey

## **References**


[11] Nassar IT, Weller TM. A novel method for improving antipodal Vivaldi antenna performance. IEEE Transactions on Antennas and Propagation. 2015;**63**:3321-3324

**Author details**

54 Antennas and Wave Propagation

Turkey

Istanbul, Turkey

**References**

TAP.2011.2109361

2143666

Sultan Aldırmaz Çolak1

and Nurhan Türker Tokan<sup>2</sup>

Propagation. 2010;**58**:2107-2112. DOI: 10.1109/TAP.2010.2046848

the SPIE Conference 4015 Radio Telescope, Munich; March 27-30 2000

antennas. Proceedings of the IEEE. 2009;**97**:372-385

Sensors. 2018;**18**:88. DOI: 10.3390/s18010088

Brighton; September 17-20, 1979. pp 101-105

\*Address all correspondence to: nurhanturker@gmail.com

\*

1 Department of Electronics and Communications Engineering, Kocaeli University, Kocaeli,

2 Department of Electronics and Communications Engineering, Yildiz Technical University,

[1] Schwarz U, Thiel F, Seifert F, Stephan R, Hein MA. Ultrawideband antennas for magnetic resonance imaging navigator techniques. IEEE Transactions on Antennas and

[2] Chahat N, Zhadobov M, Sauleau R, Ito KA. Compact UWB antenna for on-body applications. IEEE Transactions on Antennas and Propagation. 2011;**59**:1123-1131. DOI: 10.1109/

[3] Ardenne A, Smolders B, Hampson G. Active adaptive antennas for radio astronomy; results of the initial R&D program toward the square kilometer array. In: Proceedings of

[4] "First report and order," Revision of Part 15 of the Commission's Rules Regarding Ultra-Wideband Transmission Systems Federal Communications Commission; 2002

[5] Pancera E, Zwick T, Wiesbeck W. Spherical fidelity patterns of UWB antennas. IEEE Transactions on Antennas and Propagation. 2011;**59**:2111-2119. DOI: 10.1109/TAP.2011.

[6] Shlivinsky A, Heyman E, Kastner R. Antenna characterization in the time domain. IEEE Transactions on Antennas and Propagation. 1997;**45**:1140-1149. DOI: 10.1109/8.596907 [7] Wiesbeck W, Adamiuk G, Sturm C. Basic properties and design principles of UWB

[8] Yang Y, Wang BZ, Ding S. Performance comparison with different antenna properties in time reversal ultra-wideband communications for sensor system applications. Sensors,

[9] Gibson PJ. The Vivaldi aerial. In: Proceedings of 9th European Microwave Conference,

[10] Tokan NT. Performance of Vivaldi antennas in reflector feed applications. Applied

Computational Electromagnetics Society Journal. 2013;**8**:802-808


**Chapter 4**

Provisional chapter

**Teaching Transmission Line Propagation and Plane**

DOI: 10.5772/intechopen.74937

Teaching transmission lines and wave propagation is a challenging task because it involves quantities not easily observable and also because the underlying mathematical equations functions of time, distance and using complex numbers—are not prone to an easy physical interpretation in a frequent framework of a superposition of traveling waves in distinct directions. In such a context, tools with a strong visualization and easy student interaction can improve the learning outputs. We describe here a few tools and give basic exercises to

Keywords: telegrapher's equation, transmission line equations, input impedance,

The subject of propagation in transmission lines is a very important topic in analog microwave and high-speed digital circuits design. The full understanding of the circuit that models the transmission line, the line voltage and current equations, the input impedance of a transmission line and matching circuits is a basic background for any engineer dealing with circuits having sizes from an order of magnitude less than the wavelength. The wireless communications require a basic understanding of the multipath propagation channel whose basic models imply the understanding of plane wave reflection at the boundary between two infinite mediums. The phenomena described above are addressed here with very easy to use interactive tools developed in Matlab and that can assist teachers lecturing on these subjects and may also be used by the students as a virtual home laboratory. The first tools were already described in detail in [1] and made available and in http://bit.ly/2m8oBoe, whereas a new one

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

impedance matching, plane wave reflection and transmission

Teaching Transmission Line Propagation and Plane

**Wave Reflection Using Software Tools**

Wave Reflection Using Software Tools

Susana Mota and Armando Rocha

Susana Mota and Armando Rocha

http://dx.doi.org/10.5772/intechopen.74937

address the main learning topics.

Abstract

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools** Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

DOI: 10.5772/intechopen.74937

Susana Mota and Armando Rocha Susana Mota and Armando Rocha

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74937

#### Abstract

Teaching transmission lines and wave propagation is a challenging task because it involves quantities not easily observable and also because the underlying mathematical equations functions of time, distance and using complex numbers—are not prone to an easy physical interpretation in a frequent framework of a superposition of traveling waves in distinct directions. In such a context, tools with a strong visualization and easy student interaction can improve the learning outputs. We describe here a few tools and give basic exercises to address the main learning topics.

Keywords: telegrapher's equation, transmission line equations, input impedance, impedance matching, plane wave reflection and transmission

## 1. Introduction

The subject of propagation in transmission lines is a very important topic in analog microwave and high-speed digital circuits design. The full understanding of the circuit that models the transmission line, the line voltage and current equations, the input impedance of a transmission line and matching circuits is a basic background for any engineer dealing with circuits having sizes from an order of magnitude less than the wavelength. The wireless communications require a basic understanding of the multipath propagation channel whose basic models imply the understanding of plane wave reflection at the boundary between two infinite mediums. The phenomena described above are addressed here with very easy to use interactive tools developed in Matlab and that can assist teachers lecturing on these subjects and may also be used by the students as a virtual home laboratory. The first tools were already described in detail in [1] and made available and in http://bit.ly/2m8oBoe, whereas a new one

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

on plane waves that is available in http://bit.ly/2m9TvMY is now presented and comprehensively described here.

## 2. Transmission line model: TLM.m

An ideal transmission line is made of a pair of conductors that carries a signal along a path length ℓ with only a delay τ. Supposing a very high frequency and that any perturbation travels at finite speed we cannot assume that the voltage and current are constant along the line. Assuming two narrow parallel conducting strips of length ℓ = 1 m the image of a capacitor, with capacitance Ct(F), materializes immediately: an input voltage at one end will trigger a current to charge the capacitor. However, the two strips carrying the current can be seen also as a single rectangular turn of an inductor with inductance Lt(H): the current rate of change is limited. The equivalent circuit emerges as a series inductor followed by a capacitor to the ground.

This section addresses the response of a cascade of infinitesimal ladder circuits with length Δx, consisting of a series inductor with inductance LΔx and a capacitor to the GND with capacitance CΔx, where L (H/m) and C (F/m) are, respectively, a distributed inductance and a distributed capacitance.

• Frame (3) shows a Smith chart with the impedance of the ladder circuit and optionally, by activating an upper button, with the singular points at frequencies for which the transfer function is minimal. This output intends to show the trend of the discretized line input

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

59

• Frame (4) shows the schematics with the circuit parameters beneath. At the right, there is the input area allowing changing the simulation conditions (i.e., the frequency, number of

Run the script with N = 1 and the frequency range up to 1.5 GHz. Observe the results in Figure 2 (left). Check if the single cell circuit is a low-pass filter whose impedance becomes

By increasing N = 5 the filter bandwidth becomes larger and the impedance starts to exhibit a real part close to 50 Ω and an imaginary part still inductive, but, at some frequencies is null: resonances are occurring and the input impedance is Zin = 50 Ω. The response amplitude is 0 dB at these last frequencies with others exhibiting some local minima. The Smith chart and the numerical outputs allow the identification of these points easily. The trend of Zin with frequency is an infinite inductive reactance: the first input cell dominates always the full circuit

Increasing further the number of cells allows observing that the input resistance remains close to the 50 Ω and the input reactance remains close to 0 Ω for an increased bandwidth. Therefore, the circuit becomes an all pass filter with a delay, as can be observed from the linear phase exhibited

by the frequency response of the circuit, approaching that of the ideal transmission line.

impedance in a Vector Network Analyzer (VNA) format.

Figure 1. TLM Matlab tool aspect: frame description is given below.

purely reactive (inductive more specifically) and tends to an infinite value.

at a sufficiently high frequency that is related with N.

cells and ZL).

2.2. A few exercises

#### 2.1. The TLM script description and objectives

The TLM.m script analyses the frequency response of a chosen number of N cascaded cell circuits (i.e., a discretized transmission line) terminated by an arbitrary resistance RL. The total inductance and capacitance correspond to that of a 1 m commercial coaxial cable with a characteristic impedance of Z0 = 50 Ω. The frequency response of the 1 m line is also calculated and depicted together with that of the discretized line, for comparison purposes.

The tool interface is shown in Figure 1 depicting a simulation from DC to 1 GHz with 4 cells and ZL = 50 Ω. All the simulations are made assuming L = 189 nH/m, C = 76 pF/m and a generator with an internal impedance Rg = 50 Ω. The main components of the tool and corresponding objectives are:


Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools http://dx.doi.org/10.5772/intechopen.74937 59

Figure 1. TLM Matlab tool aspect: frame description is given below.


#### 2.2. A few exercises

on plane waves that is available in http://bit.ly/2m9TvMY is now presented and comprehen-

An ideal transmission line is made of a pair of conductors that carries a signal along a path length ℓ with only a delay τ. Supposing a very high frequency and that any perturbation travels at finite speed we cannot assume that the voltage and current are constant along the line. Assuming two narrow parallel conducting strips of length ℓ = 1 m the image of a capacitor, with capacitance Ct(F), materializes immediately: an input voltage at one end will trigger a current to charge the capacitor. However, the two strips carrying the current can be seen also as a single rectangular turn of an inductor with inductance Lt(H): the current rate of change is limited. The equivalent circuit emerges as a series inductor followed by a capacitor to

This section addresses the response of a cascade of infinitesimal ladder circuits with length Δx, consisting of a series inductor with inductance LΔx and a capacitor to the GND with capacitance CΔx, where L (H/m) and C (F/m) are, respectively, a distributed inductance and a

The TLM.m script analyses the frequency response of a chosen number of N cascaded cell circuits (i.e., a discretized transmission line) terminated by an arbitrary resistance RL. The total inductance and capacitance correspond to that of a 1 m commercial coaxial cable with a characteristic impedance of Z0 = 50 Ω. The frequency response of the 1 m line is also calculated

The tool interface is shown in Figure 1 depicting a simulation from DC to 1 GHz with 4 cells and ZL = 50 Ω. All the simulations are made assuming L = 189 nH/m, C = 76 pF/m and a generator with an internal impedance Rg = 50 Ω. The main components of the tool and

• Frame (1) shows the ladder circuit transfer function—the amplitude in dB (blue line) and the unwrapped phase in rad (green line)—and the transfer function of the ideal transmission line in the same format (black dotted line for the amplitude and red dotted line for the phase). The outputs intend to show that by increasing the number of cells the response of

• Frame (2) shows the ladder circuit and the transmission line input resistance and input reactance using the same trace and color codes. The bottom slider is used to select a particular frequency and to highlight the ladder circuit response, in all the graphs, with a square marker. These values are also given in a table at the right of the slider. Additionally,

and depicted together with that of the discretized line, for comparison purposes.

the discrete ladder circuit becomes close to that of the transmission line.

the particular points of the frequency response can be examined.

sively described here.

58 Antennas and Wave Propagation

the ground.

distributed capacitance.

corresponding objectives are:

2. Transmission line model: TLM.m

2.1. The TLM script description and objectives

Run the script with N = 1 and the frequency range up to 1.5 GHz. Observe the results in Figure 2 (left). Check if the single cell circuit is a low-pass filter whose impedance becomes purely reactive (inductive more specifically) and tends to an infinite value.

By increasing N = 5 the filter bandwidth becomes larger and the impedance starts to exhibit a real part close to 50 Ω and an imaginary part still inductive, but, at some frequencies is null: resonances are occurring and the input impedance is Zin = 50 Ω. The response amplitude is 0 dB at these last frequencies with others exhibiting some local minima. The Smith chart and the numerical outputs allow the identification of these points easily. The trend of Zin with frequency is an infinite inductive reactance: the first input cell dominates always the full circuit at a sufficiently high frequency that is related with N.

Increasing further the number of cells allows observing that the input resistance remains close to the 50 Ω and the input reactance remains close to 0 Ω for an increased bandwidth. Therefore, the circuit becomes an all pass filter with a delay, as can be observed from the linear phase exhibited by the frequency response of the circuit, approaching that of the ideal transmission line.

inductance and capacitance of the ladder circuit elements, was added to the PCB for comparison purposes. The impedance of both circuits was measured with a VNA and is presented in

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

As can be observed, the ladder circuit impedance follows in major lines the expected behavior, including the trend with the frequency to become a very high inductive impedance, in spite of some noticeable deviation at higher frequencies and not the perfect "ties" in general. The global similarity can be tested with TLM.m simulations using N = 6. On the other hand, the LP filter shows the expected trend: the input impedance becomes a very high inductive value at an early frequency if compared to the ladder circuit. In fact, the impedance even becomes capacitive: the anomaly is due to the non-ideality of the components, namely the serial reso-

In spite of the non-ideality of the frequency response of the components and other implementation difficulties, the prototype is very promising as a pedagogical tool to teach the circuit model of transmission line: it represents the telegrapher's equation physical model. Eventually, having a single L-C circuit, a circuit with N = 2 and another one with N = 4 may be recommended. The hardware implementation requires the use of high quality inductors and capacitors (high selfresonance frequency: small nominal values are recommended), equalization of the nominal

Another interesting and easy exercise for the student is to compute, himself, the input imped-

where n goes from 1 to N and the starting condition is Zn�<sup>1</sup> = ZL, for n = 1. With a little effort Eq. (1) can be implemented and computed in Matlab for any range of frequencies, an arbitrary number N of sections or even load impedances (other details, such as how to compute the

Wave propagation—voltage and current—in transmission lines is usually handled by employing phasor analysis. In this context, it is crucial to understand that, in general, there are two waves traveling in opposite directions and that their relative phase changes along the line. Consequently, a standing wave is produced for voltage and current. The Zin tool explores the characteristics of the line input impedance by displaying simultaneously the amplitudes of the voltage and current standing waves and the corresponding phasors. Therefore, the relative amplitude and phase between them may be associated with the standing waves. The input impedance may be observed at any arbitrary point in a line of one-wavelength long, enabling

this way, the perception of the impedance periodic behavior for lossless lines.

Zn�<sup>1</sup> 1 þ jZn�<sup>1</sup>ωC<sup>1</sup>

(1)

61

values and a careful layout of the inductors to minimize mutual inductance.

ance of the circuit with N cascaded cells, by using the recursive equation:

Zn ¼ jωL<sup>1</sup> þ

Figure 3(b).

nance of the inductor.

transfer function, can be found in [1]).

3. Transmission line input impedance: Zin.m

Figure 2. Transfer function and input impedance: N = 1 (left) and N = 16 (right).

#### 2.3. Laboratory experiment

A laboratory prototype with N = 6 sections was implemented as can be observed in Figure 3(a). The ladder circuit uses inductors with an average L1 = 36.7 nH and capacitors with an average C1 = 16.0 pF, ( ffiffiffiffiffiffiffiffiffiffiffi LC�<sup>1</sup> <sup>p</sup> ffi 50Ω). Furthermore, a lumped low-pass L-C circuit, with the integrated

Figure 3. (a) Synthetic line with N = 6 and a lumped low-pass L-C circuit; (b) Zin measured with a VNA up to 400 MHz.

inductance and capacitance of the ladder circuit elements, was added to the PCB for comparison purposes. The impedance of both circuits was measured with a VNA and is presented in Figure 3(b).

As can be observed, the ladder circuit impedance follows in major lines the expected behavior, including the trend with the frequency to become a very high inductive impedance, in spite of some noticeable deviation at higher frequencies and not the perfect "ties" in general. The global similarity can be tested with TLM.m simulations using N = 6. On the other hand, the LP filter shows the expected trend: the input impedance becomes a very high inductive value at an early frequency if compared to the ladder circuit. In fact, the impedance even becomes capacitive: the anomaly is due to the non-ideality of the components, namely the serial resonance of the inductor.

In spite of the non-ideality of the frequency response of the components and other implementation difficulties, the prototype is very promising as a pedagogical tool to teach the circuit model of transmission line: it represents the telegrapher's equation physical model. Eventually, having a single L-C circuit, a circuit with N = 2 and another one with N = 4 may be recommended. The hardware implementation requires the use of high quality inductors and capacitors (high selfresonance frequency: small nominal values are recommended), equalization of the nominal values and a careful layout of the inductors to minimize mutual inductance.

Another interesting and easy exercise for the student is to compute, himself, the input impedance of the circuit with N cascaded cells, by using the recursive equation:

$$Z\_n = j\omega L\_1 + \frac{Z\_{n-1}}{1 + jZ\_{n-1}\omega C\_1} \tag{1}$$

where n goes from 1 to N and the starting condition is Zn�<sup>1</sup> = ZL, for n = 1. With a little effort Eq. (1) can be implemented and computed in Matlab for any range of frequencies, an arbitrary number N of sections or even load impedances (other details, such as how to compute the transfer function, can be found in [1]).

## 3. Transmission line input impedance: Zin.m

2.3. Laboratory experiment

60 Antennas and Wave Propagation

ffiffiffiffiffiffiffiffiffiffiffi LC�<sup>1</sup> <sup>p</sup>

Figure 2. Transfer function and input impedance: N = 1 (left) and N = 16 (right).

C1 = 16.0 pF, (

A laboratory prototype with N = 6 sections was implemented as can be observed in Figure 3(a). The ladder circuit uses inductors with an average L1 = 36.7 nH and capacitors with an average

Figure 3. (a) Synthetic line with N = 6 and a lumped low-pass L-C circuit; (b) Zin measured with a VNA up to 400 MHz.

ffi 50Ω). Furthermore, a lumped low-pass L-C circuit, with the integrated

Wave propagation—voltage and current—in transmission lines is usually handled by employing phasor analysis. In this context, it is crucial to understand that, in general, there are two waves traveling in opposite directions and that their relative phase changes along the line. Consequently, a standing wave is produced for voltage and current. The Zin tool explores the characteristics of the line input impedance by displaying simultaneously the amplitudes of the voltage and current standing waves and the corresponding phasors. Therefore, the relative amplitude and phase between them may be associated with the standing waves. The input impedance may be observed at any arbitrary point in a line of one-wavelength long, enabling this way, the perception of the impedance periodic behavior for lossless lines.

### 3.1. Description and objectives

The application assumes a transmission line with Z0 = 50 Ω and a generator with an amplitude of 2 V and internal impedance of 50 Ω. The tool interface is depicted in Figure 4 and its description is as follows:

The picture of the reflection coefficient in a polar representation, as it is usually measured in laboratory using VNAs, intends to introduce or reinforce, in a straightforward way, the "Towards generator" or "Towards load" concepts, facilitating the understanding and use of the Smith chart. The impedance graph aims to establish the relationship between the type of impedance (with inductive or capacitive reactance) and the quadrant where the complex coefficient is represented and to establish also that the reflection coefficient and the corresponding impedance both describe a circumference (i.e., there is a conformal transforma-

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

63

Consider the lossless line with a resistive load impedance being two to five times smaller (or

• Observe the fluctuations on the voltage and current amplitudes along the line, the resulting input impedance, by inspecting both representations (the rectangular and complex plane graphs) and the refection coefficient. Notice that the impedance at d = 0.25λ is again purely resistive (but greater than the characteristic impedance), whereas at d = 0.5λ

• Also, analyze the behavior of the input impedance at the standing wave maxima and minima points, and notice that the impedance nature (capacitive/inductive) changes, that is, at these points, the line behaves as a resonant circuit. Try to identify the type of resonance (series/parallel) exhibited at each of these points. Hint: consider a small increase/reduction in the frequency (i.e., a small reduction/increase of the wavelength)

• Within the first line section of 0.5λ length, closer to the load, locate two points where the resistance of the observed impedance is equal to the line characteristic impedance and annotate the corresponding reactance. Explain how this reactance could be canceled by

Maintaining the lossless line, study other loads as the short-circuit (SC), open-circuit (OC) and purely reactive impedances. In addition to d = 0.25λ and d = 0.5λ, considered previously, also

Consider now a lossy line and repeat the simulations with the loads suggested in the previous paragraphs. The results presented in Figure 4 refer to a 12 Ω load and those in Figure 5 refer to the SC. In both cases, it was considered a transmission line with an attenuation coefficient of 0.07 Np/m. Notice that voltage and current phase relations are maintained, but their amplitude fluctuations are more significant near the load. On the other hand, near the line input (generator) these fluctuations tend to vanish, particularly for highly attenuating lines (to better observe this effect, it is advisable to increase the attenuation coefficient). As a result, both the reflection coefficient and the input line impedance trace a spiral converging to the line charac-

larger) than the characteristic impedance, for example, as 12 Ω.

it is exactly equal to the load impedance again.

and compare with the lumped resonant circuits.

using, for example, a lumped element.

observe the line impedance at d = 0.125λ.

teristic impedance.

tion between both).

3.2. A few exercises


Figure 4. The interface of the input impedance demonstrator.

The picture of the reflection coefficient in a polar representation, as it is usually measured in laboratory using VNAs, intends to introduce or reinforce, in a straightforward way, the "Towards generator" or "Towards load" concepts, facilitating the understanding and use of the Smith chart. The impedance graph aims to establish the relationship between the type of impedance (with inductive or capacitive reactance) and the quadrant where the complex coefficient is represented and to establish also that the reflection coefficient and the corresponding impedance both describe a circumference (i.e., there is a conformal transformation between both).

#### 3.2. A few exercises

3.1. Description and objectives

also presented graphically in polar coordinates.

resistance and reactance at the selected line point.

Figure 4. The interface of the input impedance demonstrator.

description is as follows:

62 Antennas and Wave Propagation

The application assumes a transmission line with Z0 = 50 Ω and a generator with an amplitude of 2 V and internal impedance of 50 Ω. The tool interface is depicted in Figure 4 and its

• On the top, it is shown the line circuit schematic, and on its right, there is a small area for entering the simulation parameters: the load impedance and the attenuation coefficient may be introduced. Above the schematic, there is an information area for results output, displaying the VSWR, the maximum and the minimum values of the resistance and reactance along the line. These results are updated every time any input is changed. • Below the schematic, there is a rectangular graph where the normalized voltage phasor, 1+Γ(d), and the normalized current phasor, 1-Γ(d), are represented (in blue and red, respectively) at one point d of the line, which is chosen by actuating on the slider below this diagram. Furthermore, the voltage and current amplitudes along the line are also shown. Closely, on the right of this graph there is a new area for results output, presenting the reflection coefficient in the chosen point of the line, using rectangular and polar coordinates. At the rightmost side, the reflection coefficient, along the line, is

• At the bottom left, there is another rectangular graph exhibiting the observed resistance and reactance along the line (in blue and red, respectively). In addition, at the bottom rightmost side, the impedance along the line is represented again, this time in the complex plane. Below this representation, there is one more area for results output displaying the Consider the lossless line with a resistive load impedance being two to five times smaller (or larger) than the characteristic impedance, for example, as 12 Ω.


Maintaining the lossless line, study other loads as the short-circuit (SC), open-circuit (OC) and purely reactive impedances. In addition to d = 0.25λ and d = 0.5λ, considered previously, also observe the line impedance at d = 0.125λ.

Consider now a lossy line and repeat the simulations with the loads suggested in the previous paragraphs. The results presented in Figure 4 refer to a 12 Ω load and those in Figure 5 refer to the SC. In both cases, it was considered a transmission line with an attenuation coefficient of 0.07 Np/m. Notice that voltage and current phase relations are maintained, but their amplitude fluctuations are more significant near the load. On the other hand, near the line input (generator) these fluctuations tend to vanish, particularly for highly attenuating lines (to better observe this effect, it is advisable to increase the attenuation coefficient). As a result, both the reflection coefficient and the input line impedance trace a spiral converging to the line characteristic impedance.

Figure 5. Input impedance of a lossy transmission line terminated in SC.

#### 3.3. Laboratory experiment

Figure 6 displays experimental results obtained for a microstrip line terminated in SC whose length is 6.2 cm, which equals one wavelength at about 2.73 GHz.

achieve quite consistent results, because, the demonstrator considers an increasing distance to the load, while, the measurements include an increasing attenuation coefficient with frequency

Figure 7. (a) Microstrip line terminated with ZL = 12 Ω; (b) the input reflection coefficient measured from 500 kHz up to

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

65

The resistance and reactance of the impedance along the line are also presented in Figure 6(c). In order to make the comparison with the results from the demonstrator easier (Figure 5), the frequency axis has been translated into electrical distance by using the frequency for which the

Figure 7 displays the same type of experimental results obtained for a microstrip line with the same physical properties and terminated with ZL = 12 Ω. This line may be viewed as a practical implementation for the input parameters used in Figure 4. In this case, similarly to the lineterminated SC, the reflection coefficient starts with an angle of 180 and its angular excursion should be the same, since the physical properties of the two lines are identical. However, the reflection coefficient in Figure 7(b) presents a larger angular excursion, suggesting a longer line or, in alternative, a load with a small inductive element. In fact, this can be due to the small patch used for welding the load to the ground plane below the substrate or the non-ideality of the resistor. Therefore, the frequency for which the line length appears to be one wavelength decreased to 2.57 GHz, which is about 6% less than that of the SC. Consequently, by using this value to obtain the electrical distance, for the abscissa axis in Figure 7(c), results in an appar-

This application helps to find a matching solution expending a single series or parallel element. The solution is not given immediately; instead, it is intended to be found by experimentation,

for a constant line length.

line length equals one wavelength.

2 GHz; and (c) the corresponding input impedance.

ently longer transmission line.

4. Impedance matching: Matching.m

The reflection coefficient, presented in Figure 6(b), was measured using a VNA, from 500 kHz (almost DC) up to 2 GHz. This line may be considered as a practical implementation for the input parameters used in Figure 5, even though, the measurements were acquired in the frequency domain, not along the line as shown by the demonstrator. In fact, it is possible to

Figure 6. (a) Microstrip line terminated in SC; (b) the input reflection coefficient measured from 500 kHz up to 2 GHz; and (c) the corresponding input impedance.

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools http://dx.doi.org/10.5772/intechopen.74937 65

Figure 7. (a) Microstrip line terminated with ZL = 12 Ω; (b) the input reflection coefficient measured from 500 kHz up to 2 GHz; and (c) the corresponding input impedance.

achieve quite consistent results, because, the demonstrator considers an increasing distance to the load, while, the measurements include an increasing attenuation coefficient with frequency for a constant line length.

The resistance and reactance of the impedance along the line are also presented in Figure 6(c). In order to make the comparison with the results from the demonstrator easier (Figure 5), the frequency axis has been translated into electrical distance by using the frequency for which the line length equals one wavelength.

Figure 7 displays the same type of experimental results obtained for a microstrip line with the same physical properties and terminated with ZL = 12 Ω. This line may be viewed as a practical implementation for the input parameters used in Figure 4. In this case, similarly to the lineterminated SC, the reflection coefficient starts with an angle of 180 and its angular excursion should be the same, since the physical properties of the two lines are identical. However, the reflection coefficient in Figure 7(b) presents a larger angular excursion, suggesting a longer line or, in alternative, a load with a small inductive element. In fact, this can be due to the small patch used for welding the load to the ground plane below the substrate or the non-ideality of the resistor. Therefore, the frequency for which the line length appears to be one wavelength decreased to 2.57 GHz, which is about 6% less than that of the SC. Consequently, by using this value to obtain the electrical distance, for the abscissa axis in Figure 7(c), results in an apparently longer transmission line.

#### 4. Impedance matching: Matching.m

3.3. Laboratory experiment

64 Antennas and Wave Propagation

and (c) the corresponding input impedance.

Figure 6 displays experimental results obtained for a microstrip line terminated in SC whose

The reflection coefficient, presented in Figure 6(b), was measured using a VNA, from 500 kHz (almost DC) up to 2 GHz. This line may be considered as a practical implementation for the input parameters used in Figure 5, even though, the measurements were acquired in the frequency domain, not along the line as shown by the demonstrator. In fact, it is possible to

Figure 6. (a) Microstrip line terminated in SC; (b) the input reflection coefficient measured from 500 kHz up to 2 GHz;

length is 6.2 cm, which equals one wavelength at about 2.73 GHz.

Figure 5. Input impedance of a lossy transmission line terminated in SC.

This application helps to find a matching solution expending a single series or parallel element. The solution is not given immediately; instead, it is intended to be found by experimentation, using the properties of the transmission line impedance, that is, the user must find an appropriate distance to the load where to insert a susceptance (parallel) or a reactance (series) and the corresponding length (for a stub) or the nominal component value (if a lumped element is used).

The demonstrator shows the transmission line together with the stub or the lumped component, and uses the Smith chart so that the user is able to evaluate the outcome of all the inputs or changes introduced, by displaying the input impedance or admittance on the Smith chart and also, numerically, next to the circuit schematic. Furthermore, assuming that the reactance of the load does not change with frequency, the circuit frequency response is presented on the Smith chart and, in addition, on one rectangular graph window (return loss, VSWR or transmission loss can be represented).

## 4.1. Description and objectives

The tool assumes a generator with internal impedance of 50 Ω and a lossless transmission line with characteristic impedance of 50 Ω, with a length that is 0.5λ at the chosen frequency. The user interface is depicted in Figure 8 and its description is as follows:


• By actuating in the slider below the circuit schematic locate the first point in line (as close to the load as possible) where the real part of Z(d) is about 50 Ω. Then move the other slider inside the parameter input panel and try to reach the center of the Smith chart. This is the solution depicted in Figure 8. Confirm that only one of the available components

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

67

• Locate a second point in the line where the real part of Z(d) is again 50 Ω, and find the lumped component which matches the system. Verify whether there is an alternative position on the line (different from those found previously) where a match can be achieved. Observe the input impedance, the VSWR and the return loss as a function of the frequency. Try to explain the behavior of the return loss when the frequency tends to

• Summarize all solutions found: two possible points per half wavelength to insert the component and, that for each position, only one type of lumped component (inductor or

• Move the slider below the schematic and find one first position on the line (as close to the load as possible) where the real part of Y(d) is about 1/50 S and notice that at this location, the equivalent impedance, Zin, involves a 50 Ω resistor in parallel with a certain reactance. Then, move the slider inside the parameter input panel and try to reach the center of the

Consider now ZL = 12.5 Ω, fp = 650 MHz and achieve the matching using a parallel stub:

(inductor or capacitor) allows matching the transmission line.

capacitor) offers a solution to the problem.

Figure 8. The interface of the matching demonstrator.

infinity.


#### 4.2. A few exercises

Consider ZL = 12 Ω and fp = 700 MHz. Start by introducing a series network using a lumped component:

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools http://dx.doi.org/10.5772/intechopen.74937 67

Figure 8. The interface of the matching demonstrator.

using the properties of the transmission line impedance, that is, the user must find an appropriate distance to the load where to insert a susceptance (parallel) or a reactance (series) and the corresponding length (for a stub) or the nominal component value (if a lumped element is used). The demonstrator shows the transmission line together with the stub or the lumped component, and uses the Smith chart so that the user is able to evaluate the outcome of all the inputs or changes introduced, by displaying the input impedance or admittance on the Smith chart and also, numerically, next to the circuit schematic. Furthermore, assuming that the reactance of the load does not change with frequency, the circuit frequency response is presented on the Smith chart and, in addition, on one rectangular graph window (return loss, VSWR or trans-

The tool assumes a generator with internal impedance of 50 Ω and a lossless transmission line with characteristic impedance of 50 Ω, with a length that is 0.5λ at the chosen frequency. The

• On the upper left side it is shown the circuit schematic and, on its right there is an area for entering the parameters, where the user can specify the load impedance, the project frequency (fp), the type of matching network (series or parallel) and the type of the matching element (inductor, capacitor or OC/SC terminated stub). Any of these options

• Below the circuit schematic, there is a slider allowing the user to choose where to place the matching component and, below the slider, a Smith chart is displayed. As the slider is moved, the line impedance for series arrangements (or admittance for parallel arrangements) is displayed on the Smith chart using a blue line. Simultaneously, the corresponding

• The matching element (stub or lumped component) may be adjusted using another slider inside the area for parameter inputs. The resulting impedance (or admittance) is immediately shown on the Smith chart using a green line. At the same time, numerical values are also displayed near this slider: stub length (or nominal value of the lumped component), corresponding reactance (or susceptance in absolute value), position of the matching

• At the bottom right side, a rectangular window presents the return loss, transmission loss or the VSWR: the user can choose one of the three which is displayed as a function of the frequency from fp/2 to 10fp. In addition, a "check box" allows, at any moment, to visualize

Consider ZL = 12 Ω and fp = 700 MHz. Start by introducing a series network using a lumped

also the impedance frequency response on the Smith chart (or remove it).

user interface is depicted in Figure 8 and its description is as follows:

may be set and changed at any time by the user.

numerical values are given next to the slider.

network, and the VSWR achieved.

4.2. A few exercises

component:

mission loss can be represented).

66 Antennas and Wave Propagation

4.1. Description and objectives


Consider now ZL = 12.5 Ω, fp = 650 MHz and achieve the matching using a parallel stub:

• Move the slider below the schematic and find one first position on the line (as close to the load as possible) where the real part of Y(d) is about 1/50 S and notice that at this location, the equivalent impedance, Zin, involves a 50 Ω resistor in parallel with a certain reactance. Then, move the slider inside the parameter input panel and try to reach the center of the Smith chart. Annotate the corresponding stub lengths for OC and SC terminations. Figure 9 presents the solution using a stub in OC.


#### 4.3. Laboratory experiment

Figure 10(a) shows a practical implementation of the solution given in Figure 8. Its input impedance was measured, from 500 kHz up to 2 GHz, using a VNA and is represented in Figure 10(b). The corresponding return loss is presented in Figure 10(c). The best return loss value (highlighted by an asterisk in the figure) is slightly better than 15 dB and it is achieved at nearly the project frequency (698 MHz).

Furthermore, Figure 11(a) depicts a practical implementation of the solution found in Figure 9. In the same way, Figure 11(b) and (c) show, respectively, the measured input impedance and the corresponding return loss. At the project frequency (650 MHz) the return loss measured is about 20.4 dB (the best return loss value is about 21 dB achieved at 637 MHz).

In both cases, the results displayed by the demonstrator are generally in good agreement with

Figure 11. (a) Matching ZL = 12.5 Ω at fp = 650 MHz, with a parallel OC stub, (b) the measured input impedance, and (c)

Figure 10. (a) Matching ZL = 12 Ω at fp = 700 MHz using a series capacitor, (b) the measured input impedance, and (c)

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

69

The reflection and transmission of electromagnetic plane waves is a fundamental resource to understand wireless multipath propagation channel models and to introduce basic concepts concerning propagation phenomena in optical fibers. Teaching this matter is even more difficult than the transmission lines one. To justify we can point out the following reasons: the phenomenon depends on the wave polarization; vectors should be used to represent electromagnetic fields (3-D visualization or similar is mandatory); the superposition of incident and

those from the measurements.

measured return loss.

measured return loss.

5. Electromagnetic waves: EM.m

Figure 9. Matching ZL = 12.5 Ω at fp = 650 MHz with a parallel stub in OC.

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools http://dx.doi.org/10.5772/intechopen.74937 69

Figure 10. (a) Matching ZL = 12 Ω at fp = 700 MHz using a series capacitor, (b) the measured input impedance, and (c) measured return loss.

Figure 11. (a) Matching ZL = 12.5 Ω at fp = 650 MHz, with a parallel OC stub, (b) the measured input impedance, and (c) measured return loss.

In both cases, the results displayed by the demonstrator are generally in good agreement with those from the measurements.

#### 5. Electromagnetic waves: EM.m

Smith chart. Annotate the corresponding stub lengths for OC and SC terminations. Figure 9

• Observe that, at certain frequency, the return loss reaches the worst value of 0 dB (meaning that the reflection coefficient touches somewhere the outer circle of the Smith chart,

• Repeat this last exercise using lumped elements and notice that for each of the above positions (those where it is possible to match the line impedance with parallel stubs) there

Figure 10(a) shows a practical implementation of the solution given in Figure 8. Its input impedance was measured, from 500 kHz up to 2 GHz, using a VNA and is represented in Figure 10(b). The corresponding return loss is presented in Figure 10(c). The best return loss value (highlighted by an asterisk in the figure) is slightly better than 15 dB and it is achieved at

Furthermore, Figure 11(a) depicts a practical implementation of the solution found in Figure 9. In the same way, Figure 11(b) and (c) show, respectively, the measured input impedance and the corresponding return loss. At the project frequency (650 MHz) the return loss measured is

about 20.4 dB (the best return loss value is about 21 dB achieved at 637 MHz).

Figure 9. Matching ZL = 12.5 Ω at fp = 650 MHz with a parallel stub in OC.

presents the solution using a stub in OC.

4.3. Laboratory experiment

68 Antennas and Wave Propagation

nearly the project frequency (698 MHz).

that is, |Γ| = 1). Try to explain this observation.

is only one lumped element that allows to match the line.

The reflection and transmission of electromagnetic plane waves is a fundamental resource to understand wireless multipath propagation channel models and to introduce basic concepts concerning propagation phenomena in optical fibers. Teaching this matter is even more difficult than the transmission lines one. To justify we can point out the following reasons: the phenomenon depends on the wave polarization; vectors should be used to represent electromagnetic fields (3-D visualization or similar is mandatory); the superposition of incident and reflected fields is more complex, because, waves do not travel in the same direction; and finally, there are particular cases, such as, the total transmission (Brewster angle) and the total reflection (above a critical angle), of incident plane waves.

#### 5.1. The EM.m script description and objectives

Let us assume an electromagnetic plane wave traveling in a medium 1 impinging on a plane infinite boundary (from now on called reflection plane) between two lossless mediums: medium 1 and medium 2, having, respectively, a dielectric permittivity ε<sup>1</sup> and ε2. Both mediums are non-magnetic, that is, μ<sup>1</sup> = μ<sup>2</sup> = μ0, with μ<sup>0</sup> being the vacuum permeability. The dielectric permittivity, ε, can be written using the relative permittivity, εr, as ε ¼ εrε0, with ε<sup>0</sup> being the free space permittivity.

The impedance of a medium is given by.

$$
\eta = \frac{120\pi}{\sqrt{\varepsilon\_r}} \text{ (\Omega)}\tag{2}
$$

with θ<sup>t</sup> given by Eq. (4) and, for the perpendicular polarization, are given by:

q

angle at which the reflection coefficient becomes null. This angle is given by:

<sup>θ</sup>iB <sup>¼</sup> sin �<sup>1</sup>

<sup>θ</sup>ic <sup>¼</sup> sin �<sup>1</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>E</sup>ioð Þ <sup>b</sup><sup>x</sup> cos <sup>θ</sup><sup>i</sup> � <sup>b</sup>z senθ<sup>i</sup> <sup>e</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>Γ</sup>kEio ð Þ <sup>b</sup><sup>x</sup> cos <sup>θ</sup><sup>i</sup> <sup>þ</sup> <sup>b</sup>z senθ<sup>i</sup> <sup>e</sup>

Eio η1 e

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>b</sup><sup>y</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εr<sup>2</sup> <sup>ε</sup>r<sup>1</sup> � sen<sup>2</sup> θ<sup>i</sup>

Figure 12. Plane wave incidence (plane XoY): 'i', 'r' and 't' means incident, reflected and transmitted waves, the unit vector along the incident, reflected and transmitted waves direction of propagation are bki,bkr,bkt, the symbol ⊙ means

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

71

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εr<sup>2</sup> <sup>ε</sup>r<sup>1</sup> � sen<sup>2</sup> θ<sup>i</sup>

It can be shown that, for parallel polarization, there is an angle of incidence, θiB, called Brewster

In addition, regardless the wave polarization there is an angle of incidence, for the case of incidence in a less dense medium, which turns the absolute value of the reflection coefficient

For the parallel polarization, the following set of electric and magnetic fields equations can be

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εr<sup>2</sup> εr<sup>1</sup> þ εr<sup>2</sup>

ffiffiffiffiffiffi εr<sup>2</sup> εr<sup>1</sup> � � r

<sup>q</sup> <sup>Τ</sup><sup>⊥</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>Γ</sup><sup>⊥</sup> (6)

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>iþ<sup>z</sup> cos <sup>θ</sup><sup>i</sup> (9)

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>i�<sup>z</sup> cos <sup>θ</sup><sup>i</sup> (11)

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>iþ<sup>z</sup> cos <sup>θ</sup><sup>i</sup> (10)

(7)

(8)

cos θ<sup>i</sup> �

cos θ<sup>i</sup> þ

Γ<sup>⊥</sup> ¼

vector pointing to the reader and ⊗ pointing downwards.

unitary. This is the critical angle, θic, given by:

Ei !

Er ! Hi !

derived using Eqs. (2)–(5):

and the wavenumber (or phase constant) is given by:

$$
\beta = \frac{2\pi\sqrt{\varepsilon\_r}}{\lambda\_0} \pmod{\text{m}} \tag{3}
$$

with λ<sup>0</sup> being the free space wavelength. The angle of incidence, θi, is defined as the angle between the direction of propagation, bki, and the normal to the reflection plane (please see Figure 12). The incidence plane is the plane defined by the direction of propagation and the normal to the reflection plane (XoZ plane). The incident wave causes a reflected wave, having an angle of reflection equal to the angle of incidence, and a transmitted wave propagating on medium 2, having an angle of transmission, θt, given by:

$$
\operatorname{sen}\,\Theta\_t = \sqrt{\frac{\varepsilon\_{r1}}{\varepsilon\_{r2}}} \operatorname{sen}\,\Theta\_i \tag{4}
$$

Two distinct linearly polarized incident plane waves, Figure 12, can be identified:


The reflection coefficient, Γ, and the transmission coefficient, T, for the parallel polarization, are given by:

$$\Gamma\_{\parallel} = \frac{\sqrt{\left(1 - \frac{\varepsilon\_{\rm 2}}{\varepsilon\_{\rm 2}} \text{sen}^{2} \theta\_{i}\right)} - \sqrt{\frac{\varepsilon\_{\rm 2}}{\varepsilon\_{\rm 1}}} \cos \,\theta\_{i}}{\sqrt{\left(1 - \frac{\varepsilon\_{\rm 2}}{\varepsilon\_{\rm 2}} \text{sen}^{2} \theta\_{i}\right)} + \sqrt{\frac{\varepsilon\_{\rm 2}}{\varepsilon\_{\rm 1}}} \cos \,\theta\_{i}} \qquad T\_{\parallel} = \left(1 + \Gamma\_{\parallel}\right) \frac{\cos \,\theta\_{i}}{\cos \,\theta\_{i}}\tag{5}$$

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools http://dx.doi.org/10.5772/intechopen.74937 71

Figure 12. Plane wave incidence (plane XoY): 'i', 'r' and 't' means incident, reflected and transmitted waves, the unit vector along the incident, reflected and transmitted waves direction of propagation are bki,bkr,bkt, the symbol ⊙ means vector pointing to the reader and ⊗ pointing downwards.

with θ<sup>t</sup> given by Eq. (4) and, for the perpendicular polarization, are given by:

reflected fields is more complex, because, waves do not travel in the same direction; and finally, there are particular cases, such as, the total transmission (Brewster angle) and the total

Let us assume an electromagnetic plane wave traveling in a medium 1 impinging on a plane infinite boundary (from now on called reflection plane) between two lossless mediums: medium 1 and medium 2, having, respectively, a dielectric permittivity ε<sup>1</sup> and ε2. Both mediums are non-magnetic, that is, μ<sup>1</sup> = μ<sup>2</sup> = μ0, with μ<sup>0</sup> being the vacuum permeability. The dielectric permittivity, ε, can be written using the relative permittivity, εr, as ε ¼ εrε0, with ε<sup>0</sup> being the

> <sup>η</sup> <sup>¼</sup> <sup>120</sup><sup>π</sup> ffiffiffiffi εr

<sup>β</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> ffiffiffiffi εr p λ0

senθ<sup>t</sup> ¼

Two distinct linearly polarized incident plane waves, Figure 12, can be identified:

�

þ

with λ<sup>0</sup> being the free space wavelength. The angle of incidence, θi, is defined as the angle between the direction of propagation, bki, and the normal to the reflection plane (please see Figure 12). The incidence plane is the plane defined by the direction of propagation and the normal to the reflection plane (XoZ plane). The incident wave causes a reflected wave, having an angle of reflection equal to the angle of incidence, and a transmitted wave propagating on

> ffiffiffiffiffiffi εr<sup>1</sup> εr<sup>2</sup> r

• A polarization with the electric field parallel to the incidence plane, Ek, having components along z and x. The orthogonal magnetic field, Hk, has a component along y. • A polarization with the electric field perpendicular to the incidence plane, E⊥. The orthog-

The reflection coefficient, Γ, and the transmission coefficient, T, for the parallel polarization, are

cos θ<sup>i</sup>

cos θ<sup>i</sup>

T<sup>k</sup> ¼ 1 þ Γ<sup>k</sup>

� � cos θ<sup>i</sup>

cos θ<sup>t</sup>

(5)

ffiffiffiffi εr<sup>2</sup> εr<sup>1</sup> q

ffiffiffiffi εr<sup>2</sup> εr<sup>1</sup> q

p ð Þ Ω (2)

ð Þ rad=m (3)

senθ<sup>i</sup> (4)

reflection (above a critical angle), of incident plane waves.

5.1. The EM.m script description and objectives

The impedance of a medium is given by.

and the wavenumber (or phase constant) is given by:

medium 2, having an angle of transmission, θt, given by:

onal magnetic field, H⊥, has components along z and x.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>ε</sup>r<sup>1</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>ε</sup>r<sup>1</sup>

<sup>ε</sup>r<sup>2</sup> sen<sup>2</sup> θ<sup>i</sup> r� �

<sup>ε</sup>r<sup>2</sup> sen<sup>2</sup> θ<sup>i</sup> r� �

free space permittivity.

70 Antennas and Wave Propagation

given by:

Γ<sup>k</sup> ¼

$$\Gamma\_{\perp} = \frac{\cos \theta\_i - \sqrt{\frac{\varepsilon\_2}{\varepsilon\_1} - \text{sen}^2 \theta\_i}}{\cos \theta\_i + \sqrt{\frac{\varepsilon\_2}{\varepsilon\_1} - \text{sen}^2 \theta\_i}} \qquad \quad \mathcal{T}\_{\perp} = 1 + \Gamma\_{\perp} \tag{6}$$

It can be shown that, for parallel polarization, there is an angle of incidence, θiB, called Brewster angle at which the reflection coefficient becomes null. This angle is given by:

$$\theta\_{i\mathcal{B}} = \sin^{-1}\sqrt{\frac{\varepsilon\_{r2}}{\varepsilon\_{r1} + \varepsilon\_{r2}}}\tag{7}$$

In addition, regardless the wave polarization there is an angle of incidence, for the case of incidence in a less dense medium, which turns the absolute value of the reflection coefficient unitary. This is the critical angle, θic, given by:

$$\theta\_{ic} = \sin^{-1}\left(\sqrt{\frac{\varepsilon\_{r2}}{\varepsilon\_{r1}}}\right) \tag{8}$$

For the parallel polarization, the following set of electric and magnetic fields equations can be derived using Eqs. (2)–(5):

$$\overrightarrow{\mathbf{E}}\_{i}(\mathbf{x},z) = \mathbf{E}\_{i\theta}(\widehat{\mathbf{x}}\cos\theta\_{i} - \widehat{\mathbf{z}}\sin\theta\_{i})e^{-j\theta\_{1}(\mathbf{x}\sin\theta\_{i} + z\cos\theta\_{i})}\tag{9}$$

$$\overrightarrow{\mathbf{H}}\_{i}\left(\mathbf{x},\mathbf{z}\right) = \hat{\mathbf{y}}\,\frac{\mathbf{E}\_{i\nu}}{\eta\_{1}}\,e^{-j\mathfrak{E}\_{1}\left(\mathbf{x}\sin\theta\_{i} + z\,\cos\theta\_{i}\right)}\tag{10}$$

$$\overrightarrow{\mathbf{E}}\_r(\mathbf{x}, z) = \Gamma\_{\parallel} \mathbf{E}\_{i\phi} \left( \widehat{\mathbf{x}} \cos \theta\_i + \widehat{\mathbf{z}} \sin \theta\_i \right) e^{-j\theta\_1 \left( \mathbf{x} \sin \theta\_i - z \cos \theta\_i \right)} \tag{11}$$

$$\overrightarrow{\mathbf{H}}\_{\prime}(\mathbf{x},z) = -\widehat{\mathbf{y}}\,\Gamma\_{\parallel}\frac{\mathbf{E}\_{i\flat}}{\eta\_{1}}\,e^{-j\theta\_{1}\left(\mathbf{x}\cdot\mathbf{sen}\,\theta\_{\prime}-z\cdot\cos\theta\_{\prime}\right)}\tag{12}$$

4. The amplitude of the reflection and transmission coefficients and their respective phase.

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

73

5. The interaction tools to choose the incident polarization, the angle of incidence, the dielec-

6. The numerical outputs: reflection and transmission coefficients, angle of transmission and

The simulations are made for a normalized electric field amplitude of 1 V/m, a free space wavelength λ<sup>0</sup> = 1 m and the z and x axis ranges are fixed to 4λ0. All the magnetic fields are multiplied by the impedance of medium 1, η1, to have all the values in the same range for a convenient pseudo-color graph with a color code around the black color. The graphs are time

The first exercise is made with εr1 = 1 and εr2 = 4 and normal incidence (θ<sup>i</sup> = 0) for perpendicular polarization. Figure 13 depicts some results shown for this case. Check the reflection and transmission coefficients (similar to a transmission line terminated by a resistive load). Verify

tric constants, the angle of incidence and the waves to be displayed (i, r or t).

animated by depicting successively several snapshots within one period.

The slider can be used to change the angle of incidence.

the critical or Brewster angle (if they exist).

Figure 13. EM.m tool screen shot: frame description is given below.

5.2. A few exercises

$$\overrightarrow{\mathbf{E}}\_{l}\left(\mathbf{x},z\right) = T\_{\parallel}\mathbf{E}\_{l\flat}\left(\widehat{\mathbf{x}}\cos\theta\_{l} - \widehat{\mathbf{z}}\sin\theta\_{l}\right)e^{-j\theta\_{2}\left(\mathbf{x}\sin\theta\_{l} + z\cos\theta\_{l}\right)}\tag{13}$$

Ht ! ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>b</sup>y T<sup>k</sup> Eio η2 e �jβ<sup>2</sup> ð Þ x sen <sup>θ</sup>tþ<sup>z</sup> cos <sup>θ</sup><sup>t</sup> (14)

with Eio being a reference incident electric field amplitude.

Similarly, using Eqs. (2)–(4) and (6) the following equations can be derived for the perpendicular polarization:

$$\stackrel{\rightarrow}{\mathbf{E}}\_{i}(\mathbf{x},z) = \widehat{\mathbf{y}} \cdot \mathbf{E}\_{i\boldsymbol{\theta}} e^{-j\beta\_{1}(\mathbf{x}\sin\theta\_{i} + z\cos\theta\_{i})} \tag{15}$$

$$\overrightarrow{\mathbf{H}}\_{i}\left(\mathbf{x},z\right) = \frac{\mathbf{E}\_{i\hat{w}}}{\eta\_{1}}\left(-\hat{\mathbf{x}}\cos\theta\_{i} + \hat{\mathbf{z}}\sin\theta\_{i}\right)e^{-j\theta\_{1}\left(\mathbf{x}\sin\theta\_{i} + z\cdot\cos\theta\_{i}\right)}\tag{16}$$

$$\overrightarrow{\mathbf{E}}\_r(\mathbf{x}, z) = \widehat{\mathbf{y}} \, \Gamma\_\perp \mathbf{E}\_{i\nu} e^{-j\beta\_1 \left(x \sin \theta\_i - z \cos \theta\_i\right)} \tag{17}$$

$$\overrightarrow{\mathbf{H}}\_{\tau}\left(\mathbf{x},z\right) = \frac{\Gamma\_{\perp}\mathbf{E}\_{\text{iv}}}{\eta\_{1}}\left(\widehat{\mathbf{x}}\cos\theta\_{i} + \widehat{\mathbf{z}}\sin\theta\_{i}\right)e^{-j\theta\_{1}\left(\mathbf{x}\sin\theta\_{i} - z\cos\theta\_{i}\right)}\tag{18}$$

$$\stackrel{\rightarrow}{\mathbf{E}}\_{l}(\mathbf{x},z) = \widehat{\mathbf{y}}\ \, ^{\leftarrow}T\_{\perp}\mathbf{E}\_{i\flat}e^{-j\oint\_{\Sigma}(\mathbf{x}\sin\theta\_{l}+z\cdot\cos\theta\_{l})}\tag{19}$$

$$\overrightarrow{\mathbf{H}}\_{l}\left(\mathbf{x},\mathbf{z}\right) = \frac{T\_{\perp}\mathbf{E}\_{i\flat}}{\eta\_{2}}\left(-\widehat{\mathbf{x}}\cos\Theta\_{l} + \widehat{\mathbf{z}}\sin\Theta\_{l}\right)e^{-j\theta\_{2}\left(\mathbf{x}\sin\Theta\_{l} + \mathbf{z}\cdot\cos\Theta\_{l}\right)}\tag{20}$$

The interpretation of Eqs. (5) and (6) and the fields given from Eqs. (9)–(20) as a function of θ<sup>i</sup> is difficult. The EM.m script addresses this problem by depicting the reflection coefficients and all the set of fields in three separate figures. A screenshot of the tool interface, that follows a similar but more complete approach of the one described in [2], is presented in Figure 13.

The interface is divided in four graphs (left part) and an input data/output area (right part). The numbered areas contain:


The simulations are made for a normalized electric field amplitude of 1 V/m, a free space wavelength λ<sup>0</sup> = 1 m and the z and x axis ranges are fixed to 4λ0. All the magnetic fields are multiplied by the impedance of medium 1, η1, to have all the values in the same range for a convenient pseudo-color graph with a color code around the black color. The graphs are time animated by depicting successively several snapshots within one period.

#### 5.2. A few exercises

Hr !

> Ht !

with Eio being a reference incident electric field amplitude.

Ei !

Er !

Et !

> T⊥Eio η2

medium 1 and 2 and along the <sup>b</sup><sup>x</sup> and <sup>b</sup><sup>z</sup> axis (written equations).

in medium 1 and Etz<sup>ǁ</sup> in medium 2 for the parallel polarization.

3. The fields along the <sup>b</sup><sup>x</sup> axis (as described above).

Eio η1

Γ⊥Eio η1

Et !

Hi !

H<sup>r</sup> !

Ht !

The numbered areas contain:

ð Þ¼ x; z

ð Þ¼ x; z

ð Þ¼ x; z

ular polarization:

72 Antennas and Wave Propagation

ð Þ¼� <sup>x</sup>; <sup>z</sup> <sup>b</sup><sup>y</sup> <sup>Γ</sup><sup>k</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>b</sup>y T<sup>k</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>T</sup>kEio ð Þ <sup>b</sup><sup>x</sup> cos <sup>θ</sup><sup>t</sup> � <sup>b</sup>z senθ<sup>t</sup> <sup>e</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>b</sup><sup>y</sup> <sup>E</sup>io <sup>e</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>b</sup><sup>y</sup> <sup>Γ</sup>⊥Eio <sup>e</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>b</sup>y T⊥Eio <sup>e</sup>

Eio η1 e

Eio η2 e

Similarly, using Eqs. (2)–(4) and (6) the following equations can be derived for the perpendic-

ð Þ �b<sup>x</sup> cos <sup>θ</sup><sup>i</sup> <sup>þ</sup> <sup>b</sup>z senθ<sup>i</sup> <sup>e</sup>

ð Þ <sup>b</sup><sup>x</sup> cos <sup>θ</sup><sup>i</sup> <sup>þ</sup> <sup>b</sup>z senθ<sup>i</sup> <sup>e</sup>

ð Þ �b<sup>x</sup> cos <sup>θ</sup><sup>t</sup> <sup>þ</sup> <sup>b</sup>z senθ<sup>t</sup> <sup>e</sup>

The interpretation of Eqs. (5) and (6) and the fields given from Eqs. (9)–(20) as a function of θ<sup>i</sup> is difficult. The EM.m script addresses this problem by depicting the reflection coefficients and all the set of fields in three separate figures. A screenshot of the tool interface, that follows a similar but more complete approach of the one described in [2], is presented in Figure 13.

The interface is divided in four graphs (left part) and an input data/output area (right part).

1. The fields with a single Cartesian component along <sup>b</sup>y: Ei⊥, Er⊥, or the total field E1<sup>⊥</sup> = Ei<sup>⊥</sup> + Er<sup>⊥</sup> in medium 1 and Et<sup>⊥</sup> in medium 2 for the perpendicular polarization; Hiǁ, Hrǁ, or the total field H1<sup>ǁ</sup> = Hi<sup>ǁ</sup> + Hr<sup>ǁ</sup> in medium 1 and Ht<sup>ǁ</sup> in medium 2 for the parallel polarization. The representation contains also the directions of the incident, reflected and transmitted waves (represented by segments as in Figure 12) and the wavelength in

2. The fields along the <sup>b</sup><sup>z</sup> axis: Hiz⊥, Hrz⊥, or the total field Htz<sup>⊥</sup> = Hiz<sup>⊥</sup> + Hrz<sup>⊥</sup> in medium 1 and Etz<sup>⊥</sup> in medium 2 for the perpendicular polarization; Eizǁ, Erzǁ, or the total field E1<sup>ǁ</sup> = Eiz<sup>ǁ</sup> + Erz<sup>ǁ</sup>

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>r�<sup>z</sup> cos <sup>θ</sup><sup>r</sup> (12)

�jβ<sup>2</sup> ð Þ x sen <sup>θ</sup>tþ<sup>z</sup> cos <sup>θ</sup><sup>t</sup> (14)

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>iþ<sup>z</sup> cos <sup>θ</sup><sup>i</sup> (15)

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>i�<sup>z</sup> cos <sup>θ</sup><sup>i</sup> (17)

�jβ<sup>2</sup> ð Þ x sen <sup>θ</sup>tþ<sup>z</sup> cos <sup>θ</sup><sup>t</sup> (19)

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>iþ<sup>z</sup> cos <sup>θ</sup><sup>i</sup> (16)

�jβ<sup>1</sup> ð Þ x sen <sup>θ</sup>i�<sup>z</sup> cos <sup>θ</sup><sup>i</sup> (18)

�jβ<sup>2</sup> ð Þ x sen <sup>θ</sup>tþ<sup>z</sup> cos <sup>θ</sup><sup>t</sup> (20)

�jβ<sup>2</sup> ð Þ x sen <sup>θ</sup>tþ<sup>z</sup> cos <sup>θ</sup><sup>t</sup> (13)

The first exercise is made with εr1 = 1 and εr2 = 4 and normal incidence (θ<sup>i</sup> = 0) for perpendicular polarization. Figure 13 depicts some results shown for this case. Check the reflection and transmission coefficients (similar to a transmission line terminated by a resistive load). Verify

Figure 13. EM.m tool screen shot: frame description is given below.

that they are the same for the parallel polarization and also that Γ<sup>⊥</sup> increases monotonically from θ<sup>i</sup> = 0 to 90 up to 1 (text frame (3)).

Observe the incident, the reflected and total fields and the standing waves in medium 1. The minima (black stripes) and the maxima (more hot colors and cold colors) occur always at planes parallel to the reflection plane, in spite of a travel in time being observed (left side of text frames (1): electric field Ey and (3): magnetic field Hx). Also the distance between maxima and minima is a quarter of the wavelength. In medium 2 the wave is traveling to the right and has half of the wavelength of that in medium 1, due to the higher dielectric constant (right side of text frames (1) and (3)). No interference is observed because only one traveling wave exists. The component of the magnetic field (text frame (2)) along z-axis does not exist for this particular geometry as can be confirmed by checking Figure 12.

The diectric constant for medium 2 can be increased to observe a larger standing wave amplitude (on the limit an infinite dielectric constant mimics a perfectly conducting medium). There is no perceived advance in medium 2 and the standing wave is depicted as flashing stripes alternating between a light blue and yellow colors. The physical situation is similar to an ideal ressonant circuit where energy is being transferred between the inductor (magnetic field) and the capacitor (electric field) with no dissipated power (transmitted).

interesting to observe the discontinuity of the electric field, Ez, at the interface between the two mediums (see Figure 15(c)), as it is expected from the boundary conditions. It is worthwhile to

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools

http://dx.doi.org/10.5772/intechopen.74937

75

The fourth exercise (εr1 = 4 and εr2 = 1) uses parallel polarization and illustrates the incidence above the critical angle θic = 30 (Eq. (8)). Start by an angle of incidence a few degrees below θic and check that the angle of transmission is close to 90. Increasing the angle of incidence a few degrees past θic allows to observe the results of Figure 16: the reflection coefficient becomes

The fields inside medium 2 assume a tooth-like shape along the z-axis, the amplitude decreases with the distance z from the reflection plane and the wave travels along x. By increasing angle of incidence, the fields become closer to the reflection plane. That is, there is no power transferred to medium 2 and the wave is contained in medium 1 by a dielectric based "container". The angle θ<sup>t</sup> becomes complex, Eq. (4), and an attenuation constant emerges from the exponentials in

Check that the phenomena occurs for both parallel and perpendicular polarization and only

verify that the Brewster angle does not exist for the perpendicular polarization.

Figure 15. (a) Γ<sup>ǁ</sup> and T<sup>ǁ</sup> (b) magnetic field Hy and (c) electric field Ez.

real and unitary (full reflection) at θic and complex (but unitary) at higher angles.

Figure 16. Incidence above θic: (a) Γ<sup>⊥</sup> and T<sup>⊥</sup> (b) electric field Ey and (c) magnetic field Hz.

Eqs. (13), (14), (19) and (20).

when medium 1 is more "dense" (εr1 < εr2).

The second exercise uses the same configuration but considering θ<sup>i</sup> = 45 and the results are presented in Figure 14. Observe the interference in medium 1: the distances of the maxima and minima are now longer and the interference is more "intense", due to a higher amplitude of the reflected wave. There is a perception of the wave in medium 1 traveling along x and the interference occurring along z. The wave in medium 2 moves away along the direction θt.

The third exercise (εr1 = 1 and εr2 = 4) uses parallel polarization and incidence with the Brewster angle, θ<sup>i</sup> = 63.43 (Eq. (7)) and the results are depicted in Figure 15. The reflection coefficient is null and, as in other cases, increases again up to 1 for tangential incidence after changing sign at the Brewster angle. By observing the total field, check that there is no reflected wave and consequently no interference in medium 1. This means that all the incident power at this polarization and angle of incidence is entirely transmitted to medium 2: an incident circular polarization would give a reflected wave with linear polarization. In addition, it is

Figure 14. Incidence: θ<sup>i</sup> = 45 (a) electric field Ey (b) magnetic field Hz and (c) magnetic field Hx.

Teaching Transmission Line Propagation and Plane Wave Reflection Using Software Tools http://dx.doi.org/10.5772/intechopen.74937 75

Figure 15. (a) Γ<sup>ǁ</sup> and T<sup>ǁ</sup> (b) magnetic field Hy and (c) electric field Ez.

that they are the same for the parallel polarization and also that Γ<sup>⊥</sup> increases monotonically

Observe the incident, the reflected and total fields and the standing waves in medium 1. The minima (black stripes) and the maxima (more hot colors and cold colors) occur always at planes parallel to the reflection plane, in spite of a travel in time being observed (left side of text frames (1): electric field Ey and (3): magnetic field Hx). Also the distance between maxima and minima is a quarter of the wavelength. In medium 2 the wave is traveling to the right and has half of the wavelength of that in medium 1, due to the higher dielectric constant (right side of text frames (1) and (3)). No interference is observed because only one traveling wave exists. The component of the magnetic field (text frame (2)) along z-axis does not exist for this

The diectric constant for medium 2 can be increased to observe a larger standing wave amplitude (on the limit an infinite dielectric constant mimics a perfectly conducting medium). There is no perceived advance in medium 2 and the standing wave is depicted as flashing stripes alternating between a light blue and yellow colors. The physical situation is similar to an ideal ressonant circuit where energy is being transferred between the inductor (magnetic

The second exercise uses the same configuration but considering θ<sup>i</sup> = 45 and the results are presented in Figure 14. Observe the interference in medium 1: the distances of the maxima and minima are now longer and the interference is more "intense", due to a higher amplitude of the reflected wave. There is a perception of the wave in medium 1 traveling along x and the interference occurring along z. The wave in medium 2 moves away along the direction θt.

The third exercise (εr1 = 1 and εr2 = 4) uses parallel polarization and incidence with the Brewster angle, θ<sup>i</sup> = 63.43 (Eq. (7)) and the results are depicted in Figure 15. The reflection coefficient is null and, as in other cases, increases again up to 1 for tangential incidence after changing sign at the Brewster angle. By observing the total field, check that there is no reflected wave and consequently no interference in medium 1. This means that all the incident power at this polarization and angle of incidence is entirely transmitted to medium 2: an incident circular polarization would give a reflected wave with linear polarization. In addition, it is

from θ<sup>i</sup> = 0 to 90 up to 1 (text frame (3)).

74 Antennas and Wave Propagation

particular geometry as can be confirmed by checking Figure 12.

field) and the capacitor (electric field) with no dissipated power (transmitted).

Figure 14. Incidence: θ<sup>i</sup> = 45 (a) electric field Ey (b) magnetic field Hz and (c) magnetic field Hx.

interesting to observe the discontinuity of the electric field, Ez, at the interface between the two mediums (see Figure 15(c)), as it is expected from the boundary conditions. It is worthwhile to verify that the Brewster angle does not exist for the perpendicular polarization.

The fourth exercise (εr1 = 4 and εr2 = 1) uses parallel polarization and illustrates the incidence above the critical angle θic = 30 (Eq. (8)). Start by an angle of incidence a few degrees below θic and check that the angle of transmission is close to 90. Increasing the angle of incidence a few degrees past θic allows to observe the results of Figure 16: the reflection coefficient becomes real and unitary (full reflection) at θic and complex (but unitary) at higher angles.

The fields inside medium 2 assume a tooth-like shape along the z-axis, the amplitude decreases with the distance z from the reflection plane and the wave travels along x. By increasing angle of incidence, the fields become closer to the reflection plane. That is, there is no power transferred to medium 2 and the wave is contained in medium 1 by a dielectric based "container". The angle θ<sup>t</sup> becomes complex, Eq. (4), and an attenuation constant emerges from the exponentials in Eqs. (13), (14), (19) and (20).

Check that the phenomena occurs for both parallel and perpendicular polarization and only when medium 1 is more "dense" (εr1 < εr2).

Figure 16. Incidence above θic: (a) Γ<sup>⊥</sup> and T<sup>⊥</sup> (b) electric field Ey and (c) magnetic field Hz.

## 6. Conclusions

Simulation tools, to explore the basic concepts about transmission line propagation and about reflection and transmission of plane waves, were described. Moreover, some theoretical exercises were given and, in addition, a few laboratory prototypes measurements depicting the theoretical exercises were presented, thus, highlighting the usefulness of the tools, as well as, the need to complement the learning by using simulations with practical laboratory experiences.

**Chapter 5**

Provisional chapter

**Radio Network Planning and Propagation Models for**

DOI: 10.5772/intechopen.75384

Radio Network Planning and Propagation Models for

**Urban and Indoor Wireless Communication Networks**

As the growing demand for mobile communications is constantly increasing, the need for better coverage, improved capacity, and higher transmission quality rises. Thus, a more efficient use of the radio spectrum and communication systems availability are required. This chapter presents EM propagation models most commonly used for the design of wireless communication systems, computer networks WLAN and WPAN for urban and/ or in indoor environments. The review of commercial or University computer codes to assist design of WLAN and WPAN networks were done. An example of computer design and simulation of indoor Bluetooth and WLAN communication systems, in the building of Wroclaw University of Science and Technology, Wroclaw, Poland is shown in Chapter 8. Keywords: EM wave propagation, urban, outdoor, indoor, deterministic model, empirical

model, one-slop, multi-wall, ray tracing, ray lunching, dominant path, mobile

In recent years, a substantial increase in the development of broadband wireless access technologies for evolving wireless Internet services and improved cellular systems has been observed. Because of them, it is widely foreseen that in the future an enormous rise in traffic will be experienced for mobile and personal communications systems. This is due to both an increased number of users and introduction of new high bit rate data services. This trend is observed for thirdgeneration systems, and it will most certainly continue for fourth- and fifth-generation systems.

Wireless communication systems, as opposed to their wireline counterparts, pose some unique

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

communication system, WLAN, Wi-Fi, Bluetooth

Urban and Indoor Wireless Communication Networks

Wojciech Jan Krzysztofik

Wojciech Jan Krzysztofik

Abstract

1. Introduction

challenge:

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75384

## Acknowledgements

The authors are grateful to the Department of Electronics, Telecommunications and Informatics for the opportunity of teaching this subjects and using the RF equipment to perform the laboratory measurements. They are also grateful to the former MSc student, C. Ribeiro, for the intense brainstorming with them and for the implementation of the simulation tools.

## Author details

Susana Mota\* and Armando Rocha

\*Address all correspondence to: smota@ua.pt

Departamento de Eletrónica, Telecomunicações e Informática, Instituto de Telecomunicações, Aveiro, Portugal

## References


#### **Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks** Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

DOI: 10.5772/intechopen.75384

Wojciech Jan Krzysztofik Wojciech Jan Krzysztofik

6. Conclusions

76 Antennas and Wave Propagation

Acknowledgements

Author details

Aveiro, Portugal

References

Susana Mota\* and Armando Rocha

\*Address all correspondence to: smota@ua.pt

Simulation tools, to explore the basic concepts about transmission line propagation and about reflection and transmission of plane waves, were described. Moreover, some theoretical exercises were given and, in addition, a few laboratory prototypes measurements depicting the theoretical exercises were presented, thus, highlighting the usefulness of the tools, as well as, the need to

The authors are grateful to the Department of Electronics, Telecommunications and Informatics for the opportunity of teaching this subjects and using the RF equipment to perform the laboratory measurements. They are also grateful to the former MSc student, C. Ribeiro, for the

Departamento de Eletrónica, Telecomunicações e Informática, Instituto de Telecomunicações,

[1] Rocha A, Mota S, Ribeiro C. Software tools for teaching wave propagation in transmission lines [education corner]. IEEE Antennas and Propagation Magazine. 2017;59(3):118-127 [2] Trintinalia LC. Simulation tool for the visualization of EM wave reflection and refraction.

IEEE Antennas and Propagation Magazine. 2013;55(1):203-211

complement the learning by using simulations with practical laboratory experiences.

intense brainstorming with them and for the implementation of the simulation tools.

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75384

#### Abstract

As the growing demand for mobile communications is constantly increasing, the need for better coverage, improved capacity, and higher transmission quality rises. Thus, a more efficient use of the radio spectrum and communication systems availability are required. This chapter presents EM propagation models most commonly used for the design of wireless communication systems, computer networks WLAN and WPAN for urban and/ or in indoor environments. The review of commercial or University computer codes to assist design of WLAN and WPAN networks were done. An example of computer design and simulation of indoor Bluetooth and WLAN communication systems, in the building of Wroclaw University of Science and Technology, Wroclaw, Poland is shown in Chapter 8.

Keywords: EM wave propagation, urban, outdoor, indoor, deterministic model, empirical model, one-slop, multi-wall, ray tracing, ray lunching, dominant path, mobile communication system, WLAN, Wi-Fi, Bluetooth

## 1. Introduction

In recent years, a substantial increase in the development of broadband wireless access technologies for evolving wireless Internet services and improved cellular systems has been observed. Because of them, it is widely foreseen that in the future an enormous rise in traffic will be experienced for mobile and personal communications systems. This is due to both an increased number of users and introduction of new high bit rate data services. This trend is observed for thirdgeneration systems, and it will most certainly continue for fourth- and fifth-generation systems.

Wireless communication systems, as opposed to their wireline counterparts, pose some unique challenge:

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In addition, cellular wireless communication systems have to cope with interference due to frequency reuse. Research efforts investigating effective technologies to mitigate such effects have been going on for the past years, as wireless communications are experiencing rapid growth.

The rise in traffic will put a demand on both manufacturers and operators to provide sufficient capacity in the networks. This becomes a major challenging problem for the service providers to solve, since there exist certain negative factors in the radiation environment contributing to the limit in capacity.

A major limitation in capacity is co-channel interference caused by the increasing number of users. The other impairments contributing to the reduction of system performance and capacity are multipath fading and delay spread caused by signals being reflected from structures (e.g., buildings and mountains) and users traveling on vehicles (Figure 1).

To aggravate further the capacity problem, in 1990s, the Internet gave the people the tool to get data on-demand (e.g., stock quotes, news, weather reports, e-mails, etc.) and share information in real time. This resulted in an increase in airtime usage and in the number of subscribers, thus saturating the systems' capacity.

Wireless carriers have begun to explore new ways to maximize the spectral efficiency of their networks and improve their return on investment. Research efforts investigating methods of improving wireless systems performance are currently being conducted worldwide.

In the design of wireless networks for companies and institutions, there is no room for chance. Failure to even potentially the smallest factor can cause errors that will make our project become useless. To organize the design process, the concept of splitting the process into three phases is introduced [1]: initial phase—collects information about the requirements and expectations of the client; design phase—identifying the best access point, AP, location by application of simulation based on the EM wave propagation models; and measurement phase —implementation of the project and the introduction of any amendments arising from the difference between the results of measurements (the real behavior of the network), and simulation.

Presented models allow streamlining the design process for wireless networks. On the market,

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

79

Wireless network design requires that you specify the size and shape of the areas covered by

where PRx is the received power (dBm), PTx is the broadcast power (dBm), GTx is the power gain of transmitting antenna (dB), GRx is the power gain of receiving antenna (dB), LTx is transmitting antenna cable attenuation, LRx is receiving antenna cable attenuation, and L is

PRx ¼ PTx þ GTx þ GRx � ð Þ LTx þ LRx þ L (1)

there are a lot of solutions, commercial computer codes, based on these concepts.

Figure 1. Wireless systems impairments in: indoor (a) and outdoor (b) environments.

the access points. To this end, the link budget is performed:

the route of EM wave propagation attenuation (dB).

2. Radio link budget

In the circumstances in which radio waves are many phenomena, the designation of a useful signal level is extremely complex and requires the introduction of appropriate propagation models. Such model is a collection of mathematical expressions, graphs, and algorithms used to produce propagation characteristics of EM waves in chosen environment. Propagation models can be divided into empirical (statistical) such as One-Slope (O-S), Multi-Wall (M-W) —linear and nonlinear, dominant path model (DPM), etc., the deterministic (e.g., Ray Tracing, RT, Intelligent Ray Tracing, IRT, Ray Lunching, RL, etc.). There are also models which are a combination of both of these types.

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 79

Figure 1. Wireless systems impairments in: indoor (a) and outdoor (b) environments.

Presented models allow streamlining the design process for wireless networks. On the market, there are a lot of solutions, commercial computer codes, based on these concepts.

### 2. Radio link budget

a. allocated spectrum limitation results in a limited levels of system capacity;

c. expired battery life at the mobile device poses DC power constraints.

(e.g., buildings and mountains) and users traveling on vehicles (Figure 1).

spreading in time, space, and frequency; and

the limit in capacity.

78 Antennas and Wave Propagation

simulation.

saturating the systems' capacity.

combination of both of these types.

b. radio propagation environment and the mobility of users give rise to signal fading and

In addition, cellular wireless communication systems have to cope with interference due to frequency reuse. Research efforts investigating effective technologies to mitigate such effects have been going on for the past years, as wireless communications are experiencing rapid growth.

The rise in traffic will put a demand on both manufacturers and operators to provide sufficient capacity in the networks. This becomes a major challenging problem for the service providers to solve, since there exist certain negative factors in the radiation environment contributing to

A major limitation in capacity is co-channel interference caused by the increasing number of users. The other impairments contributing to the reduction of system performance and capacity are multipath fading and delay spread caused by signals being reflected from structures

To aggravate further the capacity problem, in 1990s, the Internet gave the people the tool to get data on-demand (e.g., stock quotes, news, weather reports, e-mails, etc.) and share information in real time. This resulted in an increase in airtime usage and in the number of subscribers, thus

Wireless carriers have begun to explore new ways to maximize the spectral efficiency of their networks and improve their return on investment. Research efforts investigating methods of

In the design of wireless networks for companies and institutions, there is no room for chance. Failure to even potentially the smallest factor can cause errors that will make our project become useless. To organize the design process, the concept of splitting the process into three phases is introduced [1]: initial phase—collects information about the requirements and expectations of the client; design phase—identifying the best access point, AP, location by application of simulation based on the EM wave propagation models; and measurement phase —implementation of the project and the introduction of any amendments arising from the difference between the results of measurements (the real behavior of the network), and

In the circumstances in which radio waves are many phenomena, the designation of a useful signal level is extremely complex and requires the introduction of appropriate propagation models. Such model is a collection of mathematical expressions, graphs, and algorithms used to produce propagation characteristics of EM waves in chosen environment. Propagation models can be divided into empirical (statistical) such as One-Slope (O-S), Multi-Wall (M-W) —linear and nonlinear, dominant path model (DPM), etc., the deterministic (e.g., Ray Tracing, RT, Intelligent Ray Tracing, IRT, Ray Lunching, RL, etc.). There are also models which are a

improving wireless systems performance are currently being conducted worldwide.

Wireless network design requires that you specify the size and shape of the areas covered by the access points. To this end, the link budget is performed:

$$P\_{Rx} = P\_{Tx} + G\_{Tx} + G\_{Rx} - (L\_{Tx} + L\_{Rx} + L) \tag{1}$$

where PRx is the received power (dBm), PTx is the broadcast power (dBm), GTx is the power gain of transmitting antenna (dB), GRx is the power gain of receiving antenna (dB), LTx is transmitting antenna cable attenuation, LRx is receiving antenna cable attenuation, and L is the route of EM wave propagation attenuation (dB).

The most difficult to determine the part of the link budget is the attenuation loss L of the propagation route. Typical wireless systems environment is located inside the buildings filled with walls, furniture, peoples, and other objects. In such conditions, the mechanism of propagation of the EM waves is very complex. A number of EM waves distributed inside buildings belong to different physical phenomena [2]:

p xð Þ¼ <sup>1</sup>

deviation of the distribution, and m is the mean value of the signal.

p xð Þ¼ <sup>x</sup>

<sup>σ</sup><sup>2</sup> � <sup>e</sup>

component amplitude, I0 is a modified Bessel function of the first kind of order 0.

In order to facilitate the description of the Rice distribution, the k parameter is used [8]:

p xð Þ¼ <sup>x</sup>

standard deviation of the lognormal distribution with mean value m.

calculation time that empirical models are not lost in popularity [8].

the Rice distribution with a value of σ = 7/14 dB [8]:

distribution [8]:

by about 6 dB from the mean value.

3. Empirical models

<sup>x</sup> � <sup>σ</sup> � ffiffiffiffiffiffiffiffiffi <sup>2</sup> � <sup>π</sup> <sup>p</sup> � <sup>e</sup>

where p(x) is the probability of the appearance of the signal with a value of x, σ is the standard

The quick-exchange dropouts shall be understood as fluctuations in the average value of the signal caused by changes in the propagation environment, for example, the movement of people in the building. These fluctuations at the small number of moving people best describes

> � <sup>x</sup>2þA<sup>2</sup> ð Þ<sup>2</sup> <sup>2</sup>�σ<sup>2</sup> � I<sup>0</sup>

where p(x) is the probability of the appearance of the signal with a value of x, σ is the standard deviation of the lognormal distribution with mean value m, and A is the strongest signal

<sup>k</sup> <sup>¼</sup> 10log <sup>A</sup><sup>2</sup>

In the case of a large movement of people in buildings, it is advisable to use the Rayleigh

where p(x) is the probability of the appearance of the signal with a value of x and σ is the

Analysis of the introduced distributions has allowed determining that the value of the signal exceeded at a given point in the 90% of the time in the conditions inside the building is smaller

Empirical models are based on measurements and observations made under different conditions. Their accuracy depends not only on the results of measurements but also from the similarity of the present environment and the environment in which the modeling measurements were conducted. The propagation environments inside buildings are strongly differentiated. The use of empirical models in such conditions can lead to less accurate results relative to other promotional environments. Therefore, for indoor environments, more accurate models are considered to be deterministic models. Despite this, the easy implementation and fast

<sup>σ</sup><sup>2</sup> � <sup>e</sup> �x<sup>2</sup> <sup>2</sup>�σ<sup>2</sup> � �

�ð Þ logx�log<sup>m</sup> <sup>2</sup>

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

A � x σ2

<sup>2</sup>�σ<sup>2</sup> (2)

http://dx.doi.org/10.5772/intechopen.75384

� � (3)

<sup>2</sup> � <sup>σ</sup><sup>2</sup> (4)

(5)

81


The designation of the attenuation of a route in such conditions is extremely difficult. Having regard to these mechanisms of propagation, the propagation models have been developed. The propagation model is an algorithm to analyze the propagation of radio waves in the environment taking into account the mechanisms described above. The algorithm described it in the proper order by means of the specified mathematical expressions, charts, and tables of some coefficients, and they are most frequently served in the form of recommendations of the ITU, IEEE and others worldwide standardizing institutions.

The propagation models permit to determine the average value of the propagation loss in a proper place. For the complete modeling of the propagation environment, the statistics of the received signal should be provided, which lets you include slowly variable and quickexchange of signal dropouts. The slowly variable dropouts are understood as fluctuations in the average value of a signal over a distance of several wavelengths. Studies have shown that the lognormal distribution with standard deviation of values σ = 2.7/5.3 dB is the best model of the phenomenon of slow signal fade [8]:

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 81

$$p(\mathbf{x}) = \frac{1}{\mathbf{x} \cdot \sigma \cdot \sqrt{2 \cdot \pi}} \cdot \overline{\mathcal{e}}^{\frac{-(\log x - \log m)^2}{2\sigma^2}} \tag{2}$$

where p(x) is the probability of the appearance of the signal with a value of x, σ is the standard deviation of the distribution, and m is the mean value of the signal.

The quick-exchange dropouts shall be understood as fluctuations in the average value of the signal caused by changes in the propagation environment, for example, the movement of people in the building. These fluctuations at the small number of moving people best describes the Rice distribution with a value of σ = 7/14 dB [8]:

$$p(\mathbf{x}) = \frac{\mathbf{x}}{\sigma^2} \cdot e^{-\frac{(\mathbf{x}^2 + A^2)^2}{2\sigma^2}} \cdot I\_0\left(\frac{A \cdot \mathbf{x}}{\sigma^2}\right) \tag{3}$$

where p(x) is the probability of the appearance of the signal with a value of x, σ is the standard deviation of the lognormal distribution with mean value m, and A is the strongest signal component amplitude, I0 is a modified Bessel function of the first kind of order 0.

In order to facilitate the description of the Rice distribution, the k parameter is used [8]:

$$k = 10 \log \frac{A^2}{2 \cdot \sigma^2} \tag{4}$$

In the case of a large movement of people in buildings, it is advisable to use the Rayleigh distribution [8]:

$$p(\mathbf{x}) = \frac{\mathbf{x}}{\sigma^2} \cdot e^{\left(\frac{\mathbf{x}^2}{2\sigma^2}\right)} \tag{5}$$

where p(x) is the probability of the appearance of the signal with a value of x and σ is the standard deviation of the lognormal distribution with mean value m.

Analysis of the introduced distributions has allowed determining that the value of the signal exceeded at a given point in the 90% of the time in the conditions inside the building is smaller by about 6 dB from the mean value.

#### 3. Empirical models

The most difficult to determine the part of the link budget is the attenuation loss L of the propagation route. Typical wireless systems environment is located inside the buildings filled with walls, furniture, peoples, and other objects. In such conditions, the mechanism of propagation of the EM waves is very complex. A number of EM waves distributed inside buildings

• Diffraction: when signal encounters on the road, an impermeable barrier, whose dimensions are larger than the wavelength. At the edges of the obstacles is the deflection of the wave causing the attenuation, dispersion, and a change in the direction of EM wave

• Dispersion: when on the road the wave contains obstacles, whose dimensions are comparable to the wavelength. In this case, the radio waves are directed in more directions,

• Reflection: when radio wave on the way encounters an obstacle, whose dimensions are much larger than the wavelength of the incident EM wave, they reflect itself from the obstacle. In cases when at the receiving end, there are many of reflected waves signal can

• Penetration over obstacles: when radio wave encounters an obstacle, which is to some extent transparent for radio waves, allowing the reception of radio signals inside build-

• Absorption: caused by the appearance of the plants on the propagation route, peoples with high absorption capacity. Radio waves are absorbed also by other obstacles, such as

• Running along the tunnels and corridors: guided wave phenomenon can be dealt with as a special mechanism to describe the propagation in tunnels or corridors, arising as a result

The designation of the attenuation of a route in such conditions is extremely difficult. Having regard to these mechanisms of propagation, the propagation models have been developed. The propagation model is an algorithm to analyze the propagation of radio waves in the environment taking into account the mechanisms described above. The algorithm described it in the proper order by means of the specified mathematical expressions, charts, and tables of some coefficients, and they are most frequently served in the form of recommendations of the

The propagation models permit to determine the average value of the propagation loss in a proper place. For the complete modeling of the propagation environment, the statistics of the received signal should be provided, which lets you include slowly variable and quickexchange of signal dropouts. The slowly variable dropouts are understood as fluctuations in the average value of a signal over a distance of several wavelengths. Studies have shown that the lognormal distribution with standard deviation of values σ = 2.7/5.3 dB is the best model of

walls, furniture, and painting of the walls, curtains on the windows, etc.,

of multiple reflections and interference of the EM waves along the route.

belong to different physical phenomena [2]:

which is difficult to predict and model,

ings (waves penetrate through walls and ceilings),

ITU, IEEE and others worldwide standardizing institutions.

the phenomenon of slow signal fade [8]:

propagation,

80 Antennas and Wave Propagation

be very unstable,

Empirical models are based on measurements and observations made under different conditions. Their accuracy depends not only on the results of measurements but also from the similarity of the present environment and the environment in which the modeling measurements were conducted. The propagation environments inside buildings are strongly differentiated. The use of empirical models in such conditions can lead to less accurate results relative to other promotional environments. Therefore, for indoor environments, more accurate models are considered to be deterministic models. Despite this, the easy implementation and fast calculation time that empirical models are not lost in popularity [8].

#### 3.1. The one-slope model

The one-slope model is the simplest model used in the indoor environment. It does not take into account the details of the structure of the building but only the distance and n parameter describing an environment [8]:

$$L = L\_0 + 10 \cdot n \cdot \log(d) \tag{6}$$

<sup>L</sup> <sup>¼</sup> <sup>L</sup><sup>0</sup> <sup>þ</sup> <sup>10</sup> � <sup>n</sup> � logð Þþ <sup>d</sup> <sup>X</sup>

<sup>L</sup> <sup>¼</sup> LFS <sup>þ</sup> <sup>L</sup><sup>0</sup> <sup>þ</sup><sup>X</sup>

bricks, perforated in the form of holes in their design), and b = 0.46.

I

i¼1

n = 2.

tions [4]:

receiver (km).

walls (c).

attenuation of the i-th walls- kind, and Lfj is the attenuation of j-th ceilings value.

I

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

Nwi � Lwi <sup>þ</sup><sup>X</sup>

J

j¼1

Nfj � Lfj (7)

83

http://dx.doi.org/10.5772/intechopen.75384

� Lf (8)

i¼1

where L0 is the attenuation at distance of 1 m, n is the index that depends on the adopted model, d is the distance between the transmitter and receiver, Nwi is the number of walls of equal attenuation value, Nfj is the number of floors with equal attenuation values, Lwi is the

Based on measurements, the parameter values are designated as [8]: L0 = 37 dB, Lf = 20 dB (for typical slabs thickness < 30 cm), Lw = 3.5 dB (for plaster walls), and 7 dB (for concrete walls),

Conditions of the propagation inside the building more accurately reflects, so called, nonlinear model of multi-walls, which was developed within the framework of the European project COST 231 and approved by the ITU as recommended for third generation of cellular mobile systems projects. Attenuation of radio link referred to the model is given by following equa-

Nwi � Lwi þ Nf

where LFS is the free space loss, L0 is the attenuation at distance of 1 m, Nwi is the number of itype walls on the propagation route, Nf is the number of floors on the propagation route, Lwi is the attenuation of the i-type walls, Lf is the slab attenuation, b is the empirical parameter, f is the channel center frequency (MHz), and d is the distance between the transmitter and the

Some of model parameters were determined empirically [4]: L0 = 37 dB, Lf = 18.3 dB (for ceilings constructed of reinforced concrete with a thickness < 30 cm), Lw1 = 3.4 dB (for plaster walls with lots of window openings), Lw2 = 6.9 dB (for concrete walls and constructed from

The M-W models are used in computer programs (commercial and academic). Examples of computer codes are: WinProp (AWE Communication), ACTIX Analyzer, and I-Prop (University of Prague). Figure 2 shows a comparison of the simulation results obtained in the building type 1 by applying the model of one-slope (b) and nonlinear model of multi-

Simulations were made using the demo version of the I-Prop program for the transmitter with a power of 10 dBm and at a frequency of 2.45 GHz [3]. The M-W model in indoor environments takes into account the impact of the walls on the suppression of radio wave but only on the direct of transmission transmitter to receiver route. As mentioned earlier, the specificity of phenomena in indoor environments causes if a direct signal is often not the

Nf þ2 Nf <sup>þ</sup>1�<sup>b</sup> � �

LFS <sup>¼</sup> <sup>32</sup>, <sup>4</sup> <sup>þ</sup> 20log <sup>f</sup> ½ � MHz � � <sup>þ</sup> 20log <sup>d</sup>½ � km � � (9)

where L0 is the attenuation at a distance of 1 m, n is the coefficient of the power distribution, and d is the distance between the transmitter and the receiver.

The parameters of the L0 and n are empirical parameters assigned to the specified environment. By appropriate selection of their values, we can adjust the model to a specific type of building, for which we design a wireless network. Examples of model values for different propagation environments are shown in Table 1.

Building 1 represents buildings introducing the large attenuation (with high density of users, furniture, walls, and other obstacles). Building 2 represents the typical office buildings of medium attenuation. Building 3 represents the big empty spaces of exhibition halls, warehouses, and large offices with a small amount of furniture and other obstacles of the generally small damping. Parameter n in the case of propagation in the corridor is only 1, 2—what is apparent from the account of the phenomenon of driving in the tunnel (waveguide), which amplifies the signal. Due to the small accuracy of the model O-S has been used in a small number of computer tool applications: I-PROP (Technical University of Prague) and WinProp (AWE).


Table 1. Sample parameters of the one-slope model.

#### 3.2. The multi-wall model

The most popular models of empirical studies to take into account the effect of the walls and ceilings are multi-wall models. Linear multi-wall model (also called the Motley-Keenan model) specifies the attenuation on the direct route between transmitter and receiver, taking into account the attenuation by walls and ceilings of the building [8]:

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 83

$$L = L\_0 + 10 \cdot n \cdot \log(d) + \sum\_{i=1}^{l} N\_{wi} \cdot L\_{wi} + \sum\_{j=1}^{l} N\_{\tilde{f}\tilde{}} \cdot L\_{\tilde{f}} \tag{7}$$

where L0 is the attenuation at distance of 1 m, n is the index that depends on the adopted model, d is the distance between the transmitter and receiver, Nwi is the number of walls of equal attenuation value, Nfj is the number of floors with equal attenuation values, Lwi is the attenuation of the i-th walls- kind, and Lfj is the attenuation of j-th ceilings value.

3.1. The one-slope model

82 Antennas and Wave Propagation

describing an environment [8]:

3.2. The multi-wall model

Table 1. Sample parameters of the one-slope model.

The one-slope model is the simplest model used in the indoor environment. It does not take into account the details of the structure of the building but only the distance and n parameter

where L0 is the attenuation at a distance of 1 m, n is the coefficient of the power distribution,

The parameters of the L0 and n are empirical parameters assigned to the specified environment. By appropriate selection of their values, we can adjust the model to a specific type of building, for which we design a wireless network. Examples of model values for different

Building 1 represents buildings introducing the large attenuation (with high density of users, furniture, walls, and other obstacles). Building 2 represents the typical office buildings of medium attenuation. Building 3 represents the big empty spaces of exhibition halls, warehouses, and large offices with a small amount of furniture and other obstacles of the generally small damping. Parameter n in the case of propagation in the corridor is only 1, 2—what is apparent from the account of the phenomenon of driving in the tunnel (waveguide), which amplifies the signal. Due to the small accuracy of the model O-S has been used in a small number of computer

The most popular models of empirical studies to take into account the effect of the walls and ceilings are multi-wall models. Linear multi-wall model (also called the Motley-Keenan model) specifies the attenuation on the direct route between transmitter and receiver, taking into

2.45 40 1.2 Corridor in the building

5.25 46.8 1.2 Corridor in the building

account the attenuation by walls and ceilings of the building [8]:

tool applications: I-PROP (Technical University of Prague) and WinProp (AWE).

f, GHz L0, dB n Application 2.45 40 3.5 Building 1 2.45 40 4.5 Building 2 2.45 40 2 Building 3

5.25 46.8 3.5 Building 1 5.25 46.8 4.5 Building 2 5.25 46.8 2 Building 3

and d is the distance between the transmitter and the receiver.

propagation environments are shown in Table 1.

L ¼ L<sup>0</sup> þ 10 � n � logð Þd (6)

Based on measurements, the parameter values are designated as [8]: L0 = 37 dB, Lf = 20 dB (for typical slabs thickness < 30 cm), Lw = 3.5 dB (for plaster walls), and 7 dB (for concrete walls), n = 2.

Conditions of the propagation inside the building more accurately reflects, so called, nonlinear model of multi-walls, which was developed within the framework of the European project COST 231 and approved by the ITU as recommended for third generation of cellular mobile systems projects. Attenuation of radio link referred to the model is given by following equations [4]:

$$L = L\_{\rm FS} + L\_0 + \sum\_{i=1}^{I} \mathcal{N}\_{wi} \cdot L\_{wi} + \mathcal{N}\_f \binom{\binom{N\_f+2}{N\_f+1}}{} \cdot L\_f \tag{8}$$

$$L\_{FS} = 32\,4 + 20\log\left(f\_{\left[MHz\right]}\right) + 20\log\left(d\_{\left[km\right]}\right) \tag{9}$$

where LFS is the free space loss, L0 is the attenuation at distance of 1 m, Nwi is the number of itype walls on the propagation route, Nf is the number of floors on the propagation route, Lwi is the attenuation of the i-type walls, Lf is the slab attenuation, b is the empirical parameter, f is the channel center frequency (MHz), and d is the distance between the transmitter and the receiver (km).

Some of model parameters were determined empirically [4]: L0 = 37 dB, Lf = 18.3 dB (for ceilings constructed of reinforced concrete with a thickness < 30 cm), Lw1 = 3.4 dB (for plaster walls with lots of window openings), Lw2 = 6.9 dB (for concrete walls and constructed from bricks, perforated in the form of holes in their design), and b = 0.46.

The M-W models are used in computer programs (commercial and academic). Examples of computer codes are: WinProp (AWE Communication), ACTIX Analyzer, and I-Prop (University of Prague). Figure 2 shows a comparison of the simulation results obtained in the building type 1 by applying the model of one-slope (b) and nonlinear model of multiwalls (c).

Simulations were made using the demo version of the I-Prop program for the transmitter with a power of 10 dBm and at a frequency of 2.45 GHz [3]. The M-W model in indoor environments takes into account the impact of the walls on the suppression of radio wave but only on the direct of transmission transmitter to receiver route. As mentioned earlier, the specificity of phenomena in indoor environments causes if a direct signal is often not the

The losses on selected routes shall be, according to [5]:

Figure 3. Routes of the wave propagation in dominant path model [5].

the transition of the wave by j-wall (or ceiling).

continuity of the phenomenon of the conduct of the wave.

<sup>L</sup> <sup>¼</sup> <sup>20</sup> � <sup>n</sup> � logð Þþ <sup>d</sup> <sup>X</sup>

I

i¼1

where n is the factor depending on propagation conditions for direct radius, d is the length of the propagation route, W is the factor of the conduct of wave which grows for the smaller distance between the faces on the propagation route, f (φi,i) is the function determining the loss depending on the number of phenomena, i specifies the following phenomena, φ<sup>i</sup> is the angle between the new and the previous direction of propagation, and Lj is the attenuation caused by

The D-P model allows you to take into account the situation, when the dominant radius is not the dominant one, it means the attenuation of EM waves is not lowest one. So, on the beginning the attenuations for some few routes are determined, for which are expected to be the smallest. On further analysis, only those routes have been taken into account. It is important that the exact designation of the losses is associated with some encountered phenomena (reflection and diffraction) and strengthening related to the phenomenon of the driving waves in the corridors (the waveguide effect). The designation of a wave driving factor W and suppression caused by diffraction and reflection can greatly vary depending on the type of building. For each environment before the calculations, it is recommended that measurements should be arranged, the results of which will allow for the designation of a function f (φi, i) and parameter W. Parameter W that specifies the contribution to the phenomenon of the conduct of the wave in the D-P model depends on: the width of the corridor/tunnel, the material from which the walls are built, the walls' orientation relative to the designated path, and the

f φ<sup>i</sup>

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

; <sup>i</sup> � � <sup>þ</sup><sup>X</sup>

J

j¼1

Lj � W (10)

http://dx.doi.org/10.5772/intechopen.75384

85

Figure 2. The projection of floors and location of the access point (a), the distribution of the signal for one-slope model (b), and for nonlinear model of multi-walls (c) [3].

strongest one. In that cases in the received signal the reflected as well as the diffraction rays should also be considered [4].

#### 3.3. Dominant path model

In the dominant path model (D-P), the losses shall be determined for several wave propagation routes. As shows experience, in most cases, only two or three rays carry more than 95% of the signal energy. Analysis of the results of measurements also showed that the adjacent rays decompose the same phenomena and are almost identical. This fact was used in the D-P model, which is looking for the routes with the smallest attenuations. In Figure 3, the concept of the D-P model is shown.

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 85

Figure 3. Routes of the wave propagation in dominant path model [5].

The losses on selected routes shall be, according to [5]:

strongest one. In that cases in the received signal the reflected as well as the diffraction rays

Figure 2. The projection of floors and location of the access point (a), the distribution of the signal for one-slope model (b),

In the dominant path model (D-P), the losses shall be determined for several wave propagation routes. As shows experience, in most cases, only two or three rays carry more than 95% of the signal energy. Analysis of the results of measurements also showed that the adjacent rays decompose the same phenomena and are almost identical. This fact was used in the D-P model, which is looking for the routes with the smallest attenuations. In Figure 3, the concept

should also be considered [4].

and for nonlinear model of multi-walls (c) [3].

3.3. Dominant path model

84 Antennas and Wave Propagation

of the D-P model is shown.

$$L = 20 \cdot n \cdot \log(d) + \sum\_{i=1}^{l} f(\varphi\_i, \mathbf{i}) + \sum\_{j=1}^{l} L\_j - W \tag{10}$$

where n is the factor depending on propagation conditions for direct radius, d is the length of the propagation route, W is the factor of the conduct of wave which grows for the smaller distance between the faces on the propagation route, f (φi,i) is the function determining the loss depending on the number of phenomena, i specifies the following phenomena, φ<sup>i</sup> is the angle between the new and the previous direction of propagation, and Lj is the attenuation caused by the transition of the wave by j-wall (or ceiling).

The D-P model allows you to take into account the situation, when the dominant radius is not the dominant one, it means the attenuation of EM waves is not lowest one. So, on the beginning the attenuations for some few routes are determined, for which are expected to be the smallest. On further analysis, only those routes have been taken into account. It is important that the exact designation of the losses is associated with some encountered phenomena (reflection and diffraction) and strengthening related to the phenomenon of the driving waves in the corridors (the waveguide effect). The designation of a wave driving factor W and suppression caused by diffraction and reflection can greatly vary depending on the type of building. For each environment before the calculations, it is recommended that measurements should be arranged, the results of which will allow for the designation of a function f (φi, i) and parameter W. Parameter W that specifies the contribution to the phenomenon of the conduct of the wave in the D-P model depends on: the width of the corridor/tunnel, the material from which the walls are built, the walls' orientation relative to the designated path, and the continuity of the phenomenon of the conduct of the wave.

Determination of the coefficient W starts the analysis of three parameters for each of the walls and at the point of x. The factor wi(x) that represents the interim impact of the i th wall on the conduct of the wave at the point x, can be appointed from [5]:

$$
\Delta w\_i(\mathbf{x}) = A\_i \cdot D\_i(\mathbf{x}) \cdot L\_i \tag{11}
$$

$$A\_i = \begin{cases} 1 - \left(\wp\_i/45^{\circ}\right) & 0 \le \wp \le 45^{\circ} \\ 0 & \wp > 45^{\circ} \end{cases} \tag{12}$$

$$D\_i(\mathbf{x}) = \begin{cases} 1 - (d\_i(\mathbf{x})/3m) & 0 \le d\_i \le 3m \\ 0 & d\_i > 3m \end{cases} \tag{13}$$

$$L\_i = \begin{cases} 1 - (L\_{Ri}/10dB) & 0 \le L\_{Ri} \le 10dB \\ 0 & L\_{Ri} > 10dB \end{cases} \tag{14}$$

where di is the distance between the wall and the path of propagation; LRi are losses resulting from reflection, depending on the material of the wall is built; and φ<sup>i</sup> is the angle between the wall and the path of propagation.

The factor w(x) that represents the partial effects of all N walls in section x specifies the pattern [5]:

$$w(\mathbf{x}) = \frac{1}{w\_0} \min \left( w\_0, \sum\_{i=1}^{N} w\_i(\mathbf{x}) \right) \tag{15}$$

The factor is normalized relative to the w0—maximum value of the conducted wave. Resultant factor wave driving is the superposition of all partial coefficients in the way of propagation [5]:

$$\mathcal{W} = \frac{1}{\mathbf{x}\_{\mathcal{R}} - \mathbf{x}\_{T}} \cdot \int\_{\mathbf{x} = \mathbf{x}\_{T}}^{\mathbf{x}\_{\mathcal{R}}} w(\mathbf{x}) d\mathbf{x} \tag{16}$$

Both results were obtained by means of ACTIX Analyzer for the transmitter with a power of 10 dBm at frequency of 2.45 GHz [6]. In this example, the direct radius is weaker than the strongest radius of about 5.66 dB, or almost 3.7 times. This result proves the superiority of the D-P model over the M-W model. Unfortunately, the very complex initial calculations make D-P

Figure 4. Simulation results obtained using: only the direct radius (nonlinear of M-W model) (a), of the strongest ray (the

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

87

model difficult to implement.

dominant path D-P model) (b), and ray tracing, 3D RT, approach (c).

where xR is the receiver position and xT-the position of the transmitter.

The value of the coefficient W can range between a value of 0 (the conducting of EM wave phenomenon does not exist) and 1 (the ideal case of the conducting EM wave phenomenon).

The second essential and difficult to model parameter of the D-P model is the attenuation caused by the changing direction of propagation of the angle φi. This parameter is determined as a factor W of the wave driving.

Besides, you should also know the suppression resulting from the transition of the wave by an obstacle and the distance between the transmitter and the receiver. Based on the difficulties in the designation f (φi,i) and W in the result, this model is used in a small number of computer codes (e.g., WinProp) [5].

Figure 4 shows a comparison of the simulation results obtained from: (a) only the direct radius (nonlinear of M-W model, (b) the strongest ray (the dominant path D-P model), and (c) the ray tracing, 3D RT, approach (c).

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 87

Determination of the coefficient W starts the analysis of three parameters for each of the walls

=45 � ��

1 � ð Þ dið Þx =3m

where di is the distance between the wall and the path of propagation; LRi are losses resulting from reflection, depending on the material of the wall is built; and φ<sup>i</sup> is the angle between the

The factor w(x) that represents the partial effects of all N walls in section x specifies the pattern [5]:

min w0;

The factor is normalized relative to the w0—maximum value of the conducted wave. Resultant factor wave driving is the superposition of all partial coefficients in the way of propagation [5]:

The value of the coefficient W can range between a value of 0 (the conducting of EM wave phenomenon does not exist) and 1 (the ideal case of the conducting EM wave phenomenon). The second essential and difficult to model parameter of the D-P model is the attenuation caused by the changing direction of propagation of the angle φi. This parameter is determined

Besides, you should also know the suppression resulting from the transition of the wave by an obstacle and the distance between the transmitter and the receiver. Based on the difficulties in the designation f (φi,i) and W in the result, this model is used in a small number of computer

Figure 4 shows a comparison of the simulation results obtained from: (a) only the direct radius (nonlinear of M-W model, (b) the strongest ray (the dominant path D-P model), and (c) the ray

x¼xT

X N

wið Þx !

w xð Þdx (16)

i¼1

wið Þ¼ x Ai � Dið Þ� x Li (11)

0 ≤ φ≤ 45� φ > 45�

> 0 ≤ di ≤ 3m di > 3m

0 ≤ LRi ≤ 10dB LRi > 10dB

th wall on the

(12)

(13)

(14)

(15)

and at the point of x. The factor wi(x) that represents the interim impact of the i

Ai <sup>¼</sup> <sup>1</sup> � <sup>φ</sup><sup>i</sup>

(

(

0

0

Li <sup>¼</sup> <sup>1</sup> � ð Þ LRi=10dB 0

> w xð Þ¼ <sup>1</sup> w0

> > <sup>W</sup> <sup>¼</sup> <sup>1</sup> xR � xT � xðR

where xR is the receiver position and xT-the position of the transmitter.

conduct of the wave at the point x, can be appointed from [5]:

Dið Þ¼ x

wall and the path of propagation.

86 Antennas and Wave Propagation

as a factor W of the wave driving.

codes (e.g., WinProp) [5].

tracing, 3D RT, approach (c).

(

Figure 4. Simulation results obtained using: only the direct radius (nonlinear of M-W model) (a), of the strongest ray (the dominant path D-P model) (b), and ray tracing, 3D RT, approach (c).

Both results were obtained by means of ACTIX Analyzer for the transmitter with a power of 10 dBm at frequency of 2.45 GHz [6]. In this example, the direct radius is weaker than the strongest radius of about 5.66 dB, or almost 3.7 times. This result proves the superiority of the D-P model over the M-W model. Unfortunately, the very complex initial calculations make D-P model difficult to implement.

## 4. Deterministic models

Deterministic models are based on physics laws and allow for precise modeling of the propagation of electromagnetic waves. These models take into account the phenomena such as reflection, diffraction, absorption, and wave running, which are essential in the conditions inside buildings as well as in the outdoor area of dense building centers. Unlike the empirical models, the deterministic models are not based on measurements and thus it accuracy does not depend on the similarity of the standard environment and concerned one. To accurately model the phenomenon of propagation of electromagnetic waves, deterministic models require accurate rendering of environment propagation periods. In addition to the level of the model of the environment and on the quality of results, calculation has the effect of repeat accuracy phenomena, which are subject to the electromagnetic waves. The EM waves propagation environments inside buildings are strongly differentiated. Therefore, for these environments, deterministic models are considered to be more accurate in comparison with models of empirical studies.

If it encounters obstacles, the wave is: reflected, refracted, diffracted, or dispersed. At the beginning, all possible routes between the transmitter and the receiver are determined. In several recent years, the largest popularity won two techniques to determine the possible paths of propagation: method of images (image method, IM) and method of the rays shooting (ray lunching, RL). These methods allow you to recreate the three-dimensional spread of the waves by taking into account of losses on the transmitter-receiver road. The accuracy of the two methods depends largely on the number of phenomena that are included in the models. The more the phenomena will be included, we can get more accu-

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

89

The image method IM lets you designate all possible routes of the signal propagation between the transmitter and the receiver (the center of the pixel) with regard to the phenomena of reflection and transmission. To find a ray route, the mirrors of transmitter relative to the illuminated surrounding surfaces are created. The intersection of the straight line connecting the virtual source and receiving point of the illuminated plane designates the place of reflec-

In the case of multiple reflections, the following mirrors of virtual source are generated and the algorithm is repeated. The analysis can be carried out in three dimensions, or separately in the vertical and horizontal planes. Method of images allows you to designate the exact routes of the reflected rays. The direction of the rays invading to the obstacle shall be determined according to the Snell's law. Method of images can be used to find the routes

In the RL method, the created rays are shooting directly from the source. The azimuth (eleva-

For the 3D model at 1�, discretization of both shooting angles more than 32,000 rays are generated. The route of each of the ray is tracked independently of the other. For each of ray

In the simplest solution, an excerpt of the sphere is approximated by a circle. Then, the circle creates the so-called received sphere (Figure 6a), in which the ray for 2D model is expressed by

RS <sup>¼</sup> <sup>γ</sup> � <sup>d</sup>

RS <sup>¼</sup> <sup>γ</sup> � <sup>d</sup> ffiffiffi 3

where γ is the angle between the directions of launching rays and d is the distance the center of

This method introduces errors of interpretation arising from the limited geometrical possibility of approximation of a sphere by concentric circles. In order to increase the accuracy, portions of a sphere are approximated by the rectangles (Figure 6). The sizes of the υ<sup>i</sup> and ψ<sup>i</sup> determine the coordinates of the launched ray, and Δψ<sup>i</sup> and Δυ<sup>i</sup> are the sides of the rectangle and are described

<sup>2</sup> (17)

p (18)

tion) angle of rays increases gradually at constant values (Figure 5b).

is allocated a part of the sphere forming the wave front (Figure 6b).

rate results.

tion of the rays (Figure 5a).

of diffracted rays [8].

(17), and for the 3D model by (18) [8]:

the sphere to the signal source.

by [8]:

In order to wave propagation modeling, two concepts were developed. The first is a technique for FDTD (finite-difference time-domain) and the other is a technique GO (geometrical optics) also known under the name of ROM (ray optical method).

#### 4.1. FDTD method

The FDTD method is to solve the boundary conditions of the Maxwell's equations for the analyzed area by means of frequency differences method in the time domain. The method requires a transition from continuous space and time distribution of electromagnetic field to a discrete spatially grid that contains in its nodes of the field values in a certain moment of time. The transition from derivatives for odds ratios differential allows you to create an algorithm to calculate the distribution of the field in the next time step of the fields at the time of the previous one. The distribution of the structure given by the material constants is specified in each of the nodes of the grid. This algorithm has been provided by Kane Yee. In accordance with the algorithm EM field value in the node of the grid depends only on the values of the EM field at this point in the previous time step, as well as of the values of other EM fields in the adjacent nodes of the grid, and of the known features of magnetic and electrical sources. FDTD simulation results are more precise relative to GO. It requires a large amount of CPU memory and of the computer microprocessor power for fast calculation of the large Maxwell's equations. For this reason, it is rarely used in computer programs that support design of the wireless networks. An example of such a program is the Wireless InSite of the Remcom Company. The FDTD method is limited in suburban environments, in which the structures are situated close to antennas with complex material properties [7, 23–27].

#### 4.2. Method of geometrical optics GO

Deterministic models are based primarily on methods of geometrical optics (ray optical method (RO), or geometrical optics (GO)), which are based on the assumption of rectilinear propagation of electromagnetic waves. Each of the rays carries the part of the radiated power. If it encounters obstacles, the wave is: reflected, refracted, diffracted, or dispersed. At the beginning, all possible routes between the transmitter and the receiver are determined. In several recent years, the largest popularity won two techniques to determine the possible paths of propagation: method of images (image method, IM) and method of the rays shooting (ray lunching, RL). These methods allow you to recreate the three-dimensional spread of the waves by taking into account of losses on the transmitter-receiver road. The accuracy of the two methods depends largely on the number of phenomena that are included in the models. The more the phenomena will be included, we can get more accurate results.

4. Deterministic models

88 Antennas and Wave Propagation

empirical studies.

4.1. FDTD method

[7, 23–27].

4.2. Method of geometrical optics GO

Deterministic models are based on physics laws and allow for precise modeling of the propagation of electromagnetic waves. These models take into account the phenomena such as reflection, diffraction, absorption, and wave running, which are essential in the conditions inside buildings as well as in the outdoor area of dense building centers. Unlike the empirical models, the deterministic models are not based on measurements and thus it accuracy does not depend on the similarity of the standard environment and concerned one. To accurately model the phenomenon of propagation of electromagnetic waves, deterministic models require accurate rendering of environment propagation periods. In addition to the level of the model of the environment and on the quality of results, calculation has the effect of repeat accuracy phenomena, which are subject to the electromagnetic waves. The EM waves propagation environments inside buildings are strongly differentiated. Therefore, for these environments, deterministic models are considered to be more accurate in comparison with models of

In order to wave propagation modeling, two concepts were developed. The first is a technique for FDTD (finite-difference time-domain) and the other is a technique GO (geometrical optics)

The FDTD method is to solve the boundary conditions of the Maxwell's equations for the analyzed area by means of frequency differences method in the time domain. The method requires a transition from continuous space and time distribution of electromagnetic field to a discrete spatially grid that contains in its nodes of the field values in a certain moment of time. The transition from derivatives for odds ratios differential allows you to create an algorithm to calculate the distribution of the field in the next time step of the fields at the time of the previous one. The distribution of the structure given by the material constants is specified in each of the nodes of the grid. This algorithm has been provided by Kane Yee. In accordance with the algorithm EM field value in the node of the grid depends only on the values of the EM field at this point in the previous time step, as well as of the values of other EM fields in the adjacent nodes of the grid, and of the known features of magnetic and electrical sources. FDTD simulation results are more precise relative to GO. It requires a large amount of CPU memory and of the computer microprocessor power for fast calculation of the large Maxwell's equations. For this reason, it is rarely used in computer programs that support design of the wireless networks. An example of such a program is the Wireless InSite of the Remcom Company. The FDTD method is limited in suburban environments, in which the structures are situated close to antennas with complex material properties

Deterministic models are based primarily on methods of geometrical optics (ray optical method (RO), or geometrical optics (GO)), which are based on the assumption of rectilinear propagation of electromagnetic waves. Each of the rays carries the part of the radiated power.

also known under the name of ROM (ray optical method).

The image method IM lets you designate all possible routes of the signal propagation between the transmitter and the receiver (the center of the pixel) with regard to the phenomena of reflection and transmission. To find a ray route, the mirrors of transmitter relative to the illuminated surrounding surfaces are created. The intersection of the straight line connecting the virtual source and receiving point of the illuminated plane designates the place of reflection of the rays (Figure 5a).

In the case of multiple reflections, the following mirrors of virtual source are generated and the algorithm is repeated. The analysis can be carried out in three dimensions, or separately in the vertical and horizontal planes. Method of images allows you to designate the exact routes of the reflected rays. The direction of the rays invading to the obstacle shall be determined according to the Snell's law. Method of images can be used to find the routes of diffracted rays [8].

In the RL method, the created rays are shooting directly from the source. The azimuth (elevation) angle of rays increases gradually at constant values (Figure 5b).

For the 3D model at 1�, discretization of both shooting angles more than 32,000 rays are generated. The route of each of the ray is tracked independently of the other. For each of ray is allocated a part of the sphere forming the wave front (Figure 6b).

In the simplest solution, an excerpt of the sphere is approximated by a circle. Then, the circle creates the so-called received sphere (Figure 6a), in which the ray for 2D model is expressed by (17), and for the 3D model by (18) [8]:

$$R\_S = \frac{\mathcal{V} \cdot d}{2} \tag{17}$$

$$R\_S = \frac{\mathcal{V} \cdot d}{\sqrt{3}}\tag{18}$$

where γ is the angle between the directions of launching rays and d is the distance the center of the sphere to the signal source.

This method introduces errors of interpretation arising from the limited geometrical possibility of approximation of a sphere by concentric circles. In order to increase the accuracy, portions of a sphere are approximated by the rectangles (Figure 6). The sizes of the υ<sup>i</sup> and ψ<sup>i</sup> determine the coordinates of the launched ray, and Δψ<sup>i</sup> and Δυ<sup>i</sup> are the sides of the rectangle and are described by [8]:

$$
\Delta\psi\_i(\mathfrak{d}\_i) = \frac{\Delta\mathfrak{d}}{\sin\mathfrak{d}\_i} \tag{19}
$$

the obstacle shall be determined with the law of Snell. The RL method can be used to find the routes of diffracted rays. The source of diffracted rays may be found when the rectangle (or

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

91

The RL method has a disadvantage in relation to the IM method because it may neglect the very narrow obstacles lying on the extension of transmitter, which can be placed between two rays. To ensure satisfactory resolution of the simulation, the technique of rays splitting RS was introduced, which allows you to split the rays when the radius of the receiving sphere (or side

The RL method is easier to implement with respect to the IM method. It is characterized by the weaker resolution and longer time of computation. Currently, the IM and RL methods are

In several recent years, research to refine the method of RT has been conducted. Thus the smart method of ray tracing, that is, intelligent ray tracing (IRT) (Figure 8b) has been developed. It is based on simple assumptions [10, 14, 15]: (1) only a few rays takes an essential part of the energy of the electromagnetic field, (2) visibility of faces and edges do not depend on the position of the of transmitter antenna, (3) often the bordering receiving points (pixels) are reachable by the rays with a very similar the properties. In accordance with the objectives the calculation were optimized. Pretreatment process of database processing with a collection of information about the obstacles encountered in the model shall be carried out only once. The idea is that each of these obstacles is divided into small pieces called tiles and on the borders in

Mutual relations of visibility between objects are calculated once and stored in the database. The tiles and the edges are represented only by their center points. The idea of the relationship of visibility center points of the edges or tiles is that if the two center points are in the zone, direct visibility defines the rays from the center of the first tile to the corners of the next one

circle) associated with the ray crosses the diffraction edge [8].

Figure 6. Cross-section of the receiving sphere (a) and rays launching method concept (b) [11].

of the rectangle) reaches its maximum size (Figure 6).

called the ray tracing, RT, method.

the form of episodes.

(Figure 7a).

$$
\Delta\theta\_i = \frac{\Delta\theta}{2} + (i - 1) \cdot \Delta\theta\_\prime \qquad \qquad i = 1...N\_\partial\,, \tag{20}
$$

The intensity of the electric field at the receiving end is the sum of the intensities of all rays, whose distances from a receiving point are less than RS. The direction of the rays, penetrating

Figure 5. Routing of the reflected rays in the method of images IM (a), and routing of the reflected rays in the rays launching R-L method [1].

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 91

Figure 6. Cross-section of the receiving sphere (a) and rays launching method concept (b) [11].

Δψ<sup>i</sup>

<sup>ϑ</sup><sup>i</sup> <sup>¼</sup> <sup>Δ</sup><sup>ϑ</sup>

90 Antennas and Wave Propagation

launching R-L method [1].

ð Þ¼ ϑ<sup>i</sup>

The intensity of the electric field at the receiving end is the sum of the intensities of all rays, whose distances from a receiving point are less than RS. The direction of the rays, penetrating

Figure 5. Routing of the reflected rays in the method of images IM (a), and routing of the reflected rays in the rays

Δϑ sinϑ<sup>i</sup>

<sup>2</sup> <sup>þ</sup> ð Þ� <sup>i</sup> � <sup>1</sup> <sup>Δ</sup>ϑ, i <sup>¼</sup> <sup>1</sup>…Nϑ, <sup>Δ</sup><sup>ϑ</sup> <sup>¼</sup> const: (20)

(19)

the obstacle shall be determined with the law of Snell. The RL method can be used to find the routes of diffracted rays. The source of diffracted rays may be found when the rectangle (or circle) associated with the ray crosses the diffraction edge [8].

The RL method has a disadvantage in relation to the IM method because it may neglect the very narrow obstacles lying on the extension of transmitter, which can be placed between two rays. To ensure satisfactory resolution of the simulation, the technique of rays splitting RS was introduced, which allows you to split the rays when the radius of the receiving sphere (or side of the rectangle) reaches its maximum size (Figure 6).

The RL method is easier to implement with respect to the IM method. It is characterized by the weaker resolution and longer time of computation. Currently, the IM and RL methods are called the ray tracing, RT, method.

In several recent years, research to refine the method of RT has been conducted. Thus the smart method of ray tracing, that is, intelligent ray tracing (IRT) (Figure 8b) has been developed. It is based on simple assumptions [10, 14, 15]: (1) only a few rays takes an essential part of the energy of the electromagnetic field, (2) visibility of faces and edges do not depend on the position of the of transmitter antenna, (3) often the bordering receiving points (pixels) are reachable by the rays with a very similar the properties. In accordance with the objectives the calculation were optimized. Pretreatment process of database processing with a collection of information about the obstacles encountered in the model shall be carried out only once. The idea is that each of these obstacles is divided into small pieces called tiles and on the borders in the form of episodes.

Mutual relations of visibility between objects are calculated once and stored in the database. The tiles and the edges are represented only by their center points. The idea of the relationship of visibility center points of the edges or tiles is that if the two center points are in the zone, direct visibility defines the rays from the center of the first tile to the corners of the next one (Figure 7a).

Figure 7. The division of the rays in the RT method [12] (a) and the geometry of the inteligent method of ray tracing IRT [10].

These rays and the throw of their angles are data indicating the relationship prevailing between the two center points. Such relationships are created also between edges, as well as in the case of the edge of the tiles. Important are the angles that define the angular distance of possible diffraction and reflection. During the process of routing, the propagation links the information about the relationship of visibility are readily available and you do not need to set dependencies between the obstacles for each ray, which greatly increase the computation speeds [10].

coefficient of mth edge for the i

according to the formula [8]:

phenomena used in the RT method.

th border for the i

Table 2. Division of algorithms due to the number of the relevant phenomena.

ray, and dni is the length of the route of the i

the coefficient for k

Class of algorithm Direct ray

Maximum number of reflections in the absence of

diffraction

th ray, Rji is reflection coefficient from j

Maximum number of diffraction in the absence of

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

reflections

1 Yes ——— 2 Yes 1 — — 3 Yes 2 — — 4 Yes 2 1 — 5 Yes 3 1 — 6 Yes 3 1 1 o + 1 d 7 Yes 3 2 1 o + 1 d 8 Yes 3 2 2 o + 1 d 9 Yes 4 2 2 o + 1 d 10 Yes 5 2 2 o + 1 d 11 Yes 6 2 2 o + 1 d

The intensity of the electric field from the all rays arriving at the receiver is calculated

<sup>E</sup> <sup>¼</sup> <sup>X</sup><sup>n</sup> i¼1

Power PR delivered by the receiving antenna to receiver depends on the effective area (or the

Z0

The coefficients in the expression (21) are defined by the modeling techniques of propagation

In the methods of geometrical optics, GO, reflection phenomenon of reflection of electromagnetic waves describes the Fresnel coefficients, expressing ratios of electromagnetic fields strengths of the reflected (R) and incident (I) waves. The two Fresnel coefficients are different for the EM vectors intensities, namely the parallel and perpendicular components to the plane of incidence. The degree of reflection and transmission of both Fresnel vectors are varying quite different. The plane of incidence is defined as the plane determined by the wave vector of

� X i

th ray.

effective length) ASK of the receiving antenna and power density S [8]:

where GRi is a energetic gain of receiving antenna for the i

PR <sup>¼</sup> <sup>S</sup> � ASK <sup>¼</sup> j j <sup>E</sup> <sup>2</sup> � ASK

th object for the i

The maximum number of reflections (o) and diffraction

http://dx.doi.org/10.5772/intechopen.75384

(d)

Ei (22)

GRi <sup>θ</sup>Ri;φRi � � (23)

th ray, αni is attenuation constant of the nth object for the i

th ray by nth object of attenuation constant of αni.

th ray, Tki is

th

93

The introduction of the 3D model is associated with the growth of databases and considerable complexity of algorithms. This causes prolongation of the calculation time. To reduce this time, instead of 3D method, the 2 � 2D, so called 2.5D method of modeling has been introduced. In this solution, two independent analyses are carried out: in vertical and horizontal planes [8]. In order to further reduce the calculation time proposed the division of the algorithms in the class due to the number of the relevant phenomena for each of the waves (Table 2).

Further increasing the maximum number of, at issue, propagation phenomena takes longer simulation time. In order to reduce calculation time the receiving coverage surface is divided into smaller portions, called the pixels. Each pixel is represented by its center point. The route of the rays is determined only between source and any center point. Increasing the size of the pixels will reduce the resolution of the simulation and the calculation time.

After you specify all the routes between the transmitter and receiver, the components of the electrical field strength Ei at the receiving end originating in from each of the rays are determined [9]:

$$E\_i = \frac{\sqrt{Z\_0 \cdot P\_0 \cdot G\_{Ti}(\theta\_{Ti}, \varphi\_{Ti})}}{4\pi \cdot d\_i \cdot L\_{Di}(d\_i)} \cdot \prod\_m D\_{mi} \cdot \prod\_j R\_{ji} \cdot \prod\_k T\_{ki} \cdot \prod\_n e^{-a\_{ni}d\_{ni}} \cdot e^{-\frac{2\pi d\_i}{\lambda}} \tag{21}$$

where GTi is the energetic gain of receiving antenna for the i th ray, Z0 is the characteristic impedance (≈ 377 Ω), P0 is the power input to the transmitting antenna, λ is the of the transmitted wavelength, di is the length of the route and of the i th ray, Ldi is the geometric coefficient depending on the position of the diffraction edge on the route of the i th ray, Dmi is the diffraction



Table 2. Division of algorithms due to the number of the relevant phenomena.

These rays and the throw of their angles are data indicating the relationship prevailing between the two center points. Such relationships are created also between edges, as well as in the case of the edge of the tiles. Important are the angles that define the angular distance of possible diffraction and reflection. During the process of routing, the propagation links the information about the relationship of visibility are readily available and you do not need to set dependencies between the obstacles for each ray, which greatly increase the computation

Figure 7. The division of the rays in the RT method [12] (a) and the geometry of the inteligent method of ray tracing IRT [10].

The introduction of the 3D model is associated with the growth of databases and considerable complexity of algorithms. This causes prolongation of the calculation time. To reduce this time, instead of 3D method, the 2 � 2D, so called 2.5D method of modeling has been introduced. In this solution, two independent analyses are carried out: in vertical and horizontal planes [8]. In order to further reduce the calculation time proposed the division of the algorithms in the class

Further increasing the maximum number of, at issue, propagation phenomena takes longer simulation time. In order to reduce calculation time the receiving coverage surface is divided into smaller portions, called the pixels. Each pixel is represented by its center point. The route of the rays is determined only between source and any center point. Increasing the size of the

After you specify all the routes between the transmitter and receiver, the components of the electrical field strength Ei at the receiving end originating in from each of the rays are deter-

impedance (≈ 377 Ω), P0 is the power input to the transmitting antenna, λ is the of the transmit-

�j 2π�di

th ray, Z0 is the characteristic

th ray, Dmi is the diffraction

th ray, Ldi is the geometric coefficient

<sup>λ</sup> (21)

due to the number of the relevant phenomena for each of the waves (Table 2).

pixels will reduce the resolution of the simulation and the calculation time.

� Y m Dmi � Y j Rji � Y k Tki � Y n e �αni�dni � <sup>e</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z<sup>0</sup> � P<sup>0</sup> � GTi θTi;φTi q � �

where GTi is the energetic gain of receiving antenna for the i

depending on the position of the diffraction edge on the route of the i

4π � di � LDið Þ di

ted wavelength, di is the length of the route and of the i

speeds [10].

92 Antennas and Wave Propagation

mined [9]:

Ei ¼

coefficient of mth edge for the i th ray, Rji is reflection coefficient from j th object for the i th ray, Tki is the coefficient for k th border for the i th ray, αni is attenuation constant of the nth object for the i th ray, and dni is the length of the route of the i th ray by nth object of attenuation constant of αni.

The intensity of the electric field from the all rays arriving at the receiver is calculated according to the formula [8]:

$$E = \sum\_{i=1}^{n} E\_i \tag{22}$$

Power PR delivered by the receiving antenna to receiver depends on the effective area (or the effective length) ASK of the receiving antenna and power density S [8]:

$$P\_R = S \cdot A\_{SK} = \frac{|E|^2 \cdot A\_{SK}}{Z\_0} \cdot \sum\_i G\_{Ri} \{\theta\_{Ri}, \varphi\_{Ri}\} \tag{23}$$

where GRi is a energetic gain of receiving antenna for the i th ray.

The coefficients in the expression (21) are defined by the modeling techniques of propagation phenomena used in the RT method.

In the methods of geometrical optics, GO, reflection phenomenon of reflection of electromagnetic waves describes the Fresnel coefficients, expressing ratios of electromagnetic fields strengths of the reflected (R) and incident (I) waves. The two Fresnel coefficients are different for the EM vectors intensities, namely the parallel and perpendicular components to the plane of incidence. The degree of reflection and transmission of both Fresnel vectors are varying quite different. The plane of incidence is defined as the plane determined by the wave vector of the incident and the normal to the boundary of separation. Figure 8 shows the mechanism of the reflection phenomenon [8].

Coefficients of reflection (R) and transmission (T) of both components are expressed by the following expressions [11]:


$$R\_{ll} = \frac{\text{tg}\left(\theta\_1 - \theta\_2\right)}{\text{tg}\left(\theta\_1 + \theta\_2\right)}\tag{24}$$

In real terms, the walls are constructed of several layers. On each border of the wall layers the wave is splitting on two components, the reflected wave from the surface of layer and transmitted wave to the layer. The result is that the actual coefficients of transmission and reflection

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

The incident wave is dissipated as a result of surface roughness. To determine the surface

<sup>Δ</sup><sup>φ</sup> <sup>¼</sup> <sup>π</sup>

For small angles, the height H of Rayleigh and minimum distance between inequalities S

<sup>H</sup> <sup>¼</sup> <sup>λ</sup>

<sup>S</sup> <sup>¼</sup> <sup>λ</sup>

where λ is the length of a scattered wave and θ is the angle between the incident wave and

According to the Rayleigh criterion, if the height surface roughness is greater than H and the distance between them is greater than S, then consider the surface that causes the dispersion of

The phenomenon of EM wave scattering can be taken into account by reduction of the reflection coefficients (transmission coefficients are not changed). These coefficients can be multiplied by the value of slightly less than unity, where the exact value depends exponentially on

II,<sup>⊥</sup> <sup>¼</sup> RII,⊥exp �<sup>8</sup> <sup>π</sup> � <sup>H</sup> � cos<sup>θ</sup>

λ

the level of surface roughness calculated in accordance with the theory of Rayleigh:

R0

Figure 9. Determination of surface roughness [12].

where Δφ is the difference of phase between the two rays as shown in Figure 9.

<sup>2</sup> (30)

http://dx.doi.org/10.5772/intechopen.75384

95

<sup>8</sup> � <sup>θ</sup> (31)

<sup>4</sup> � <sup>θ</sup><sup>2</sup> (32)

� �<sup>2</sup> " # (33)

depend on the angle of incidence [9].

determine the following equations [12]:

reflecting plane.

incident wave [12].

roughness, the criterion of Rayleigh is used [12]:


$$R\_{\perp} = -\frac{\sin(\theta\_1 - \theta\_2)}{\sin(\theta\_1 + \theta\_2)}\tag{25}$$


$$T\_{II} = \frac{2 \cdot \cos(\theta\_1) \cdot \sin(\theta\_2)}{\sin(\theta\_1 + \theta\_2) \cdot \cos(\theta\_1 - \theta\_2)}\tag{26}$$


$$T\_{\perp} = \frac{2 \cdot \cos(\theta\_1) \cdot \sin(\theta\_2)}{\sin(\theta\_1 + \theta\_2)} \tag{27}$$

Knowing the material properties of both media: conductivity σ, permeability μ, and permittivity ε for an angular frequency ω, we can designate an angle θ<sup>2</sup> using Snell's law [11]:

$$\theta\_2 = \arcsin(\sin(\theta\_2))\tag{28}$$

$$\frac{\sin(\theta\_1)}{\sin(\theta\_2)} = \sqrt{\frac{\mu\_2 \cdot \left(\varepsilon\_2 - j\frac{\phi\_2}{\omega}\right)}{\mu\_1 \cdot \left(\varepsilon\_1 - j\frac{\phi\_1}{\omega}\right)}}\tag{29}$$

Figure 8. The EM wave reflection and transmission phenomenon.

In real terms, the walls are constructed of several layers. On each border of the wall layers the wave is splitting on two components, the reflected wave from the surface of layer and transmitted wave to the layer. The result is that the actual coefficients of transmission and reflection depend on the angle of incidence [9].

The incident wave is dissipated as a result of surface roughness. To determine the surface roughness, the criterion of Rayleigh is used [12]:

$$
\Delta \varphi = \frac{\pi}{2} \tag{30}
$$

where Δφ is the difference of phase between the two rays as shown in Figure 9.

the incident and the normal to the boundary of separation. Figure 8 shows the mechanism of

Coefficients of reflection (R) and transmission (T) of both components are expressed by the

<sup>R</sup>ΙΙ <sup>¼</sup> tg ð Þ <sup>θ</sup><sup>1</sup> � <sup>θ</sup><sup>2</sup> tg ð Þ θ<sup>1</sup> þ θ<sup>2</sup>

<sup>R</sup><sup>⊥</sup> ¼ � sinð Þ <sup>θ</sup><sup>1</sup> � <sup>θ</sup><sup>2</sup> sinð Þ θ<sup>1</sup> þ θ<sup>2</sup>

<sup>T</sup>ΙΙ <sup>¼</sup> <sup>2</sup> � cosð Þ� <sup>θ</sup><sup>1</sup> sinð Þ <sup>θ</sup><sup>2</sup>

<sup>T</sup><sup>⊥</sup> <sup>¼</sup> <sup>2</sup> � cosð Þ� <sup>θ</sup><sup>1</sup> sinð Þ <sup>θ</sup><sup>2</sup> sinð Þ θ<sup>1</sup> þ θ<sup>2</sup>

Knowing the material properties of both media: conductivity σ, permeability μ, and permittiv-

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>μ</sup><sup>2</sup> � <sup>ε</sup><sup>2</sup> � <sup>j</sup> <sup>σ</sup><sup>2</sup>

<sup>μ</sup><sup>1</sup> � <sup>ε</sup><sup>1</sup> � <sup>j</sup> <sup>σ</sup><sup>1</sup>

ω � �

ω � �

θ<sup>2</sup> ¼ arcsin sin ð Þ ð Þ θ<sup>2</sup> (28)

ity ε for an angular frequency ω, we can designate an angle θ<sup>2</sup> using Snell's law [11]:

¼

s

sinð Þ θ<sup>1</sup> sinð Þ θ<sup>2</sup>

sinð Þ� θ<sup>1</sup> þ θ<sup>2</sup> cosð Þ θ<sup>1</sup> � θ<sup>2</sup>

(24)

(25)

(26)

(27)

(29)

the reflection phenomenon [8].





Figure 8. The EM wave reflection and transmission phenomenon.

following expressions [11]:

94 Antennas and Wave Propagation

For small angles, the height H of Rayleigh and minimum distance between inequalities S determine the following equations [12]:

$$H = \frac{\lambda}{8 \cdot \theta} \tag{31}$$

$$S = \frac{\lambda}{4 \cdot \theta^2} \tag{32}$$

where λ is the length of a scattered wave and θ is the angle between the incident wave and reflecting plane.

According to the Rayleigh criterion, if the height surface roughness is greater than H and the distance between them is greater than S, then consider the surface that causes the dispersion of incident wave [12].

The phenomenon of EM wave scattering can be taken into account by reduction of the reflection coefficients (transmission coefficients are not changed). These coefficients can be multiplied by the value of slightly less than unity, where the exact value depends exponentially on the level of surface roughness calculated in accordance with the theory of Rayleigh:

$$R'\_{ll,\perp} = R\_{ll,\perp} \exp\left[-8\left(\frac{\pi \cdot H \cdot \cos\theta}{\lambda}\right)^2\right] \tag{33}$$

Figure 9. Determination of surface roughness [12].

where RII,<sup>⊥</sup> is the reflection coefficient from the perfectly flat surface, R'II,<sup>⊥</sup> is the reflection coefficient taking into account the scattering phenomenon, λ is the length of the wave, H is the Rayleigh's height, and θ is the angle of incidence.

Studies have proven that the absorption of the wave by different types of objects (first of all living beings) is very difficult to model. Knowing the obstacle dimensions and material properties: conductivity σ, permeability μ, and ε = ε'- jε", given the frequency of the EM wave, using a general description of the EM plane wave, we can determine the impact of absorption on changing the wave parameters [11]:

$$\mathcal{U} \cdot e^{i \cdot a \cdot t} = A \cdot e^{-a \cdot d} \cdot e^{i \left(a \cdot t - \beta \cdot d\right)} \tag{34}$$

expression (21), and D<sup>⊥</sup>

where <sup>R</sup><sup>⊥</sup>,ΙΙ <sup>n</sup> and <sup>R</sup><sup>⊥</sup>,ΙΙ

following expressions [13]:

D<sup>⊥</sup>

ΙΙ <sup>¼</sup> <sup>e</sup>�jπ=<sup>4</sup>

D<sup>1</sup> ¼ cot

D<sup>2</sup> ¼ cot

<sup>D</sup><sup>4</sup> <sup>¼</sup> <sup>R</sup><sup>⊥</sup>,ΙΙ <sup>n</sup> � cot

<sup>0</sup> � cot

a� β

<sup>D</sup><sup>3</sup> <sup>¼</sup> <sup>R</sup><sup>⊥</sup>,ΙΙ

<sup>2</sup> � <sup>n</sup> � sinθ<sup>0</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>π</sup> <sup>þ</sup> <sup>φ</sup> � <sup>φ</sup> � �<sup>0</sup> 2 � n !

Figure 10. Diffraction cone in the UTD method (a) and an example of the radius diffracted on the edge (b) [13].

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

<sup>π</sup> � <sup>φ</sup> � <sup>φ</sup> � �<sup>0</sup> 2 � n !

> <sup>π</sup> � <sup>φ</sup> <sup>þ</sup> <sup>φ</sup> � �<sup>0</sup> 2 � n !

> <sup>π</sup> <sup>þ</sup> <sup>φ</sup> <sup>þ</sup> <sup>φ</sup> � �<sup>0</sup> 2 � n !

wave polarization, and a�, β�, L, and Fð Þ� (Fresnel transition function) are expressed in the

� � <sup>¼</sup> <sup>2</sup> � cos<sup>2</sup> <sup>2</sup> � <sup>n</sup> � <sup>π</sup> � <sup>N</sup>� � <sup>β</sup>

<sup>β</sup>� <sup>¼</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup>

<sup>0</sup> � sin<sup>2</sup>

<sup>L</sup> <sup>¼</sup> <sup>s</sup> � <sup>s</sup> 0 s þ s

follows [13]:

ΙΙ is the diffraction coefficient of the UTD method which is defined as

<sup>0</sup> are the reflection coefficients of sides of diffraction edges at a given EM

2

<sup>2</sup> � <sup>π</sup> � <sup>k</sup> <sup>p</sup> � ð Þ <sup>D</sup><sup>1</sup> <sup>þ</sup> <sup>D</sup><sup>2</sup> <sup>þ</sup> <sup>D</sup><sup>3</sup> <sup>þ</sup> <sup>D</sup><sup>4</sup> (38)

� F k � <sup>L</sup> � <sup>a</sup><sup>þ</sup> <sup>φ</sup> � <sup>φ</sup><sup>0</sup> � � � � (39)

http://dx.doi.org/10.5772/intechopen.75384

97

� F k � <sup>L</sup> � <sup>a</sup>� <sup>φ</sup> � <sup>φ</sup><sup>0</sup> � � � � (40)

� F k � <sup>L</sup> � <sup>a</sup>� <sup>φ</sup> <sup>þ</sup> <sup>φ</sup><sup>0</sup> � � � � (41)

� F k � <sup>L</sup> � <sup>a</sup><sup>þ</sup> <sup>φ</sup> <sup>þ</sup> <sup>φ</sup><sup>0</sup> � � � � (42)

� � (43)

θ<sup>0</sup> (45)

(44)

where α is a attenuation constant, β is a phase constant, U is the wave amplitude after absorption, A is the wave amplitude before absorption, d is the dimension of obstacles, and ω is angular frequency of the wave.

The attenuation constant α and the phase constant β are expressed in the following equations [11]:

$$\alpha = \omega \cdot \sqrt{\varepsilon' \cdot \mu'} \cdot \sqrt{\frac{1}{2} \cdot \left[ \sqrt{1 + \left( \frac{\boldsymbol{\sigma}\_{\omega} + \varepsilon''}{\varepsilon'} \right)^2} - 1 \right]} \tag{35}$$

$$\beta = \omega \cdot \sqrt{\varepsilon' \cdot \mu'} \cdot \sqrt{\frac{1}{2} \cdot \left[ \sqrt{1 + \left( \frac{\sigma/\omega}{\varepsilon'} \right)^2} + 1 \right]} \tag{36}$$

The ray tracing method RT does not take diffraction into account. In order to model this phenomenon, the deterministic models based on ray tracing method are enhanced with correct techniques. There are several methods for modeling a phenomenon of diffraction: PAW (perfectly absorbing wedge), GTD (geometrical theory of diffraction) and UTD (uniform theory of diffraction). One of the biggest problems of modeling of diffraction is the precise definition of diffraction edges of the buildings and other objects. The most commonly used method is the UTD, which takes into account the wave polarization and the material properties of the diffraction edge. The EM wave coming on the diffraction edge is scattered on a cone whose vertex is at the diffraction point. In Figure 10a, the example of a diffraction cone is shown [8].

Figure 10b shows the transmission of the ray, which includes the phenomenon of diffraction by means of the UTD method.

The intensity of the electric field at the point of reception can be designated from [13]:

$$E\_{\rm ITD} = E\_0 \frac{e^{-\text{jks}'}}{\text{s}'} \cdot D\_{\text{II}}^\perp \cdot \sqrt{\frac{\text{s}'}{\text{s} + \text{s}'}} \cdot e^{-\text{jks}} \tag{37}$$

where E0 is the wave field intensity emitted at the transmitting point, <sup>k</sup> <sup>¼</sup> <sup>2</sup>�<sup>π</sup> <sup>λ</sup> is the wavenumber, ffiffiffiffiffiffiffi s 0 sþs 0 q sets the geometric relationship necessary for the determination of LDi in an

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 97

where RII,<sup>⊥</sup> is the reflection coefficient from the perfectly flat surface, R'II,<sup>⊥</sup> is the reflection coefficient taking into account the scattering phenomenon, λ is the length of the wave, H is the

Studies have proven that the absorption of the wave by different types of objects (first of all living beings) is very difficult to model. Knowing the obstacle dimensions and material properties: conductivity σ, permeability μ, and ε = ε'- jε", given the frequency of the EM wave, using a general description of the EM plane wave, we can determine the impact of absorption

�α�<sup>d</sup> � <sup>e</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffi s 0 s þ s 0

sets the geometric relationship necessary for the determination of LDi in an

� e

�jks (37)

<sup>λ</sup> is the wave-

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>σ</sup>=<sup>ω</sup> <sup>þ</sup> <sup>ε</sup>

ε 0 � �<sup>2</sup>

00

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>σ</sup>=<sup>ω</sup> <sup>þ</sup> <sup>ε</sup><sup>00</sup> ε0 � �<sup>2</sup>

� 1

þ 1

3 5 vuuut (35)

3 5 vuuut (36)

where α is a attenuation constant, β is a phase constant, U is the wave amplitude after absorption, A is the wave amplitude before absorption, d is the dimension of obstacles, and ω

The attenuation constant α and the phase constant β are expressed in the following equations [11]:

2 4

s

s

2 4

The ray tracing method RT does not take diffraction into account. In order to model this phenomenon, the deterministic models based on ray tracing method are enhanced with correct techniques. There are several methods for modeling a phenomenon of diffraction: PAW (perfectly absorbing wedge), GTD (geometrical theory of diffraction) and UTD (uniform theory of diffraction). One of the biggest problems of modeling of diffraction is the precise definition of diffraction edges of the buildings and other objects. The most commonly used method is the UTD, which takes into account the wave polarization and the material properties of the diffraction edge. The EM wave coming on the diffraction edge is scattered on a cone whose vertex is at the diffraction point. In Figure 10a, the example of a diffraction cone is shown [8]. Figure 10b shows the transmission of the ray, which includes the phenomenon of diffraction

<sup>j</sup>ð Þ <sup>ω</sup>�t�β�<sup>d</sup> (34)

Rayleigh's height, and θ is the angle of incidence.

α ¼ ω �

β ¼ ω �

U � e

ffiffiffiffiffiffiffiffiffiffiffi ε0 � μ0

ffiffiffiffiffiffiffiffiffiffiffi ε 0 � μ0

�

�

1 2 �

1 2 �

The intensity of the electric field at the point of reception can be designated from [13]:

e�jks<sup>0</sup> s <sup>0</sup> � <sup>D</sup><sup>⊥</sup> ΙΙ �

where E0 is the wave field intensity emitted at the transmitting point, <sup>k</sup> <sup>¼</sup> <sup>2</sup>�<sup>π</sup>

EUTD ¼ E<sup>0</sup>

q

q

<sup>j</sup>�ω�<sup>t</sup> <sup>¼</sup> <sup>A</sup> � <sup>e</sup>

on changing the wave parameters [11]:

96 Antennas and Wave Propagation

is angular frequency of the wave.

by means of the UTD method.

ffiffiffiffiffiffiffi s 0 sþs 0 q

number,

Figure 10. Diffraction cone in the UTD method (a) and an example of the radius diffracted on the edge (b) [13].

expression (21), and D<sup>⊥</sup> ΙΙ is the diffraction coefficient of the UTD method which is defined as follows [13]:

$$D\_{ll}^{\perp} = \frac{e^{-j\pi/4}}{2 \cdot n \cdot \sin\theta\_0 \cdot \sqrt{2 \cdot \pi \cdot k}} \times (D\_1 + D\_2 + D\_3 + D\_4) \tag{38}$$

$$D\_1 = \cot\left(\frac{\pi + \left(\varphi - \varphi'\right)}{2 \cdot n}\right) \cdot F\left(k \cdot L \cdot a^+ \left(\varphi - \varphi'\right)\right) \tag{39}$$

$$D\_2 = \cot\left(\frac{\pi - \left(\varphi - \varphi'\right)}{2 \cdot n}\right) \cdot F\left(k \cdot L \cdot a^-\left(\varphi - \varphi'\right)\right) \tag{40}$$

$$D\_3 = R\_0^{\perp, II} \cdot \cot\left(\frac{\pi - \left(\varphi + \phi'\right)}{2 \cdot n}\right) \cdot F\left(k \cdot L \cdot a^- \left(\varphi + \phi'\right)\right) \tag{41}$$

$$D\_4 = R\_n^{\perp, II} \cdot \cot\left(\frac{\pi + \left(\varphi + \phi'\right)}{2 \cdot n}\right) \cdot F\left(k \cdot L \cdot a^+ \left(\varphi + \phi'\right)\right) \tag{42}$$

where <sup>R</sup><sup>⊥</sup>,ΙΙ <sup>n</sup> and <sup>R</sup><sup>⊥</sup>,ΙΙ <sup>0</sup> are the reflection coefficients of sides of diffraction edges at a given EM wave polarization, and a�, β�, L, and Fð Þ� (Fresnel transition function) are expressed in the following expressions [13]:

$$a^{\pm}(\beta) = 2 \cdot \cos^2\left(\frac{2 \cdot n \cdot \pi \cdot N^{\pm} - \beta}{2}\right) \tag{43}$$

$$
\boldsymbol{\beta}^{\pm} = \boldsymbol{\varphi} \pm \boldsymbol{\varphi}^{'} \tag{44}
$$

$$L = \frac{\mathbf{s} \cdot \mathbf{s'}}{\mathbf{s} + \mathbf{s'}} \cdot \sin^2 \theta\_0 \tag{45}$$

$$F(\mathbf{x}) = 2 \cdot j \cdot \sqrt{\mathbf{x}} \cdot e^{j \cdot \mathbf{x}} \int\_{\sqrt{\mathbf{x}}}^{\infty} e^{-j \cdot \mathbf{r}^2} \cdot d\tau \tag{46}$$

where N� is the integer closest to the fulfillment of the following equations [13]:

$$2\cdot \pi \cdot n \cdot N^+ - \left(\varphi \pm \varphi'\right) = \pi \tag{47}$$

$$2\cdot \pi \cdot n \cdot N^- - \left(\boldsymbol{\varrho} \pm \boldsymbol{\varrho}'\right) = -\pi \tag{48}$$

$$n = \frac{2 \cdot \pi - a}{\pi} \tag{49}$$

The EM wave propagation models presented allow you to streamline the process of designing wireless networks of mobile communication systems like WLAN, Bluetooth RFhome, ZigBee, and WPAN inside buildings as well as the outdoor urban environment. Allow you to specify

Table 3. Basic characteristics of mostly used the EM wave propagation modeling computer programs.

Wireless InSite WiSE WinProp I-Prop

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

Communications

http://dx.doi.org/10.5772/intechopen.75384






points




Czech Technical University in Prague

99






Remcom Lucent Technologies AWE

0.3/300 0.05/40 \*/ >0.6 \*/







time - Delay

points

models

ACTIX Analyzer

Amdocs Company





own propagation models - Data base of people and trees models - To take into account the depolarization - Dynamic simulations (of receivers mobility)








in time - Delay

models - Incorporate of phase of received

rays

time


Firm ACTIX An

Frequency (GHz)

Propagation models

Modeled phenomenon

Parameters of materials

Result of the simulation

The characteristics of the codes

\* No details.

Figure 4c shows the results obtained by RT 3D method. Results were obtained by means of the ACTIX Analyzer computer code [6] for the 10 dBm transmitter power at a frequency of 2.45 GHz.

Method of geometrical optics, GO, is used to model the distribution of electromagnetic fields inside the buildings. On the basis of GO, the computer programs developed are: ACTIX Analyzer (by ACTIX) [6], WinProp (by AWE) [10], WiSE (by Lucent Technologies), Wireless InSite (by Remcom) [33], Cindoor (by Cantaberia University, Spain), Volcano (by Siradel) [28], and SitePlanner (by Motorola), and others for designing the indoor and outdoor wireless communication systems.

#### 5. Review of computer programs for wireless networks planning

The EM wave propagation models presented previously have been used in computer programs for planning wireless communication networks. The most commonly used method is the ray tracing RT. Methods based directly on the Maxwell's equations (e.g., FDTD) have not adopted in the design of wireless networks inside buildings, they are rather more attractive for outdoor propagation scenes simulation (e.g., Wireless InSite by Remcom [33]). An empirical model most commonly used in computer programs is a non-linear model of multi-walls MW developed within the framework of the COST 231 project and adopted in the ITU Recommendation. The one slope I-S model is not the exact and therefore rarely used, while the dominant path model (D-P) is difficult to implement. Showcased programs allow you to simulate for the frequency range from 0.05 to about 300 GHz. Table 3 summarizes the basic characteristics of the mostly used computer programs.

In addition to the presented programs on the market, there are several third-party products. These include, for example, Volcano (Siradel) [28], ADTI ICS online [29], iBWAVE DE [30], TAP™MAPPER [31], SitePlanner (Motorola), EDX Signal Pro (EDX Wireless Technologies) [32], CelPlanner™ [34], etc. The commercial computer solvers of the EM wave propagation models are complemented by the computer tools implemented by academic centers. The examples are the program I-Prop (Technical University of Prague, Czech Republic) [3], Cindoor (University of Cantaberia, Spain), etc.


No details.

F xð Þ¼ <sup>2</sup> � <sup>j</sup> � ffiffiffi

where N� is the integer closest to the fulfillment of the following equations [13]:

2.45 GHz.

communication systems.

98 Antennas and Wave Propagation

the mostly used computer programs.

Cindoor (University of Cantaberia, Spain), etc.

<sup>x</sup> <sup>p</sup> � <sup>e</sup> j�x ð ∞

<sup>2</sup> � <sup>π</sup> � <sup>n</sup> � <sup>N</sup><sup>þ</sup> � <sup>φ</sup> � <sup>φ</sup><sup>0</sup> � �

<sup>2</sup> � <sup>π</sup> � <sup>n</sup> � <sup>N</sup>� � <sup>φ</sup> � <sup>φ</sup><sup>0</sup> � �

<sup>n</sup> <sup>¼</sup> <sup>2</sup> � <sup>π</sup> � <sup>α</sup>

Figure 4c shows the results obtained by RT 3D method. Results were obtained by means of the ACTIX Analyzer computer code [6] for the 10 dBm transmitter power at a frequency of

Method of geometrical optics, GO, is used to model the distribution of electromagnetic fields inside the buildings. On the basis of GO, the computer programs developed are: ACTIX Analyzer (by ACTIX) [6], WinProp (by AWE) [10], WiSE (by Lucent Technologies), Wireless InSite (by Remcom) [33], Cindoor (by Cantaberia University, Spain), Volcano (by Siradel) [28], and SitePlanner (by Motorola), and others for designing the indoor and outdoor wireless

The EM wave propagation models presented previously have been used in computer programs for planning wireless communication networks. The most commonly used method is the ray tracing RT. Methods based directly on the Maxwell's equations (e.g., FDTD) have not adopted in the design of wireless networks inside buildings, they are rather more attractive for outdoor propagation scenes simulation (e.g., Wireless InSite by Remcom [33]). An empirical model most commonly used in computer programs is a non-linear model of multi-walls MW developed within the framework of the COST 231 project and adopted in the ITU Recommendation. The one slope I-S model is not the exact and therefore rarely used, while the dominant path model (D-P) is difficult to implement. Showcased programs allow you to simulate for the frequency range from 0.05 to about 300 GHz. Table 3 summarizes the basic characteristics of

In addition to the presented programs on the market, there are several third-party products. These include, for example, Volcano (Siradel) [28], ADTI ICS online [29], iBWAVE DE [30], TAP™MAPPER [31], SitePlanner (Motorola), EDX Signal Pro (EDX Wireless Technologies) [32], CelPlanner™ [34], etc. The commercial computer solvers of the EM wave propagation models are complemented by the computer tools implemented by academic centers. The examples are the program I-Prop (Technical University of Prague, Czech Republic) [3],

5. Review of computer programs for wireless networks planning

ffiffi x p e �j�τ<sup>2</sup>

� dτ (46)

¼ π (47)

¼ �π (48)

<sup>π</sup> (49)

Table 3. Basic characteristics of mostly used the EM wave propagation modeling computer programs.

The EM wave propagation models presented allow you to streamline the process of designing wireless networks of mobile communication systems like WLAN, Bluetooth RFhome, ZigBee, and WPAN inside buildings as well as the outdoor urban environment. Allow you to specify the most appropriate location of access points and their minimum number of required covering a given area. In the case of a large diversification of the propagation environment (e.g., shopping center) it is recommended that you use the more accurate but at the same time dealing with more time simulation based on deterministic models.

## 6. Planning for wireless communication networks

Planning of the wireless communication systems, for example, LAN and PAN is not simple. Committed design errors can significantly reduce its final performance. Each project is influenced by many factors, which makes the implementation of ready-made scenarios not useful and, as a result, affecting the final result. Due to the complexity of the implementation of projects, short-range wireless networks can be divided into three stages: preparation of the data to the project, the design and implementation of the project and the measurements. Each phase requires the use of an appropriate approach, since errors made in any one of them can undermine efforts in working on the project. Figure 11 shows the proposed design algorithm [16, 17].

#### 6.1. Data preparation for the project

Preparation of the data to the project is an important part of planning wireless networks, because the mistakes made at this stage influence the next steps to the implementation of a wireless network. At the stage of the initial project, you must obtain the following information: range of communication system, number and locations of the supported users (or devices), their mobility, and types of used applications. You need to collect and draw up the technical documentation, the design area and parameters the obstacles which prevent the propagation of electromagnetic waves. Knowledge of the infrastructure of an existing wired network will allow the use of it in a wireless network in the project as a bridge/transition to the users of different communication systems. Based on these data and the financial capacity of the client, we select the appropriate wireless network technology. Most popular solutions for WPAN is the Bluetooth system, and in the case of the WLAN, the IEEE 802.11 standard [16].

#### 6.2. Collection and development of technical documentation

The first step in the initial stage is to collect and develop the technical documentation of the design area. On its basis in the design phase, the environment model used in simulations of the EM wave distribution power is built. You must specify the obstacles parameters properties which prevent the propagation of electromagnetic waves, such as location, geometrical shape and dimensions, and material properties (conductivity σ, permeability μ, and permittivity ε).

wireless network project such as the alarm system or multimedia communication, you should refer to the specification of a specific system. On the basis of specific requirements and expectations, you must specify the maximum number of simultaneous connections, which is to ensure the proposed network. Fixed number should be greater than the estimated number of calls occurring most of the time (e.g., by 95% of the time within 1 hour). In order to facilitate the analysis and take into account the diversity of needs of all traffic, it is divided into three categories: video a1/a2/a3-duplex services (video conferencing – MPEG4

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

101

Figure 11. Design algorithm of wireless communication networks.

#### 6.3. Define assumptions of the project

In the initial stage, we need to gather information about the range of the proposed network, estimated the number and location of users (or devices) and their mobility as well as a traffic profile. To do this, you must carry out a survey or environmental interview. In the case of a Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 101

Figure 11. Design algorithm of wireless communication networks.

the most appropriate location of access points and their minimum number of required covering a given area. In the case of a large diversification of the propagation environment (e.g., shopping center) it is recommended that you use the more accurate but at the same time

Planning of the wireless communication systems, for example, LAN and PAN is not simple. Committed design errors can significantly reduce its final performance. Each project is influenced by many factors, which makes the implementation of ready-made scenarios not useful and, as a result, affecting the final result. Due to the complexity of the implementation of projects, short-range wireless networks can be divided into three stages: preparation of the data to the project, the design and implementation of the project and the measurements. Each phase requires the use of an appropriate approach, since errors made in any one of them can undermine efforts in working on the project. Figure 11 shows the proposed design algorithm [16, 17].

Preparation of the data to the project is an important part of planning wireless networks, because the mistakes made at this stage influence the next steps to the implementation of a wireless network. At the stage of the initial project, you must obtain the following information: range of communication system, number and locations of the supported users (or devices), their mobility, and types of used applications. You need to collect and draw up the technical documentation, the design area and parameters the obstacles which prevent the propagation of electromagnetic waves. Knowledge of the infrastructure of an existing wired network will allow the use of it in a wireless network in the project as a bridge/transition to the users of different communication systems. Based on these data and the financial capacity of the client, we select the appropriate wireless network technology. Most popular solutions for WPAN is

the Bluetooth system, and in the case of the WLAN, the IEEE 802.11 standard [16].

The first step in the initial stage is to collect and develop the technical documentation of the design area. On its basis in the design phase, the environment model used in simulations of the EM wave distribution power is built. You must specify the obstacles parameters properties which prevent the propagation of electromagnetic waves, such as location, geometrical shape and dimensions, and material properties (conductivity σ, permeability μ, and permittivity ε).

In the initial stage, we need to gather information about the range of the proposed network, estimated the number and location of users (or devices) and their mobility as well as a traffic profile. To do this, you must carry out a survey or environmental interview. In the case of a

6.2. Collection and development of technical documentation

dealing with more time simulation based on deterministic models.

6. Planning for wireless communication networks

6.1. Data preparation for the project

100 Antennas and Wave Propagation

6.3. Define assumptions of the project

wireless network project such as the alarm system or multimedia communication, you should refer to the specification of a specific system. On the basis of specific requirements and expectations, you must specify the maximum number of simultaneous connections, which is to ensure the proposed network. Fixed number should be greater than the estimated number of calls occurring most of the time (e.g., by 95% of the time within 1 hour). In order to facilitate the analysis and take into account the diversity of needs of all traffic, it is divided into three categories: video a1/a2/a3-duplex services (video conferencing – MPEG4


7.1. Selecting the type of network

available on the network.

7.2. Specifying the minimum number of access points

the value of the OB = 7 for one Master Bluetooth system [19]:

X 6

wi � ai ≤ 7 (50)

i¼4

IEEE 802.11 networks analysis is more complicated [20–22]. In the case of this type, the effective capacity decreases with each client connected. The DCF access mechanism provides equitable access to BSS. To ensure the quality of the services offered on the network based on IEEE 802.11 g comes down to minimize the probability of overloading. It is implemented by

This value results from the specification of the Bluetooth system (7 slots).

The first step in the design phase is to select the type of network. There are two main types of network: range coverage and capacitive. Range coverage wireless network designs due to the largest coverage with the fewest number of access points. In this type of networks not optimizes to QoS parameters. It is assumed that users will benefit from the services of low-speed of packet transmission (e.g., barcode scanning and database query). These installations are used in warehouses or retail stores for inventory control and purchasing in real time purchases. An example of a range coverage network is a wireless sensors network of anti-theft system. You can deploy the range coverage WLAN networks in small- and medium-sized companies instead of wired Ethernet. When designing the capacitive network to ensure QoS parameters is required, that is, capacity, duration, and variation of delay as well as a bit error rate, BER. The sizes of the cells in the capacity networks are smaller, which means that the system must be equipped with a greater number of access points. The size of the cells is determined based on the number of users supported by a single access point. The maximum number of users that are associated with the access point is determined on the basis of the type of services to be

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

103

Specifying the minimum number of access points is extremely difficult. The problem is due to complications related to the precision of the estimates of the anticipated traffic generated by the users in the proposed wireless network. This, in turn, makes it difficult to estimate the impact of the number of users on the access point load, and thus the determination of areas of coverage. The Bluetooth system is designed for the implementation of broadcast audio and data. It provides a guarantee of quality of the services offered by granting higher priority connections audio relative to data transmission. Moreover a movement within pico-network is coordinated by the Master device, which eliminates the problem of collisions of packages. Therefore, to ensure the quality of the services offered on the network based on Bluetooth system comes down to minimize the probability of overloading. It is implemented by specifying the maximum number of connections the devices associated with one access point on the basis of the type of generated traffic. For this purpose, referred to in the previous stage of the types of traffic are attributed to the following the weight: audio w4/w5–2/1, � date w6–1. The sum of the products of consecutive weights by the number of connections does not exceed

Table 4. A set of project assumptions of sample wireless communication system [17].

640 480 25 frames per sec)/unilateral (streaming MPEG-4, 640 480, 25 frames/sec)/ unilateral (streaming-MPEG 2, 720 576, 25 frames/sec); audio a4/a5-duplex services (voice chat-g.711)/unilateral (streaming-G. 711), and data a6-FTP, mail, Web, p2p [18]. Mobility users understand the need to move to the area covered by another cell of the network while preserving the connection. This is possible only when both access points are on the same subnet. Often it is not possible to provide coverage of the entire area or associated it with a big financial effort. Therefore, network design requires that you specify the degree of its coverage. In the situation of the area covered with a mesh wireless sensors (e.g., alarm burglar alarm or fire), 100% coverage is required. In the case of the area in which users can easily move (e.g., WLAN at the airport) you can accept less coverage, for example, 90%. It happens that the collected information is not the same throughout the area. This is due to the characteristic features of the area (e.g., conference room or the library of the University). You then need to separately specify the design intent for each of the specific areas.

Table 4 provides a set of project assumptions sample wireless system for the Bank. The project provides for the 3 areas: (1) area 1 – lobby, corridors, rooms (providing access to the network of the Bank); (2) area 2 – Conference Room (multimedia presentations setup); and (3) area 3 around doors and windows (network sensors of high security locks).

#### 6.4. Analysis of exiting wire network

In most cases, wireless networks are designed as an extension to the wired LAN. Then you should be familiar with the topology and the capabilities of the network. It is important to specify the ability to increase her workload resulting from the join of the proposed wireless network, and the ability to support its security mechanisms (e.g., VPN network, IEEE 802.1 X authentication). Analysis of the wired network topology allows you to specify the possible spots to join the proposed network. On the basis of the initial information, we select for each of the areas of wireless technology.

## 7. Network design and simulation

The next step in the process of deploying a wireless network is a design phase. At this stage, on the basis of the information gathered in the initial phase of the project is created. The design phase is discussed on the example of Bluetooth and IEEE 802.11 b/g wireless systems.

#### 7.1. Selecting the type of network

640 480 25 frames per sec)/unilateral (streaming MPEG-4, 640 480, 25 frames/sec)/ unilateral (streaming-MPEG 2, 720 576, 25 frames/sec); audio a4/a5-duplex services (voice chat-g.711)/unilateral (streaming-G. 711), and data a6-FTP, mail, Web, p2p [18]. Mobility users understand the need to move to the area covered by another cell of the network while preserving the connection. This is possible only when both access points are on the same subnet. Often it is not possible to provide coverage of the entire area or associated it with a big financial effort. Therefore, network design requires that you specify the degree of its coverage. In the situation of the area covered with a mesh wireless sensors (e.g., alarm burglar alarm or fire), 100% coverage is required. In the case of the area in which users can easily move (e.g., WLAN at the airport) you can accept less coverage, for example, 90%. It happens that the collected information is not the same throughout the area. This is due to the characteristic features of the area (e.g., conference room or the library of the University). You

1 90 1/2/0 8/6 34 + + IEEE 802.11 g 2 90 0/0/0 0/2 8 + Bluetooth 3 100 0/0/0 0/0 22 ZigBee

Mobility Communication with another network

System selection

then need to separately specify the design intent for each of the specific areas.

around doors and windows (network sensors of high security locks).

Number of concurrent connections

> Audio a4/a5

Table 4. A set of project assumptions of sample wireless communication system [17].

Data a6

Video a1/a2/a3

6.4. Analysis of exiting wire network

Area Degree of coverage [%]

102 Antennas and Wave Propagation

the areas of wireless technology.

7. Network design and simulation

Table 4 provides a set of project assumptions sample wireless system for the Bank. The project provides for the 3 areas: (1) area 1 – lobby, corridors, rooms (providing access to the network of the Bank); (2) area 2 – Conference Room (multimedia presentations setup); and (3) area 3-

In most cases, wireless networks are designed as an extension to the wired LAN. Then you should be familiar with the topology and the capabilities of the network. It is important to specify the ability to increase her workload resulting from the join of the proposed wireless network, and the ability to support its security mechanisms (e.g., VPN network, IEEE 802.1 X authentication). Analysis of the wired network topology allows you to specify the possible spots to join the proposed network. On the basis of the initial information, we select for each of

The next step in the process of deploying a wireless network is a design phase. At this stage, on the basis of the information gathered in the initial phase of the project is created. The design

phase is discussed on the example of Bluetooth and IEEE 802.11 b/g wireless systems.

The first step in the design phase is to select the type of network. There are two main types of network: range coverage and capacitive. Range coverage wireless network designs due to the largest coverage with the fewest number of access points. In this type of networks not optimizes to QoS parameters. It is assumed that users will benefit from the services of low-speed of packet transmission (e.g., barcode scanning and database query). These installations are used in warehouses or retail stores for inventory control and purchasing in real time purchases. An example of a range coverage network is a wireless sensors network of anti-theft system. You can deploy the range coverage WLAN networks in small- and medium-sized companies instead of wired Ethernet. When designing the capacitive network to ensure QoS parameters is required, that is, capacity, duration, and variation of delay as well as a bit error rate, BER. The sizes of the cells in the capacity networks are smaller, which means that the system must be equipped with a greater number of access points. The size of the cells is determined based on the number of users supported by a single access point. The maximum number of users that are associated with the access point is determined on the basis of the type of services to be available on the network.

#### 7.2. Specifying the minimum number of access points

Specifying the minimum number of access points is extremely difficult. The problem is due to complications related to the precision of the estimates of the anticipated traffic generated by the users in the proposed wireless network. This, in turn, makes it difficult to estimate the impact of the number of users on the access point load, and thus the determination of areas of coverage. The Bluetooth system is designed for the implementation of broadcast audio and data. It provides a guarantee of quality of the services offered by granting higher priority connections audio relative to data transmission. Moreover a movement within pico-network is coordinated by the Master device, which eliminates the problem of collisions of packages. Therefore, to ensure the quality of the services offered on the network based on Bluetooth system comes down to minimize the probability of overloading. It is implemented by specifying the maximum number of connections the devices associated with one access point on the basis of the type of generated traffic. For this purpose, referred to in the previous stage of the types of traffic are attributed to the following the weight: audio w4/w5–2/1, � date w6–1. The sum of the products of consecutive weights by the number of connections does not exceed the value of the OB = 7 for one Master Bluetooth system [19]:

$$\sum\_{i=4}^{6} w\_i \cdot a\_i \le 7 \tag{50}$$

This value results from the specification of the Bluetooth system (7 slots).

IEEE 802.11 networks analysis is more complicated [20–22]. In the case of this type, the effective capacity decreases with each client connected. The DCF access mechanism provides equitable access to BSS. To ensure the quality of the services offered on the network based on IEEE 802.11 g comes down to minimize the probability of overloading. It is implemented by specifying the maximum number of connections the devices associated with one access point on the basis of the type of generated traffic. For this purpose, referred to in the previous stage of the types of traffic is attributed to the following weights: video w1/w2/w3–7/3.5/10, audio w4/w5–2/1.5, data w6–1. The sum of the products of consecutive weights by the number of connections does not exceed the value of the OI = 22 for one AP of IEEE 802.11 g network:

$$\sum\_{i=1}^{6} w\_i \cdot a\_i \le 22\tag{51}$$

The minimum number of devices needed to cover a given area determines the quotient of the sum of the weights by the numbers subsequent connections in that area by the maximum load of the access point. For the Bluetooth (LB) and WLAN (LI) systems the minimum number of access points is an expression (rounding up):

$$L\_{Bor} = \frac{\sum\_{i=4 \text{ or } 1}^{6} w\_i \cdot a\_i}{O\_{B \text{ or } I}} \tag{52}$$

Simulations were performed on a computer with an AMD Athlon processor, XP 2000+, and 768 MB of RAM. The duration of the simulation depends on the specific case and propagation model parameters. With the same propagation models parameters settings, the shortest time 1 s of simulation was obtained for MW model, 7 minutes and 14 seconds for the 2.5 RT model, and 16 minutes and 33 seconds for the 3D RT model. The ratio between the durations of the simulation the tested models also depends on the specific example. In the present environment MW model due to metal elevator shaft cannot be used, which is an obstacle for electromagnetic wave (Figure 13). In this case, the direct rays of the electromagnetic waves in the MW model are shielded. Comparison of simulation results 2.5D and 3D RT models has shown that theoretically worse 2.5D RT model almost in no way inferior the 3D version, but the computation time is two times longer. In highly complex environments with a large number of obstacles, the 2.5 RT model can be too big simplification, then you must apply the 3D model.

Figure 12. Photo of the Wroclaw University of Science and Technology building C-5 (a) and the numerical model of the

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

105

A number of simulation based on 2.5D and 3D RT models, the objective of which was to find the optimal deployment of WLAN AP points in the room 1, and the Bluetooth access points in rooms 2 and 3 implementing the project. The 2.5D RT model used in power distribution simulations on the ground floor. On the basis of simulation, it was found that for the area on one floor you must put only one AP of 10 dBm radiated power, mounted at a height of 3 m. The WLAN AP locations on the ground floor and the simulation results the EM wave power

8.2. Deployment of network devices

ground floor with use of defined construction materials an example (b).

distribution is shown in Figure 14.

#### 8. Implementation of the project

Determination of the optimal location for the deployment of the wireless system access points is a result of the consideration of the position resulting from the numerical simulation of the EM propagation models, as well as taking into account the experimental verification, under real conditions, of the implemented project. In most cases, wireless networks are designed as an extension to the wired LAN. Typically, the access points connect to the LAN network through a combination of their Ethernet cable UTP 5e with the appropriate port of the switch. This combination allows, in addition to access to this network, extension of the existing security in wireless network on the security available in a wired network above all security protocols related to virtual private networks VPN and authentication protocols such as 802.1 X [16, 17].

The information provided has been used to design a wireless network multimedia transmission in the building C-5 Faculty of Electronics the Wroclaw University of Science and Technology (Figure 12).

#### 8.1. Propagation model selection

Before start of the simulation examined, the usefulness of various propagation models available in the ACTIX ANALYZER computer code [6], in order to select the most suitable for the present environment, compares the results of the propagation models: non-linear multi-wall, MW (COST 231), 2.5D, and 3D models.

Figure 12 shows the model of the ground floor of the building C-5 of the selected location for the access point, and Figure 13a–c – the EM radio wave propagation simulation results using different propagation models for the transmitted power � 10 dBm, and frequency - 2.45 GHz.

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 105

Figure 12. Photo of the Wroclaw University of Science and Technology building C-5 (a) and the numerical model of the ground floor with use of defined construction materials an example (b).

Simulations were performed on a computer with an AMD Athlon processor, XP 2000+, and 768 MB of RAM. The duration of the simulation depends on the specific case and propagation model parameters. With the same propagation models parameters settings, the shortest time 1 s of simulation was obtained for MW model, 7 minutes and 14 seconds for the 2.5 RT model, and 16 minutes and 33 seconds for the 3D RT model. The ratio between the durations of the simulation the tested models also depends on the specific example. In the present environment MW model due to metal elevator shaft cannot be used, which is an obstacle for electromagnetic wave (Figure 13). In this case, the direct rays of the electromagnetic waves in the MW model are shielded. Comparison of simulation results 2.5D and 3D RT models has shown that theoretically worse 2.5D RT model almost in no way inferior the 3D version, but the computation time is two times longer. In highly complex environments with a large number of obstacles, the 2.5 RT model can be too big simplification, then you must apply the 3D model.

#### 8.2. Deployment of network devices

specifying the maximum number of connections the devices associated with one access point on the basis of the type of generated traffic. For this purpose, referred to in the previous stage of the types of traffic is attributed to the following weights: video w1/w2/w3–7/3.5/10, audio w4/w5–2/1.5, data w6–1. The sum of the products of consecutive weights by the number of connections does not exceed the value of the OI = 22 for one AP of IEEE 802.11 g network:

The minimum number of devices needed to cover a given area determines the quotient of the sum of the weights by the numbers subsequent connections in that area by the maximum load of the access point. For the Bluetooth (LB) and WLAN (LI) systems the minimum number of

> P 6 i¼4 or 1

Determination of the optimal location for the deployment of the wireless system access points is a result of the consideration of the position resulting from the numerical simulation of the EM propagation models, as well as taking into account the experimental verification, under real conditions, of the implemented project. In most cases, wireless networks are designed as an extension to the wired LAN. Typically, the access points connect to the LAN network through a combination of their Ethernet cable UTP 5e with the appropriate port of the switch. This combination allows, in addition to access to this network, extension of the existing security in wireless network on the security available in a wired network above all security protocols related

OB or I

wi � ai

wi � ai ≤ 22 (51)

(52)

X 6

i¼1

LB or I ¼

to virtual private networks VPN and authentication protocols such as 802.1 X [16, 17].

The information provided has been used to design a wireless network multimedia transmission in the building C-5 Faculty of Electronics the Wroclaw University of Science and Technol-

Before start of the simulation examined, the usefulness of various propagation models available in the ACTIX ANALYZER computer code [6], in order to select the most suitable for the present environment, compares the results of the propagation models: non-linear multi-wall,

Figure 12 shows the model of the ground floor of the building C-5 of the selected location for the access point, and Figure 13a–c – the EM radio wave propagation simulation results using different propagation models for the transmitted power � 10 dBm, and frequency - 2.45 GHz.

access points is an expression (rounding up):

104 Antennas and Wave Propagation

8. Implementation of the project

ogy (Figure 12).

8.1. Propagation model selection

MW (COST 231), 2.5D, and 3D models.

A number of simulation based on 2.5D and 3D RT models, the objective of which was to find the optimal deployment of WLAN AP points in the room 1, and the Bluetooth access points in rooms 2 and 3 implementing the project. The 2.5D RT model used in power distribution simulations on the ground floor. On the basis of simulation, it was found that for the area on one floor you must put only one AP of 10 dBm radiated power, mounted at a height of 3 m. The WLAN AP locations on the ground floor and the simulation results the EM wave power distribution is shown in Figure 14.

Figure 13. The simulation results of signal power distribution in a MW model (a), the 2.5 D RT model (b), and the 3D RT model (c) for radiated power 10 dBm at f = 2.45 GHz on the ground floor of building C-5 of WRUoS&T.

You have selected the points where the power values shown in Table 5 were read.

Empty areas in the grid of receivers (e.g., inside an elevator shaft) indicate areas where signal strength is lower than the established system sensitivity (94 dBm for IEEE 802.11 g). Model of a lift shaft turned out to be a screen for electromagnetic waves. Analysis of the simulation results showed that the AP does not cover the whole floor area. AP coverage areas on adjacent floors enough overlap to provide users with the ability to change the AP without losing the connection to the network. Analysis of the results of the simulation of power distribution showed that the vast majority of the coverage area, it is possible to obtain the maximum transmission speed of 54 MB/s.

network, which will consist of a device with 3th pico network. Because the area of these networks covers part of the corridor in front of the entrance to each of the rooms, it is possible

Figure 14. Distribution of AP IEEE 802.11 g (a), the simulation results of EM wave power distribution (b), the arrangement of the master point of Bluetooth system (c), and the simulation results for the room 3 on the ground floor of building C-5.

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

107

Selected Bluetooth and IEEE 802.11 g systems work in this same frequency 2.4 GHz band ISM (Industry Scientific Medicine). In addition, in IEEE 802.11 g are the only three non-overlapping

to move users between meeting rooms without losing the connection.

8.3. Calculation of the interference impact

Bluetooth network simulations were carried out for the room 3 (lecture halls on the ground floor). Assume that the master devices are PCs placed on desks lecture halls at a height of 1 m above the floor. The Bluetooth access point locations in the room 1, on the ground floor, and the simulation results of the EM wave power distribution are shown in Figure 14b. The power values selected read at points are shown in Table 5. Simulations have shown that full coverage has been achieved already at the radiated power 4 dBm of access points (class 2). Pico networks areas overlap themselves which allows you to combine them to form a distributed

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks http://dx.doi.org/10.5772/intechopen.75384 107

Figure 14. Distribution of AP IEEE 802.11 g (a), the simulation results of EM wave power distribution (b), the arrangement of the master point of Bluetooth system (c), and the simulation results for the room 3 on the ground floor of building C-5.

network, which will consist of a device with 3th pico network. Because the area of these networks covers part of the corridor in front of the entrance to each of the rooms, it is possible to move users between meeting rooms without losing the connection.

#### 8.3. Calculation of the interference impact

You have selected the points where the power values shown in Table 5 were read.

model (c) for radiated power 10 dBm at f = 2.45 GHz on the ground floor of building C-5 of WRUoS&T.

transmission speed of 54 MB/s.

106 Antennas and Wave Propagation

Empty areas in the grid of receivers (e.g., inside an elevator shaft) indicate areas where signal strength is lower than the established system sensitivity (94 dBm for IEEE 802.11 g). Model of a lift shaft turned out to be a screen for electromagnetic waves. Analysis of the simulation results showed that the AP does not cover the whole floor area. AP coverage areas on adjacent floors enough overlap to provide users with the ability to change the AP without losing the connection to the network. Analysis of the results of the simulation of power distribution showed that the vast majority of the coverage area, it is possible to obtain the maximum

Figure 13. The simulation results of signal power distribution in a MW model (a), the 2.5 D RT model (b), and the 3D RT

Bluetooth network simulations were carried out for the room 3 (lecture halls on the ground floor). Assume that the master devices are PCs placed on desks lecture halls at a height of 1 m above the floor. The Bluetooth access point locations in the room 1, on the ground floor, and the simulation results of the EM wave power distribution are shown in Figure 14b. The power values selected read at points are shown in Table 5. Simulations have shown that full coverage has been achieved already at the radiated power 4 dBm of access points (class 2). Pico networks areas overlap themselves which allows you to combine them to form a distributed

Selected Bluetooth and IEEE 802.11 g systems work in this same frequency 2.4 GHz band ISM (Industry Scientific Medicine). In addition, in IEEE 802.11 g are the only three non-overlapping


channels. Therefore, in order to evaluate the possibility of the coexistence of Bluetooth and IEEE 802.11 g systems, and the impact of inter-system interference in both systems, a number of simulations were carried out. For 10 WLAN AP in the network, each frequency channel repeats at least three times. For one AP per each floor on the same frequency channels, AP will work every 3rd floor. The ACTIX ANALYZER program allows you to estimate the level of interference on a given area. Selected is the dominant transmitter and then calculated the ratio of the power of its signal to the power of the signals of the other transmitters. The result is expressed in dB. The simulations allowed said there are no significant interference inside the

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

109

Interference between Bluetooth and IEEE 802.11 g systems can affect the work of the two systems. Figure 15 shows the interference between AP (EIRP = 10 dBm) located on the ground floor and

The analysis of the results indicates a strong interference signal, above all within the hall 2 and 3. In view of the above, the project should examine methods that co-exist in both systems at the

The ACTIX Analyzer program allows you to simulate taking into account the impact of the people on the EM wave propagation phenomenon. To do this, in the propagation model of the environment are inserted the human models (Figure 16a) recognized as the additional obstacles for the EM waves. In ACTIX Analyzer, it is possible to scale the standard human model in order to obtain models representing people of different growth. The man is a complicated living organism and it is difficult to simulate its effects on electromagnetic field. However, studies have shown that the best substance simulating human tissue (a large generalization) is a solution of water with salt in varying degrees of saturation depending on the conditions in question. In the project to simulate human tissue, the solution about the contents of the 4 gram NaCl per liter of water has been chosen. The properties of the substance at frequency 2.4 GHz are ε' = 77 and ε" = 13 (ε = 77-j13) [14]. Figure 16b shows the arrangement of a dozen models of people on the ground level of building C-5 to simulate their effects on the AP IEEE 802.11 g (area 1) and the Bluetooth master device in the areas 2 and 3. Figure 16c shows the results of the simulation taking into account and without taking into account the impact of people on the network coverage in the area 1 with the active AP IEEE 802.11 g networks on the ground floor of the C-5. The AP parameters are set as in previous simulations. The analysis of the results showed that the presence of people on the route of the transmitter-receiver can significantly degrade received power. In order to reduce the impact of humans on the power level of the received signal, install access points so that the route of the transceiver as little as possible was

Figure 16d shows the results of the simulation, taking into account the impact of people on the network coverage in the area 3 with the active Master device in the Bluetooth system # 3. The analysis of the results shows that the people have a strong effect on the propagation of the signal in the pico-network. In the case of a large number of people in the room radiated power

three Bluetooth master devices (EIRP = 4 dBm) forming 3 piconets in the halls 1, 2, and 3.

8.4. Impact of humans in the area of Bluetooth and WLAN systems operation

divided by areas with a high concentration of people.

WLAN system for room one.

same time and space.

Table 5. The2.5D and 3D RT propagation models parameters applied in the simulation.

Figure 15. Levels of interference between Bluetooth and IEEE 802.11 g systems on the ground floor of the C-5 building.

channels. Therefore, in order to evaluate the possibility of the coexistence of Bluetooth and IEEE 802.11 g systems, and the impact of inter-system interference in both systems, a number of simulations were carried out. For 10 WLAN AP in the network, each frequency channel repeats at least three times. For one AP per each floor on the same frequency channels, AP will work every 3rd floor. The ACTIX ANALYZER program allows you to estimate the level of interference on a given area. Selected is the dominant transmitter and then calculated the ratio of the power of its signal to the power of the signals of the other transmitters. The result is expressed in dB. The simulations allowed said there are no significant interference inside the WLAN system for room one.

Parameter Value Attention RL Range of elevation () 90/90 The full range Angular resolution in elevation () 1

> Angular resolution in azimuth () 1 Maximum distance between the rays (m) 0.1

Minimum signal level (dBm) 94 IEEE 802.11 g

Transmitting antenna power gain (dBi) 2.2 Half-wave dipole Receiving antenna power gain (dBi) 0 Isotropic antenna

Table 5. The2.5D and 3D RT propagation models parameters applied in the simulation.

Maximum number of reflections 2 Maximum number of penetration 2 Maximum number of diffraction 1

108 Antennas and Wave Propagation

Frequency (GHz) 2.45

Suspended height of receiving/transmitting antennas (m) 1/3 Size of the receivers grid sets (m) 0.3

Azimuth angle range () 0/360 The full range

Maximum number of rays on the route of the transmitter-receiver 4 Negligible impact of the rest of the rays

Figure 15. Levels of interference between Bluetooth and IEEE 802.11 g systems on the ground floor of the C-5 building.

70 Bluetooth

Interference between Bluetooth and IEEE 802.11 g systems can affect the work of the two systems. Figure 15 shows the interference between AP (EIRP = 10 dBm) located on the ground floor and three Bluetooth master devices (EIRP = 4 dBm) forming 3 piconets in the halls 1, 2, and 3.

The analysis of the results indicates a strong interference signal, above all within the hall 2 and 3. In view of the above, the project should examine methods that co-exist in both systems at the same time and space.

### 8.4. Impact of humans in the area of Bluetooth and WLAN systems operation

The ACTIX Analyzer program allows you to simulate taking into account the impact of the people on the EM wave propagation phenomenon. To do this, in the propagation model of the environment are inserted the human models (Figure 16a) recognized as the additional obstacles for the EM waves. In ACTIX Analyzer, it is possible to scale the standard human model in order to obtain models representing people of different growth. The man is a complicated living organism and it is difficult to simulate its effects on electromagnetic field. However, studies have shown that the best substance simulating human tissue (a large generalization) is a solution of water with salt in varying degrees of saturation depending on the conditions in question. In the project to simulate human tissue, the solution about the contents of the 4 gram NaCl per liter of water has been chosen. The properties of the substance at frequency 2.4 GHz are ε' = 77 and ε" = 13 (ε = 77-j13) [14]. Figure 16b shows the arrangement of a dozen models of people on the ground level of building C-5 to simulate their effects on the AP IEEE 802.11 g (area 1) and the Bluetooth master device in the areas 2 and 3. Figure 16c shows the results of the simulation taking into account and without taking into account the impact of people on the network coverage in the area 1 with the active AP IEEE 802.11 g networks on the ground floor of the C-5. The AP parameters are set as in previous simulations. The analysis of the results showed that the presence of people on the route of the transmitter-receiver can significantly degrade received power. In order to reduce the impact of humans on the power level of the received signal, install access points so that the route of the transceiver as little as possible was divided by areas with a high concentration of people.

Figure 16d shows the results of the simulation, taking into account the impact of people on the network coverage in the area 3 with the active Master device in the Bluetooth system # 3. The analysis of the results shows that the people have a strong effect on the propagation of the signal in the pico-network. In the case of a large number of people in the room radiated power

the most appropriate location of access points and their minimum number of required covering a given area. In the case of a large diversification of the propagation environment (e.g., shopping center) it is recommended that you use the more accurate but at the same time

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

111

This chapter presents the concepts, methods, and algorithms that were used to design a wireless system for indoor as well as outdoor mobile communication systems. The project supported simulations carried out using ACTIX Analyzer computer code. Indoor propagation environment is very complex (reflection, diffraction, refraction, etc.) while maintaining high accuracy digital model was built. In most cases, reduce the applied version of computer code that are not influenced significantly on the simulation results. A very interesting option of simulation is possibility taking into account of the impact of humans on the propagation of the signal. Typically, wireless networks are used in places with a high concentration of people (e.g., airports, universities, hotels, libraries, etc.) that is why skipping in simulations of the impact

Designed a network allows users to wireless access in the building of the C-5 to the Internet and Intranet networks of the Wroclaw University of Science and Technology based on the IEEE 802.11g standard. It also allows the implementation of data transfer and audio and video connections with the guarantee of quality of offered services. For conference and lectureseminar rooms, a separate network of Bluetooth system to ensure that the combination of elements of the multimedia presentations setups is designed. Part of the position includes a host computer, multimedia projectors, cameras, sound system, printers, monitors, wireless

It is my pleasure to thank of my former MSc student Piotr Horbatowski for his valuable

Department of Telecommunication and Teleinformatics, Faculty of Electronics, Wroclaw University of Science and Technology, Wybrzeze Wyspianskiego, Wroclaw, Poland

[1] Krzysztofik WJ, Horbatowski P. Design of the WLAN and WPAN wireless networks inside buildings - propagation models. Telecommunication Review & Telecommunication

microphones, and listeners equipment (e.g., laptops, palmtops, and mobile phones).

Address all correspondence to: wojciech.krzysztofik@pwr.edu.pl

News. LXXX, Nr 7/2007:221–231. ISSN 1230–3496

dealing with more time simulation based on deterministic models.

people can turn out to be a very serious mistake.

Acknowledgements

Author details

References

Wojciech Jan Krzysztofik

contribution to the presented topics.

Figure 16. The deployment of several human models (a) on the ground floor (b) of the C-5 to simulate their effect on the extent of IEEE 802.11 g (c) and Bluetooth networks (d).

of 4 dBm of master is insufficient to cover the entire area (room 3). Therefore, rooms 2 and 3 are applied the master of first class-Bluetooth system with a higher maximum radiated power.

## 9. Conclusion

The EM wave propagation models presented allow you to streamline the process of designing wireless networks of mobile communication systems like WLAN, Bluetooth RFhome, ZigBee, and WPAN inside buildings as well as the outdoor urban environment. Allow you to specify the most appropriate location of access points and their minimum number of required covering a given area. In the case of a large diversification of the propagation environment (e.g., shopping center) it is recommended that you use the more accurate but at the same time dealing with more time simulation based on deterministic models.

This chapter presents the concepts, methods, and algorithms that were used to design a wireless system for indoor as well as outdoor mobile communication systems. The project supported simulations carried out using ACTIX Analyzer computer code. Indoor propagation environment is very complex (reflection, diffraction, refraction, etc.) while maintaining high accuracy digital model was built. In most cases, reduce the applied version of computer code that are not influenced significantly on the simulation results. A very interesting option of simulation is possibility taking into account of the impact of humans on the propagation of the signal. Typically, wireless networks are used in places with a high concentration of people (e.g., airports, universities, hotels, libraries, etc.) that is why skipping in simulations of the impact people can turn out to be a very serious mistake.

Designed a network allows users to wireless access in the building of the C-5 to the Internet and Intranet networks of the Wroclaw University of Science and Technology based on the IEEE 802.11g standard. It also allows the implementation of data transfer and audio and video connections with the guarantee of quality of offered services. For conference and lectureseminar rooms, a separate network of Bluetooth system to ensure that the combination of elements of the multimedia presentations setups is designed. Part of the position includes a host computer, multimedia projectors, cameras, sound system, printers, monitors, wireless microphones, and listeners equipment (e.g., laptops, palmtops, and mobile phones).

## Acknowledgements

It is my pleasure to thank of my former MSc student Piotr Horbatowski for his valuable contribution to the presented topics.

## Author details

Wojciech Jan Krzysztofik

Address all correspondence to: wojciech.krzysztofik@pwr.edu.pl

Department of Telecommunication and Teleinformatics, Faculty of Electronics, Wroclaw University of Science and Technology, Wybrzeze Wyspianskiego, Wroclaw, Poland

## References

of 4 dBm of master is insufficient to cover the entire area (room 3). Therefore, rooms 2 and 3 are applied the master of first class-Bluetooth system with a higher maximum radiated power.

Figure 16. The deployment of several human models (a) on the ground floor (b) of the C-5 to simulate their effect on the

The EM wave propagation models presented allow you to streamline the process of designing wireless networks of mobile communication systems like WLAN, Bluetooth RFhome, ZigBee, and WPAN inside buildings as well as the outdoor urban environment. Allow you to specify

9. Conclusion

110 Antennas and Wave Propagation

extent of IEEE 802.11 g (c) and Bluetooth networks (d).

[1] Krzysztofik WJ, Horbatowski P. Design of the WLAN and WPAN wireless networks inside buildings - propagation models. Telecommunication Review & Telecommunication News. LXXX, Nr 7/2007:221–231. ISSN 1230–3496


[17] Krzysztofik WJ, Horbatowski P. Design of the WLAN and WPAN wireless networks inside buildings – examples of realization. Telecommunication Review & Telecommuni-

Radio Network Planning and Propagation Models for Urban and Indoor Wireless Communication Networks

http://dx.doi.org/10.5772/intechopen.75384

113

[18] Ganz A, Ganz Z, Wongthavarawat K. Multimedia Wireless Networks-Technologies, Stan-

[20] Riverbed Modeler (old name: OPNET Modeler®) Computer Code Users Manual [Inter-

[21] Lai Z, Villemaud G, Luo M, Zhang J. Radio propagation modelling, Chapter 2 in book. In: Chu X, Lopez-Perez D, Yang Y, Gunnarsson F, editors. Heterogeneous Cellular Networks - Theory, Simulation, and Deployment. The Edinburgh Building, Cambridge, UK: Cam-

[22] Salous S. Radio Propagation and Channel Modelling. River Street Hoboken, NJ, United States: John Wiley & Sons, Ltd. ISBN: 978-1-118-50232-7. DOI: 10.1002/9781118. March 2013, Available online: http://onlinelibrary.wiley.com/book/10.1002/9781118502280

[23] Pagani P, Talom FT, Pajusco P, Uguen B. Ultra-Wideband Radio Propagation Channel: A Practical Approach. River Street Hoboken, NJ, United States: John Wiley & Sons, Ltd.

[24] Primak S, Kontorovich V. Wireless Multi-Antenna Channels: Modeling and Simulation, Wiley Series on Wireless Communication and Mobile Computing. Vol. 28. River Street Hoboken, NJ, United States: John Wiley & Sons, Ltd. ISBN: 978-1-119-96086-7; Oct. 2011

[25] Fontain EP, Espineira PM. Modelling the Wireless Propagation Channel: A Simulation Approach with MATLAB, Wiley Series on Wireless Communication and Mobile Computing. River Street Hoboken, NJ, United States: John Wiley & Sons, Ltd. ISBN: 978-0-470-

[26] Roche G, Alayon-Glazunov A, Allen B. LTE-Advanced and Next Generation Wireless Networks: Channel Modeling and Propagation. River Street Hoboken, NJ, United States:

[27] Yin X, Cheng X. Propagation Channel Characterization, Parameter Estimation, and Modelling for Wireless Communication. River Street Hoboken, NJ, United States: John

[28] Volcano Propagation Model is a Renowned Radio Propagation Modeling Software Available Worldwide in the Leading Commercial Radio Planning Tools, and in SIRADEL's Standalone Platform Smart City Explorer. Available on: https://www.siradel.com/soft-

[29] ADTI ICS online Share your Network and Radio Planning [Internet]. Available on: http://

John Wiley & Sons, Ltd. ISBN: 978-1-118-41101-8; Nov. 2012

Wiley & Sons, Ltd., IEEE Press. ISBN: 978-1-118-18826-2; Sep. 2016

dards and QoS. USA: Prentice Hall PTR; 2003. ISBN-13: 978-0130460998

[19] Bluetooth SIG, Bluetooth Core Specification version 6.0, USA; July 2017

cation News, LXXX, Nr 10/2007:878-895. ISSN 1230–3496

net]. 2018. Available from: www.riverbed.com

bridge University Press; 2012

ISBN: 978-1-848-21084-4; Jan. 2009

75173-2; Sep. 2008

ware/

www.atdi.com/ics-online/


[2] Wesołowski K. Mobile Radio Communication Systems. Warszawa, Poland: WKŁ; 2003

https://customer.active24.com/

112 Antennas and Wave Propagation

1996

Technology; 2006

FEKO/WinProp—Indoor-and-Campus

Broadcasting, Poznan, Poland; 2006

nologies for IMT-2000, Geneva, Switzerland; 1997

[3] I-Prop Computer Code User Manual [Internet]. 2018. Available from: www.i-prop.cz;

[4] ITU-R M.1225 Recommendation, Guidelines for Evaluation of Radio Transmission Tech-

[5] Wölfle G. Propagation Models for Indoor Radio Network Planning Including Tunnels, Millennium Conference on Antennas & Propagation, Davos, Switzerland; 2000

[6] ACTIX ANALYZER, The World's First Multi-Vendor Desktop Analytics Solution for

[7] Sarkar T, Jl Z, Kim K. A survey of various propagation models for mobile communication.

[8] Cichon D, Kürner T. COST 231 Final Report, Chapter 4, Propagation Prediction Models;

[9] Staniec K. The indoor radio wave propagation modeling in ISM bands for broadband wireless systems [PhD. Thesis]. Wroclaw, Poland: Wroclaw University of Science and

[10] WinProp - Propagation Modelling, Method of Optical Rays, Different Scenarios, 2D and 3D Models, Empirical and Ray-Optical, Planning & Simulation, Coverage & Capacity, Cellular & Broadcasting. [Internet] 2018, Available on: https://altairhyperworks.com/ product/FEKO/WinProp-Propagation-Modelinghttps://altairhyperworks.com/product/

[11] Bansal R, editor. Fundamentals of Engineering Electromagnetics. Boca Raton, FL USA.

[12] Popescu I. Neural network applications for radio coverage studies in mobile communication systems [Ph.D. Thesis]. Romania: Polytechnic University Timisoara; 2003

[13] Aryanfar F. Modelling of wireless channels and validation using a scaled MM-wave measurement system [Ph.D. Thesis]. Ann Arbor, USA: University of Michigan; 2004

[14] Krzysztofik W, Horbatowski P. Propagation Models used for WLAN Design Inside Buildings, KKRRiT 2006, The National Conference on Radiocommunication, Radio- and TV-

[15] Hoppe R, Wertz P, Landstorfer FM, Wolfle G. Advanced ray optical wave propagation modelling for urban and indoor scenarios including wideband properties. Transactions on

[16] Horbatowski P. The systems of communication and multimedia services in the mobile business and tourism based on wireless access of BLUETOOTH and/or WLAN standards

[Thesis]. Wroclaw, Poland: Wroclaw University of Science and Technology; 2006

ISBN-10: 0-8493-7360-3: CRC Press, Taylor & Francis Group; 2006

Emerging Telecommunications Technologies. 2003;14(1):61-69

Mobile Networks [Internet]. 2018. Available from http://actix.com/analyzer

IEEE Antennas and Propagation Magazine. 2003;45(3):51-82


[30] iBWAVE DE iBwave Design, the Industry Standard for Designing Indoor Wireless Networks. 2018. [Internet], Available on: http://www.ibwave.com/ibwave-design

**Chapter 6**

Provisional chapter

**Multi-Elliptical Geometry of Scatterers in Modeling**

DOI: 10.5772/intechopen.75142

In the proposed chapter, the authors present a geometric-statistical propagation model that defines three groups of received signal components, i.e., direct path, delayed scattering, and local scattering components. The multi-elliptical propagation model, which represents the geometry of scatterer locations, is the basis for determining the delayed components. For the generation of the local components, a statistical distribution is used. The basis for this model is a power angular spectrum (PAS) of the received signal, which is closely related to a type of propagation environment and transmitter-receiver spatial positions. Therefore, we have an opportunity to evaluate the influence of the environment type and an object motion direction on the basic characteristics such as envelope distribution, PAS, autocorrelation function, and spectral power density. The multi-elliptical model considers the propagation phenomena occurring in the azimuth plane. In the chapter, we

will also show the 3D extension of modeling effects of propagation phenomena. Keywords: radio wave propagation, propagation modeling, channel modeling, geometric-based model, multi-elliptical model, multi-ellipsoidal model, scattering, angular power spectrum, angle spread, angular dispersion, directional antenna pattern,

A development of information and communication systems is characterized by a dynamic increase in demand for the provision of telecommunication services with the participation of wireless networks. A limitation of frequency resources forces the search for new methods of effective spectrum management. One of the solutions to this problem is spatial multiplexing of network access, which minimizes a field strength and increases access area of the network.

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Multi-Elliptical Geometry of Scatterers in Modeling

**Propagation Effect at Receiver**

Propagation Effect at Receiver

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Jan M. Kelner and Cezary Ziółkowski

Jan M. Kelner and Cezary Ziółkowski

http://dx.doi.org/10.5772/intechopen.75142

power delay profile, simulation

Abstract

1. Introduction


#### **Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver** Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

DOI: 10.5772/intechopen.75142

Jan M. Kelner and Cezary Ziółkowski Jan M. Kelner and Cezary Ziółkowski

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75142

#### Abstract

[30] iBWAVE DE iBwave Design, the Industry Standard for Designing Indoor Wireless Networks. 2018. [Internet], Available on: http://www.ibwave.com/ibwave-design

[31] TAPTMMAPPER, Terrain Analysis package, RF Path Preview and Quick Coverage Analysis, Worldwide Terrain, Land Cover, and Building Data Access, Multi-Site Simulcast and Area Coverage Analysis, [Internet]. 2017. Available on: http://www.softwright.com/ [32] EDX® SignalPro®, is a Comprehensive and Fully Featured RF Planning Software Suite Offering all the Study Types Needed to Design Wireless Networks, Including; Area Studies, Link/Point-to-Point Studies, Point-to-Multipoint and Route Studies [Internet].

[33] Wireless InSite® is a Suite of Ray-Tracing Models and High-Fidelity EM Solvers for the Analysis of Site-Specific Radio Wave Propagation and Wireless Communication Systems. The RF Propagation Software Provides Efficient and Accurate Predictions of EM Propagation and Communication Channel Characteristics in Complex Urban, Indoor, Rural and Mixed Path Environments [Internet]. 2018. Available on: https://www.remcom.com/wire

[34] CelPlanner™, Software for Planning & Designing Wireless Communication Systems. 2018

[Internet]. Available on: http://www.celplan.com/products/celplanner.asp

2018, Available on: http://edx.com/products/edx-signalpro/

less-insite-em-propagation-software/

114 Antennas and Wave Propagation

In the proposed chapter, the authors present a geometric-statistical propagation model that defines three groups of received signal components, i.e., direct path, delayed scattering, and local scattering components. The multi-elliptical propagation model, which represents the geometry of scatterer locations, is the basis for determining the delayed components. For the generation of the local components, a statistical distribution is used. The basis for this model is a power angular spectrum (PAS) of the received signal, which is closely related to a type of propagation environment and transmitter-receiver spatial positions. Therefore, we have an opportunity to evaluate the influence of the environment type and an object motion direction on the basic characteristics such as envelope distribution, PAS, autocorrelation function, and spectral power density. The multi-elliptical model considers the propagation phenomena occurring in the azimuth plane. In the chapter, we will also show the 3D extension of modeling effects of propagation phenomena.

Keywords: radio wave propagation, propagation modeling, channel modeling, geometric-based model, multi-elliptical model, multi-ellipsoidal model, scattering, angular power spectrum, angle spread, angular dispersion, directional antenna pattern, power delay profile, simulation

## 1. Introduction

A development of information and communication systems is characterized by a dynamic increase in demand for the provision of telecommunication services with the participation of wireless networks. A limitation of frequency resources forces the search for new methods of effective spectrum management. One of the solutions to this problem is spatial multiplexing of network access, which minimizes a field strength and increases access area of the network.

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This solution is based on the use of an active phased array antenna (APAA) or massive APAA and is applicable to emerging fifth generation (5G) systems. In these systems, the multiple use of the same frequency bands is qualified by no interference between individual wireless links. Fulfillment of this condition has a large impact on received signals, properties of which significantly depend on a direction of reaching an electromagnetic wave to a reception point. This means that the prediction, modeling, and evaluation of statistical properties of the receipt direction play an important role in assessing internal and external conditions of a compatible operation of wireless networks. This fact justifies the purposefulness of topics presented in this chapter.

main disadvantages of these models are the lack of consideration of antenna pattern influence

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

http://dx.doi.org/10.5772/intechopen.75142

117

These disadvantages do not have geometrical models. These models are defined by geometrical structures that determine positions of elements scattering an electromagnetic wave. Knowledge of propagation environment geometry gives the possibility to determine the signal reception directions. This task can be carried out analytically or through simulation research.

The use of geometrical optics is one of the basic methods of determining AOA, which considers the geometric structure of propagation environment. A ray tracing is practical implementation of this propagation modeling method [8–12]. The accuracy of the obtained results depends to a large extent on the accuracy of mapping electrical properties of all elements forming the electromagnetic environment and the number of generated rays. The difficulty of obtaining environmental data and the complexity of simulation procedures are the reason for limited reliability of the results obtained by this method. A concept of propagation paths [13] gives the possibility of simplifying the ray-tracing method. In this case, the analysis comes down to an evaluation of propagation paths in the presence of scattering elements (scatterers) whose position on the plane (two-dimensional (2D) modeling) or in space (three-dimensional

A shape of scatterer occurrence areas, their position relative to the Tx and Rx locations, and a density distribution are the criteria that differentiate individual models. For the 2D areas, geometric structures such as a circle [5, 6, 14–16], ellipse [15, 16], hollow disk [17], and elliptical disk [18] are most commonly used. In the case of 3D, the scatterer areas represent a sphere [19], semi-spheroid [20, 21], clipping semi-spheroid [22], semi-ellipsoid [23], cylinder [24, 25], and complex solid figures, for example, bounded ellipsoid and elliptical cylinder [26] or sphere and

The density of the scatterers is another criterion that differentiates the individual geometric models. The uniform distribution is most widely used to describe spatial concentration of the scattering elements, especially with regard to limited geometrical structures, e.g., [14, 15, 22, 28–30]. For models where the scatterer occurrence area is unrestricted, the normal distribution

The area geometry and density of the scatterer occurrence give the possibility to determine the approach directions of the propagation paths to the Rx. However, the practical use of the geometric models to assess the statistical properties of the reception angle is largely hampered. This fact results from the lack of a relationship between the geometry of these models and transmission properties of the propagation environment. These properties are described by a channel impulse response and related characteristics such as a power delay profile (PDP) or power delay spectrum (PDS). Nevertheless, in the set of all models, we can distinguish geometrical structures whose parameters are defined by the channel transmission characteristics. Multi-elliptical and multi-ellipsoidal propagation models are these special cases. Consideration of the relationship between the transmission parameters and geometrical structures ensures minimization of the approximation error between measurement data and modeling

and distance between a transmitter (Tx) and receiver (Rx).

(3D) modeling) maps specific geometric structures.

ellipsoid [27].

results.

is used, e.g. [4, 22, 31].

The goal of the chapter is to describe the methods for determining the statistical properties of the signal reception angle and its parameters. Particular attention is given to an impact analysis of directional antenna parameters on the statistical parameters and characteristics that describe dispersion of reception angle. The main purpose of this chapter is based on geometric propagation models in which scatterer locations are determined by multi-elliptical curves or multi-ellipsoidal surfaces.

Due to a method of determining the statistical characteristics of a reception angle, propagation models can be divided in accordance with the diagram presented in Figure 1.

Measurement data and standard distributions such as the Gaussian, Laplacian, logistic, and von Mises distribution are the basis for empirical models that directly describe a probability density function (PDF) of angle of arrival (AOA) [1–3]. In this case, developing a model consists in such adjustment of distribution parameters that will ensure minimization of approximation error to the measurement data. For the first three models mentioned above, the approximation problem comes down to determining the normalizing constant and parameters that define these distributions [1]. For the von Mises distribution [4], the approximation consists in adjusting a single parameter, which simplifies the procedure of creating the statistical model of the reception angle distribution [1].

In practice, complex empirical models are used. They use standard distributions to describe the statistical properties of individual angular clusters. Examples of such models are defined, i.e., by the WINNER projects [5] and 3rd Generation Partnership Project (3GPP) [6, 7]. The

Figure 1. Classification of propagation models due to method of determining statistical properties of reception angle.

main disadvantages of these models are the lack of consideration of antenna pattern influence and distance between a transmitter (Tx) and receiver (Rx).

This solution is based on the use of an active phased array antenna (APAA) or massive APAA and is applicable to emerging fifth generation (5G) systems. In these systems, the multiple use of the same frequency bands is qualified by no interference between individual wireless links. Fulfillment of this condition has a large impact on received signals, properties of which significantly depend on a direction of reaching an electromagnetic wave to a reception point. This means that the prediction, modeling, and evaluation of statistical properties of the receipt direction play an important role in assessing internal and external conditions of a compatible operation of wireless networks. This fact justifies the purposefulness of topics presented in this chapter. The goal of the chapter is to describe the methods for determining the statistical properties of the signal reception angle and its parameters. Particular attention is given to an impact analysis of directional antenna parameters on the statistical parameters and characteristics that describe dispersion of reception angle. The main purpose of this chapter is based on geometric propagation models in which scatterer locations are determined by multi-elliptical curves or

Due to a method of determining the statistical characteristics of a reception angle, propagation

Measurement data and standard distributions such as the Gaussian, Laplacian, logistic, and von Mises distribution are the basis for empirical models that directly describe a probability density function (PDF) of angle of arrival (AOA) [1–3]. In this case, developing a model consists in such adjustment of distribution parameters that will ensure minimization of approximation error to the measurement data. For the first three models mentioned above, the approximation problem comes down to determining the normalizing constant and parameters that define these distributions [1]. For the von Mises distribution [4], the approximation consists in adjusting a single parameter, which simplifies the procedure of creating the statis-

In practice, complex empirical models are used. They use standard distributions to describe the statistical properties of individual angular clusters. Examples of such models are defined, i.e., by the WINNER projects [5] and 3rd Generation Partnership Project (3GPP) [6, 7]. The

Figure 1. Classification of propagation models due to method of determining statistical properties of reception angle.

models can be divided in accordance with the diagram presented in Figure 1.

multi-ellipsoidal surfaces.

116 Antennas and Wave Propagation

tical model of the reception angle distribution [1].

These disadvantages do not have geometrical models. These models are defined by geometrical structures that determine positions of elements scattering an electromagnetic wave. Knowledge of propagation environment geometry gives the possibility to determine the signal reception directions. This task can be carried out analytically or through simulation research.

The use of geometrical optics is one of the basic methods of determining AOA, which considers the geometric structure of propagation environment. A ray tracing is practical implementation of this propagation modeling method [8–12]. The accuracy of the obtained results depends to a large extent on the accuracy of mapping electrical properties of all elements forming the electromagnetic environment and the number of generated rays. The difficulty of obtaining environmental data and the complexity of simulation procedures are the reason for limited reliability of the results obtained by this method. A concept of propagation paths [13] gives the possibility of simplifying the ray-tracing method. In this case, the analysis comes down to an evaluation of propagation paths in the presence of scattering elements (scatterers) whose position on the plane (two-dimensional (2D) modeling) or in space (three-dimensional (3D) modeling) maps specific geometric structures.

A shape of scatterer occurrence areas, their position relative to the Tx and Rx locations, and a density distribution are the criteria that differentiate individual models. For the 2D areas, geometric structures such as a circle [5, 6, 14–16], ellipse [15, 16], hollow disk [17], and elliptical disk [18] are most commonly used. In the case of 3D, the scatterer areas represent a sphere [19], semi-spheroid [20, 21], clipping semi-spheroid [22], semi-ellipsoid [23], cylinder [24, 25], and complex solid figures, for example, bounded ellipsoid and elliptical cylinder [26] or sphere and ellipsoid [27].

The density of the scatterers is another criterion that differentiates the individual geometric models. The uniform distribution is most widely used to describe spatial concentration of the scattering elements, especially with regard to limited geometrical structures, e.g., [14, 15, 22, 28–30]. For models where the scatterer occurrence area is unrestricted, the normal distribution is used, e.g. [4, 22, 31].

The area geometry and density of the scatterer occurrence give the possibility to determine the approach directions of the propagation paths to the Rx. However, the practical use of the geometric models to assess the statistical properties of the reception angle is largely hampered. This fact results from the lack of a relationship between the geometry of these models and transmission properties of the propagation environment. These properties are described by a channel impulse response and related characteristics such as a power delay profile (PDP) or power delay spectrum (PDS). Nevertheless, in the set of all models, we can distinguish geometrical structures whose parameters are defined by the channel transmission characteristics. Multi-elliptical and multi-ellipsoidal propagation models are these special cases. Consideration of the relationship between the transmission parameters and geometrical structures ensures minimization of the approximation error between measurement data and modeling results.

This chapter is devoted to the evaluation of the statistical properties of the scattering, a reception angle, and the effects of this phenomenon, which has a significant impact on correlational and spectral properties of the received signals. The multi-elliptical and multiellipsoidal propagation models are the basis for the analysis presented in the chapter.

For majority of wireless links, relations between heights of the Tx (hT)/Rx (hR) antennas and their distance, D, meet a condition, hT, hR ≪ D. In the case of ground wave propagation, this condition is the basis for reduction of the scattering areas to semi-ellipsoids. The use of the antennas, whose radiation patterns are narrow in the elevation plane, brings the 3D to 2D modeling. For these conditions, the multi-ellipsoidal model is reduced to the multi-elliptical

The delays of the individual time clusters, τ<sup>i</sup> for i ¼ 1, 2, …, N, define the parameters of the

Each time cluster is the superposition of the signal components that reach the Rx from the scatterers located on the respective semi-ellipsoids. The reception directions of these components

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cτið Þ 2D þ cτ<sup>i</sup>

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

p (1)

http://dx.doi.org/10.5772/intechopen.75142

119

ð Þ <sup>D</sup> <sup>þ</sup> <sup>c</sup>τ<sup>i</sup> , bi <sup>¼</sup> ci <sup>¼</sup> <sup>1</sup>

model of scattering areas, as shown in Figure 3.

where c is the speed of light.

Figure 3. 3D and 2D models of scattering areas.

corresponding semi-ellipsoids. For the ith semi-ellipsoid, we have

ai <sup>¼</sup> <sup>1</sup> 2

## 2. Environment transmission properties and propagation model geometry

Statistical evaluation of the radio channel transmission properties is based on the energetic measures of the received signals. PDPs and PDSs describe powers and delays of individual components that reach the Rx. An example of PDP, Pð Þτ , defined by 3GPP [7, Table 7.7.2–2] is shown in Figure 2. In this case, the PDP represents non-line-of-sight (NLOS) conditions, urban macro (UMa)-environment type with delay spread, στ ¼ 363 ns which is characteristic of the carrier frequency, f <sup>0</sup> ¼ 2 GHz.

The PDP graph shows that the received signal is a superposition of component groups that form time clusters with different delays. Measurement results presented in a literature justify the following assumptions:


Thus, all signal components that arrive at the Rx with the same delay come from the scatterers located on the same ellipsoid. This means that the number of the ellipsoids that represent the scatterer locations is equal to the number of the time clusters. Their foci determine the Tx and Rx positions.

Figure 2. Example of PDP for UMa NLOS 2 GHz environment from 3GPP.

For majority of wireless links, relations between heights of the Tx (hT)/Rx (hR) antennas and their distance, D, meet a condition, hT, hR ≪ D. In the case of ground wave propagation, this condition is the basis for reduction of the scattering areas to semi-ellipsoids. The use of the antennas, whose radiation patterns are narrow in the elevation plane, brings the 3D to 2D modeling. For these conditions, the multi-ellipsoidal model is reduced to the multi-elliptical model of scattering areas, as shown in Figure 3.

The delays of the individual time clusters, τ<sup>i</sup> for i ¼ 1, 2, …, N, define the parameters of the corresponding semi-ellipsoids. For the ith semi-ellipsoid, we have

$$a\_i = \frac{1}{2}(D + c\tau\_i), \ b\_i = c\_i = \frac{1}{2}\sqrt{c\tau\_i(2D + c\tau\_i)}\tag{1}$$

where c is the speed of light.

This chapter is devoted to the evaluation of the statistical properties of the scattering, a reception angle, and the effects of this phenomenon, which has a significant impact on correlational and spectral properties of the received signals. The multi-elliptical and multi-

2. Environment transmission properties and propagation model geometry

Statistical evaluation of the radio channel transmission properties is based on the energetic measures of the received signals. PDPs and PDSs describe powers and delays of individual components that reach the Rx. An example of PDP, Pð Þτ , defined by 3GPP [7, Table 7.7.2–2] is shown in Figure 2. In this case, the PDP represents non-line-of-sight (NLOS) conditions, urban macro (UMa)-environment type with delay spread, στ ¼ 363 ns which is characteristic of the

The PDP graph shows that the received signal is a superposition of component groups that form time clusters with different delays. Measurement results presented in a literature justify

• The probability of the scatterer occurrence seen from the Tx is the same in every direction. • For each element, the statistical properties of scattering factor module and phase are the

Thus, all signal components that arrive at the Rx with the same delay come from the scatterers located on the same ellipsoid. This means that the number of the ellipsoids that represent the scatterer locations is equal to the number of the time clusters. Their foci determine the Tx and

Figure 2. Example of PDP for UMa NLOS 2 GHz environment from 3GPP.

• Components that undergo single scattering have a dominant energetic significance.

ellipsoidal propagation models are the basis for the analysis presented in the chapter.

carrier frequency, f <sup>0</sup> ¼ 2 GHz.

118 Antennas and Wave Propagation

the following assumptions:

same.

Rx positions.

Each time cluster is the superposition of the signal components that reach the Rx from the scatterers located on the respective semi-ellipsoids. The reception directions of these components

Figure 3. 3D and 2D models of scattering areas.

are determined by the shape of the scatterer occurrence surface. This means that the powers of the individual clusters depend on the propagation path direction to the Rx. Differentiation of the cluster delays is the basis for expressing a power angular spectrum (PAS) as a sum of the component powers reaching the Rx with the delays, PR θR; φ<sup>R</sup> � �, and the component powers, PR<sup>0</sup> θR; φ<sup>R</sup> � �, whose delays are of the order of a carrier wave period. The first and second groups of components are called the delayed and local scattering components, respectively.

The delayed components are grouped in the time clusters. Therefore,

$$P\_R(\theta\_R, \varphi\_\mathbb{R}) = \sum\_{i=1}^N P\_{Ri}(\theta\_R, \varphi\_\mathbb{R}) + P\_{R0}(\theta\_R, \varphi\_\mathbb{R}) \tag{2}$$

The correctness of the adopted model is confirmed by comparative analyses with empirical data presented, among others, in [33–36]. From these analyses, it appears that the multielliptical model provides the smallest errors of PAS and PDF of AOA approximation to other geometric and empirical models. The correctness of the adopted model is confirmed by com-

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

http://dx.doi.org/10.5772/intechopen.75142

121

The use of the antennas with the narrow radiation patterns in the elevation plane limits an environment influence on the received signal properties. Propagation phenomena predominance in the azimuth plane is the premise for reducing the analysis of the reception angle statistical properties to the 2D modeling problem. In this case, the mapping of the propagation phenomena is ensured by the multi-elliptical propagation model. Precursors of this model are Parsons and Bajwa, who presented a multi-elliptical way of modeling the distribution of the

3.1. Analysis of reception angle statistical properties for omni-directional antennas

where ei ¼ D=ð Þ 2ai means the eccentricity of the ith ellipse and xij is coordinate of Sij.

D

Considering that rTij þ rRij ¼ 2ai and substituting Eq. (5) to Eq. (4), we can write

þ

After transforming Eq. (6), the formula of cosφTij versus cosφRij has the form [32]:

1 1 � ei cosφRij

cosφTij <sup>¼</sup> <sup>2</sup>ai cosφRij <sup>þ</sup> Dei cosφRij � <sup>2</sup><sup>D</sup>

2ai þ Dei � 2D cosφRij

AOA for the delayed components comes down to determining f <sup>i</sup> φ<sup>R</sup>

xij ¼ rTij cosφTij þ

1 1 þ ei cosφTij

A PDF of AOA analysis for radio links with omni-directional antennas is based on the 2D geometric structure shown in Figure 3. In this case, Eq. (3) shows that determining the PDF of

Propagation path lengths, i.e., rTij and rRij (see Figure 3), which describe the distances TxSij ,

.

<sup>2</sup> (5)

(6)

(7)

rTij ¼ ai � eixij and rRij ¼ ai þ eixij (4)

<sup>2</sup> and xij <sup>¼</sup> rRij cosφRij � <sup>D</sup>

<sup>¼</sup> <sup>4</sup>ai 2ai � Dei

parative analyses with empirical data presented in, e.g., [37].

3. Multi-elliptical propagation model

propagation paths in [38].

respectively, are

But xij is a function of φTij and φRij:

where N is the number of the time clusters (semi-ellipsoids) and PRi θR;φ<sup>R</sup> � � means the PAS of the propagation paths that reach the Rx from the ith ellipsoid.

The PAS can be presented as the product of a total power and PDF of AOA [32]. We should also note that PR<sup>0</sup> θR;φ<sup>R</sup> � � represents the sum of the powers of the direct path component and the local scattering components. The energy relationship between these components describes the Rice factor, κ. Thus, we can present the PDF of AOA, f <sup>R</sup> θR; φ<sup>R</sup> � �, which describes the statistical properties of the signal reception angle, in the following form:

$$\begin{split} f\_{R}(\boldsymbol{\theta}\_{\mathbb{R}},\boldsymbol{\varphi}\_{\mathbb{R}}) &= f\_{d}(\boldsymbol{\theta}\_{\mathbb{R}},\boldsymbol{\varphi}\_{\mathbb{R}}) + f\_{l}(\boldsymbol{\theta}\_{\mathbb{R}},\boldsymbol{\varphi}\_{\mathbb{R}}) + f\_{dp}(\boldsymbol{\theta}\_{\mathbb{R}},\boldsymbol{\varphi}\_{\mathbb{R}}) \\ &= \sum\_{i=1}^{N} \frac{P\_{i}}{P} f\_{i}(\boldsymbol{\theta}\_{\mathbb{R}},\boldsymbol{\varphi}\_{\mathbb{R}}) + \frac{\kappa}{1+\kappa} \frac{P\_{0}}{P} f\_{0}(\boldsymbol{\theta}\_{\mathbb{R}},\boldsymbol{\varphi}\_{\mathbb{R}}) + \frac{1}{1+\kappa} \frac{P\_{0}}{P} \boldsymbol{\delta}(\boldsymbol{\theta}\_{\mathbb{R}},\boldsymbol{\varphi}\_{\mathbb{R}}) \end{split} \tag{3}$$

where f <sup>d</sup> θR;φ<sup>R</sup> � �, <sup>f</sup> <sup>l</sup> <sup>θ</sup>R;φ<sup>R</sup> � �, and <sup>f</sup> dp <sup>θ</sup>R;φ<sup>R</sup> � � represent parts of <sup>f</sup> <sup>R</sup> <sup>θ</sup>R; <sup>φ</sup><sup>R</sup> � � corresponding the delayed scattering components, local scattering components, and direct path component, respectively, P is the total power of the received signal, Pi is the power of the propagation paths reaching the Rx from the ith ellipsoids, f <sup>i</sup> θR;φ<sup>R</sup> � � means PDF of AOA for the ith ellipsoids, P<sup>0</sup> represents the power of the components reaching the Rx with negligible delay, f <sup>0</sup> θR;φ<sup>R</sup> � � is PDF of AOA for the local scattering components, and <sup>δ</sup>ð Þ� is the Dirac delta function.

For the delayed components, the multi-ellipsoidal or multi-elliptical structures are the basis for the analytical or simulation determination of individual f <sup>i</sup> θR;φ<sup>R</sup> � �, <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, N. In the case of the local scattering, a large diversity of receiving antenna surroundings prevents adoption of a determined geometry of the scatterer positions. To adapt the statistical properties of reception angle, the von Mises distribution is used [4].

Construction of the multi-ellipsoidal or multi-elliptical structures based on PDP/PDS ensures adjustment of the reception angle statistical characteristics to the transmission properties of the propagation environment. In that, these models provide a mapping of the impact of these properties on the correlational and spectral characteristics of the received signals.

The correctness of the adopted model is confirmed by comparative analyses with empirical data presented, among others, in [33–36]. From these analyses, it appears that the multielliptical model provides the smallest errors of PAS and PDF of AOA approximation to other geometric and empirical models. The correctness of the adopted model is confirmed by comparative analyses with empirical data presented in, e.g., [37].

## 3. Multi-elliptical propagation model

are determined by the shape of the scatterer occurrence surface. This means that the powers of the individual clusters depend on the propagation path direction to the Rx. Differentiation of the cluster delays is the basis for expressing a power angular spectrum (PAS) as a sum of the

� �, whose delays are of the order of a carrier wave period. The first and second groups

PRi θR;φ<sup>R</sup>

The PAS can be presented as the product of a total power and PDF of AOA [32]. We should

the local scattering components. The energy relationship between these components describes

κ 1 þ κ

delayed scattering components, local scattering components, and direct path component, respectively, P is the total power of the received signal, Pi is the power of the propagation

ellipsoids, P<sup>0</sup> represents the power of the components reaching the Rx with negligible delay,

For the delayed components, the multi-ellipsoidal or multi-elliptical structures are the basis for

the local scattering, a large diversity of receiving antenna surroundings prevents adoption of a determined geometry of the scatterer positions. To adapt the statistical properties of reception

Construction of the multi-ellipsoidal or multi-elliptical structures based on PDP/PDS ensures adjustment of the reception angle statistical characteristics to the transmission properties of the propagation environment. In that, these models provide a mapping of the impact of these

properties on the correlational and spectral characteristics of the received signals.

� � is PDF of AOA for the local scattering components, and <sup>δ</sup>ð Þ� is the Dirac delta

� � <sup>þ</sup> <sup>f</sup> dp <sup>θ</sup>R;φ<sup>R</sup>

P0

� � <sup>þ</sup> PR<sup>0</sup> <sup>θ</sup>R;φ<sup>R</sup>

� � represents the sum of the powers of the direct path component and

� �

1 1 þ κ P0

� � means PDF of AOA for the ith

� �, <sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, N. In the case of

<sup>P</sup> <sup>f</sup> <sup>0</sup> <sup>θ</sup>R;φ<sup>R</sup> � � <sup>þ</sup>

� � represent parts of <sup>f</sup> <sup>R</sup> <sup>θ</sup>R; <sup>φ</sup><sup>R</sup>

of components are called the delayed and local scattering components, respectively.

N

i¼1

� �, and the component powers,

� � (2)

� � means the PAS of

� �, which describes the

<sup>P</sup> <sup>δ</sup> <sup>θ</sup>R;φ<sup>R</sup>

� � corresponding the

� � (3)

component powers reaching the Rx with the delays, PR θR; φ<sup>R</sup>

PR θR; φ<sup>R</sup>

the propagation paths that reach the Rx from the ith ellipsoid.

The delayed components are grouped in the time clusters. Therefore,

� � <sup>¼</sup> <sup>X</sup>

where N is the number of the time clusters (semi-ellipsoids) and PRi θR;φ<sup>R</sup>

the Rice factor, κ. Thus, we can present the PDF of AOA, f <sup>R</sup> θR; φ<sup>R</sup>

statistical properties of the signal reception angle, in the following form:

<sup>P</sup> <sup>f</sup> <sup>i</sup> <sup>θ</sup>R;φ<sup>R</sup> � � <sup>þ</sup>

� � <sup>þ</sup> <sup>f</sup> <sup>l</sup> <sup>θ</sup>R;φ<sup>R</sup>

PR<sup>0</sup> θR; φ<sup>R</sup>

120 Antennas and Wave Propagation

also note that PR<sup>0</sup> θR;φ<sup>R</sup>

f <sup>R</sup> θR;φ<sup>R</sup>

� �, <sup>f</sup> <sup>l</sup> <sup>θ</sup>R;φ<sup>R</sup>

angle, the von Mises distribution is used [4].

where f <sup>d</sup> θR;φ<sup>R</sup>

f <sup>0</sup> θR;φ<sup>R</sup>

function.

� � <sup>¼</sup> <sup>f</sup> <sup>d</sup> <sup>θ</sup>R;φ<sup>R</sup>

<sup>¼</sup> <sup>X</sup> N

i¼1

Pi

� �, and <sup>f</sup> dp <sup>θ</sup>R;φ<sup>R</sup>

the analytical or simulation determination of individual f <sup>i</sup> θR;φ<sup>R</sup>

paths reaching the Rx from the ith ellipsoids, f <sup>i</sup> θR;φ<sup>R</sup>

The use of the antennas with the narrow radiation patterns in the elevation plane limits an environment influence on the received signal properties. Propagation phenomena predominance in the azimuth plane is the premise for reducing the analysis of the reception angle statistical properties to the 2D modeling problem. In this case, the mapping of the propagation phenomena is ensured by the multi-elliptical propagation model. Precursors of this model are Parsons and Bajwa, who presented a multi-elliptical way of modeling the distribution of the propagation paths in [38].

#### 3.1. Analysis of reception angle statistical properties for omni-directional antennas

A PDF of AOA analysis for radio links with omni-directional antennas is based on the 2D geometric structure shown in Figure 3. In this case, Eq. (3) shows that determining the PDF of AOA for the delayed components comes down to determining f <sup>i</sup> φ<sup>R</sup> .

Propagation path lengths, i.e., rTij and rRij (see Figure 3), which describe the distances TxSij , respectively, are

$$r\_{T\vec{\imath}\vec{\jmath}} = a\_{\vec{\imath}} - e\_{\vec{\imath}} \mathbf{x}\_{\vec{\imath}\vec{\jmath}} \quad \text{and} \quad r\_{R\vec{\imath}\vec{\jmath}} = a\_{\vec{\imath}} + e\_{\vec{\imath}} \mathbf{x}\_{\vec{\imath}\vec{\jmath}} \tag{4}$$

where ei ¼ D=ð Þ 2ai means the eccentricity of the ith ellipse and xij is coordinate of Sij.

But xij is a function of φTij and φRij:

$$\mathbf{x}\_{i\dot{\jmath}} = r\_{T\dot{\imath}\dot{\jmath}} \cos \varphi\_{T\dot{\imath}\dot{}} + \frac{D}{2} \quad \text{and} \quad \mathbf{x}\_{i\dot{\jmath}} = r\_{R\dot{\imath}\dot{}} \cos \varphi\_{R\dot{\jmath}} - \frac{D}{2} \tag{5}$$

Considering that rTij þ rRij ¼ 2ai and substituting Eq. (5) to Eq. (4), we can write

$$\frac{1}{1 + e\_i \cos \varphi\_{T\bar{\eta}}} + \frac{1}{1 - e\_i \cos \varphi\_{R\bar{\eta}}} = \frac{4a\_i}{2a\_i - \mathrm{De}\_i} \tag{6}$$

After transforming Eq. (6), the formula of cosφTij versus cosφRij has the form [32]:

$$\cos\varphi\_{T\dot{i}j} = \frac{2a\_i\cos\varphi\_{R\dot{i}j} + De\_i\cos\varphi\_{R\dot{i}j} - 2D}{2a\_i + De\_i - 2D\cos\varphi\_{R\dot{i}j}}\tag{7}$$

According to the assumptions, the statistical properties of angle of departure (AOD) describe a uniform distribution, i.e., f <sup>i</sup> φ<sup>T</sup> � � <sup>¼</sup> ð Þ <sup>2</sup><sup>π</sup> �<sup>1</sup> for <sup>φ</sup><sup>T</sup> <sup>∈</sup>h�π; <sup>π</sup>Þ. Hence, PDF of cos <sup>φ</sup><sup>T</sup> is

$$f\_i(\cos \varphi\_T) = f\_i(\varphi\_T) \left| \frac{\mathbf{d} \varphi\_T}{\mathbf{d}(\cos \varphi\_T)} \right| = \frac{1}{2\pi} \frac{1}{\sqrt{1 - \cos^2 \varphi\_T}} \quad \text{for} \quad \varphi\_T \in ( -\pi, \pi) \tag{8}$$

180 ð �

g2 <sup>T</sup> φ<sup>T</sup>

ized pattern of the transmitting antenna is used as the PDF of AOD, f <sup>T</sup> φ<sup>T</sup>

<sup>2</sup><sup>π</sup> <sup>g</sup><sup>2</sup> <sup>T</sup> φ<sup>T</sup>

� �dφ<sup>T</sup> <sup>¼</sup> 1 and <sup>g</sup><sup>2</sup>

This means that the normalized power radiation pattern meets PDF axioms. Thus, the normal-

Figure 4. Procedure for determining AOAs and powers of propagation paths for multi-elliptical (2D) and multi-

ellipsoidal (3D) propagation model.

<sup>T</sup> φ<sup>T</sup>

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

� � ≥ 0 (12)

http://dx.doi.org/10.5772/intechopen.75142

123

� �:

� � for <sup>φ</sup><sup>T</sup> <sup>∈</sup>h�π;π<sup>Þ</sup> (13)

�180�

f <sup>T</sup> φ<sup>T</sup> � � <sup>¼</sup> <sup>1</sup>

For the Gaussian model of the Tx antenna radiation pattern [39]:

Because cosφ<sup>T</sup> is a function of cosφR, so after considering Eq. (7) and Eq. (8), we get [32]:

$$f\_i(\cos \varphi\_R) = f\_i(\cos \varphi\_T) \left| \frac{\mathbf{d}(\cos \varphi\_T)}{\mathbf{d}(\cos \varphi\_R)} \right| = \frac{1}{2\pi} \frac{1}{\left| \sin \varphi\_R \right|} \frac{\sqrt{\left(2a\_i + De\_i\right)^2 - 4D^2}}{2a\_i + De\_i - 2D \cos \varphi\_R} \quad \text{for} \quad \varphi\_R \in ( -\pi, \pi) \tag{9}$$

Hence, the demanded form of f <sup>i</sup> φ<sup>R</sup> � � is

$$f\_i(\boldsymbol{\varphi}\_R) = f\_i(\cos \boldsymbol{\varphi}\_R) \left| \frac{\mathbf{d}(\cos \boldsymbol{\varphi}\_R)}{\mathbf{d}\boldsymbol{\varphi}\_R} \right| = \frac{1}{2\pi} \frac{1 - e\_i^2}{1 + e\_i^2 - 2e\_i \cos \boldsymbol{\varphi}\_R} \quad \text{for} \quad \boldsymbol{\varphi}\_R \in (-\pi, \pi) \tag{10}$$

Eventually, the PDF of AOA for all delayed components, f <sup>d</sup> φ<sup>R</sup> � �, takes the form [37]:

$$f\_d(\boldsymbol{\varphi}\_R) = \sum\_{i=1}^N \frac{P\_i}{P - P\_0} f\_i(\boldsymbol{\varphi}\_R) = \frac{1}{2\pi} \sum\_{i=1}^N \frac{P\_i}{P - P\_0} \frac{1 - e\_i^2}{1 + e\_i^2 - 2e\_i \cos \boldsymbol{\varphi}\_R} \quad \text{for} \quad \boldsymbol{\varphi}\_R \in ( -\pi, \pi) \tag{11}$$

From Eq. (11), it follows that f <sup>d</sup> φ<sup>R</sup> � � depends significantly on Pi and ai, i.e., on the major axis of each ellipse. This means that properties of this function are determined by the power distribution of the individual time clusters, which is closely related to the transmission properties of a given propagation environment.

#### 3.2. Reception angle dispersion for directional antennas

For directional antennas used in radio links, the evaluation of the reception angle statistical properties is based on simulation tests. In this case, an input data processing algorithm is the basis for the research procedure, which ensures the determining basic parameters and statistical characteristics of AOA. The purpose of simulation studies is to determine a set of pairs <sup>φ</sup>Rij; pRij � � that represent the angles and powers of the individual propagation paths reaching the Rx. The analysis of the obtained set is the basis for the assessment of the AOA statistical properties. The relationship between the multi-elliptical structure of the scatterer positions and the environmental transmission characteristics ensures that the simulation results coincide with empirical results.

A procedure scheme of determining φRij and pRij is shown in Figure 4.

In the first step of the procedure, the multi-elliptical model parameters are determined based on PDP/PDS. In the next step, the propagation path AODs are generated using the power radiation pattern of the transmitting antenna. For the normalized power pattern, g<sup>2</sup> <sup>T</sup> φ<sup>T</sup> � � is

$$\int\_{-180^{\circ}}^{180^{\circ}} g\_T^2(\varphi\_T) \, \mathrm{d}\varphi\_T = 1 \quad \text{and} \quad g\_T^2(\varphi\_T) \ge 0 \tag{12}$$

This means that the normalized power radiation pattern meets PDF axioms. Thus, the normalized pattern of the transmitting antenna is used as the PDF of AOD, f <sup>T</sup> φ<sup>T</sup> � �:

$$f\_T(\boldsymbol{\wp}\_T) = \frac{1}{2\pi} \boldsymbol{\mathfrak{g}}\_T^2(\boldsymbol{\wp}\_T) \quad \text{for} \quad \boldsymbol{\wp}\_T \in (-\pi, \pi) \tag{13}$$

For the Gaussian model of the Tx antenna radiation pattern [39]:

According to the assumptions, the statistical properties of angle of departure (AOD) describe a

� � � � � ¼ 1 2π

Because cosφ<sup>T</sup> is a function of cosφR, so after considering Eq. (7) and Eq. (8), we get [32]:

1 sin φ<sup>R</sup> � � � �

� � <sup>¼</sup> ð Þ <sup>2</sup><sup>π</sup> �<sup>1</sup> for <sup>φ</sup><sup>T</sup> <sup>∈</sup>h�π; <sup>π</sup>Þ. Hence, PDF of cos <sup>φ</sup><sup>T</sup> is

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � cos <sup>2</sup>φ<sup>T</sup>

q

<sup>1</sup> � <sup>e</sup><sup>2</sup> i

<sup>i</sup> � 2ei cosφ<sup>R</sup>

<sup>1</sup> � <sup>e</sup><sup>2</sup> i

<sup>i</sup> � 2ei cosφ<sup>R</sup>

� � depends significantly on Pi and ai, i.e., on the major axis of

<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>2</sup>

each ellipse. This means that properties of this function are determined by the power distribution of the individual time clusters, which is closely related to the transmission properties of a

For directional antennas used in radio links, the evaluation of the reception angle statistical properties is based on simulation tests. In this case, an input data processing algorithm is the basis for the research procedure, which ensures the determining basic parameters and statistical characteristics of AOA. The purpose of simulation studies is to determine a set of pairs

the Rx. The analysis of the obtained set is the basis for the assessment of the AOA statistical properties. The relationship between the multi-elliptical structure of the scatterer positions and the environmental transmission characteristics ensures that the simulation results coincide

In the first step of the procedure, the multi-elliptical model parameters are determined based on PDP/PDS. In the next step, the propagation path AODs are generated using the power

radiation pattern of the transmitting antenna. For the normalized power pattern, g<sup>2</sup>

that represent the angles and powers of the individual propagation paths reaching

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 2ai þ Dei

2ai þ Dei � 2D cosφ<sup>R</sup>

<sup>2</sup> � <sup>4</sup>D<sup>2</sup>

<sup>p</sup> for <sup>φ</sup><sup>T</sup> <sup>∈</sup> h�π;π<sup>Þ</sup> (8)

� �, takes the form [37]:

for φ<sup>R</sup> ∈h�π;πÞ

for φ<sup>R</sup> ∈h�π;πÞ (10)

for φ<sup>R</sup> ∈h�π;πÞ (11)

<sup>T</sup> φ<sup>T</sup> � � is (9)

uniform distribution, i.e., f <sup>i</sup> φ<sup>T</sup>

122 Antennas and Wave Propagation

f <sup>i</sup> cosφ<sup>T</sup>

� � <sup>¼</sup> <sup>f</sup> <sup>i</sup> cosφ<sup>T</sup>

f <sup>i</sup> φ<sup>R</sup>

f <sup>d</sup> φ<sup>R</sup> � � <sup>¼</sup> <sup>X</sup> N

φRij; pRij � �

with empirical results.

Hence, the demanded form of f <sup>i</sup> φ<sup>R</sup>

i¼1

From Eq. (11), it follows that f <sup>d</sup> φ<sup>R</sup>

given propagation environment.

� � <sup>¼</sup> <sup>f</sup> <sup>i</sup> cosφ<sup>R</sup>

Pi P � P<sup>0</sup>

f <sup>i</sup> cosφ<sup>R</sup>

� � <sup>¼</sup> <sup>f</sup> <sup>i</sup> <sup>φ</sup><sup>T</sup>

� � d cos <sup>φ</sup><sup>T</sup>

� � � � � � � <sup>d</sup>φ<sup>T</sup>

� � � � �

� � d cosφ<sup>R</sup> � �

� � d cosφ<sup>R</sup>

Eventually, the PDF of AOA for all delayed components, f <sup>d</sup> φ<sup>R</sup>

f <sup>i</sup> φ<sup>R</sup> � � <sup>¼</sup> <sup>1</sup> 2π X N

3.2. Reception angle dispersion for directional antennas

A procedure scheme of determining φRij and pRij is shown in Figure 4.

� � � � d cosφ<sup>T</sup> � �

> � � � � � ¼ 1 2π

� � is

� � dφ<sup>R</sup>

� � � � ¼ 1 2π

i¼1

Pi P � P<sup>0</sup>

Figure 4. Procedure for determining AOAs and powers of propagation paths for multi-elliptical (2D) and multiellipsoidal (3D) propagation model.

$$\log\_T(\rho\_T) = \mathbb{C}\_0 \exp\left(-\frac{\rho\_T^2}{2\sigma\_{T\rho}}\right) \quad \text{for} \quad \rho\_T \in (-\pi, \pi), (\mathbb{C}\_0-\text{normalized constant}) \tag{14}$$

we have

$$f\_T(\boldsymbol{\varphi}\_T) = \mathbb{C}\_{T\boldsymbol{\varphi}} \exp\left(-\frac{\boldsymbol{\varphi}\_T^2}{\sigma\_{T\boldsymbol{\varphi}}}\right) \quad \text{for} \quad \boldsymbol{\varphi}\_T \in (-\pi, \pi) \tag{15}$$

Therefore, pij generated using by the uniform distribution, i.e., Eq. (18), should be multiplied by the value of the power pattern which corresponds to the AOA with the same indexes, i.e.,

where pij is the so-called power at the reception point and does not consider the receiving antenna pattern, while pRij is the power seen at the output of the receiving antenna or at the Rx

If the receiving antenna is isotropic or omni-directional in the azimuth plane, then it should be

In the case of using antennas, whose pattern width in the vertical plane exceeds several degrees, the scattering in this plane also determines the direction of reaching the propagation paths. Then, we obtain the multi-ellipsoidal propagation model by extending the multielliptical model to 3D space. If the antenna heights meet the conditions of hT, hR ≪ D, then this model is reduced to the set of the semi-ellipsoids. As in the case of the multi-elliptical model, the parameters of the individual semi-ellipsoids, i.e., their semi-axes, are defined by Eq. (1) based on PDP/PDS. The reception angle statistical properties are determined based on simulation studies. In this case, we use a procedure for the 3D modeling shown in Figure 4. This

Similar to the multi-elliptical model, the properties of the normalized power radiation pattern

� �, are used to generate AODs [42]:

In this case, we use the assumption of the independence of the scatterer position in the azimuth

� � <sup>¼</sup> <sup>f</sup> <sup>T</sup>ð Þ� <sup>θ</sup><sup>T</sup> <sup>f</sup> <sup>T</sup> <sup>φ</sup><sup>T</sup>

� � <sup>¼</sup> CTθexp � ð Þ <sup>θ</sup><sup>T</sup> � <sup>π</sup>=<sup>2</sup> <sup>2</sup>

� � sin <sup>θ</sup><sup>T</sup> for <sup>θ</sup><sup>T</sup> <sup>∈</sup>h0;π=2Þ,φ<sup>T</sup> <sup>∈</sup> h�π;π<sup>Þ</sup> (21)

� �, we can describe the PDF of AOD for the multi-

� � are the one-dimensional PDFs of AOD in the elevation and azimuth

σ<sup>T</sup><sup>θ</sup> !

� � (22)

sin ð Þ� <sup>θ</sup><sup>T</sup> CTφexp � <sup>φ</sup><sup>2</sup>

T σ<sup>T</sup><sup>φ</sup> � � (23)

� � <sup>¼</sup> 1 for <sup>φ</sup><sup>R</sup> <sup>∈</sup>h�π;πÞ.

<sup>R</sup> <sup>φ</sup>Rij � � (20)

http://dx.doi.org/10.5772/intechopen.75142

125

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

pRij <sup>¼</sup> pijg<sup>2</sup>

input [41].

assumed that GR <sup>¼</sup> 1 and <sup>g</sup><sup>2</sup>

<sup>R</sup> φ<sup>R</sup>

4. Multi-ellipsoidal propagation model

procedure also includes the elevation plane.

<sup>T</sup> θT; φ<sup>T</sup>

4πg<sup>2</sup>

<sup>T</sup> θT; φ<sup>T</sup>

f <sup>T</sup> θT; φ<sup>T</sup>

of the transmit antenna, g<sup>2</sup>

where f <sup>T</sup>ð Þ θ<sup>T</sup> and f <sup>T</sup> φ<sup>T</sup>

planes, respectively.

ellipsoidal model as

f <sup>T</sup> θT;φ<sup>T</sup>

f <sup>T</sup> θT;φ<sup>T</sup> � � <sup>¼</sup> <sup>1</sup>

and elevation planes. Hence, we have

Using the Gaussian model [39] for gT θT;φ<sup>T</sup>

� � <sup>¼</sup> <sup>f</sup> <sup>T</sup>ð Þ� <sup>θ</sup><sup>T</sup> <sup>f</sup> <sup>T</sup> <sup>φ</sup><sup>T</sup>

where CT<sup>φ</sup> <sup>¼</sup> <sup>Ð</sup> 180� �180� f <sup>T</sup> φ<sup>T</sup> � �dφ<sup>T</sup> !�<sup>1</sup> <sup>¼</sup> ffiffiffi <sup>π</sup> <sup>p</sup> <sup>σ</sup><sup>T</sup>φerf <sup>π</sup>=σ<sup>T</sup><sup>φ</sup> � � � � �<sup>1</sup> , <sup>σ</sup><sup>T</sup><sup>φ</sup> <sup>¼</sup> HPBWTφ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffi ln 2 � � <sup>p</sup> ffi <sup>0</sup>:<sup>6</sup>

HPBWTφ, and HPBWT<sup>φ</sup> is half power beam width (HPBW) of the transmitting antenna in the azimuth plane.

Eq. (15) is the basis for the AOD generation. A relationship between AOD and AOA results from the ellipse properties [32]:

$$\cos\varphi\_{R\dot{i}j} = \frac{2e\_i + \left(1 + e\_i^2\right)\cos\varphi\_{T\dot{i}j}}{1 + e\_i^2 + 2e\_i\cos\varphi\_{T\dot{i}j}}\tag{16}$$

Hence [40]:

$$\varphi\_{R\bar{\eta}} = \text{sgn}\left(\varphi\_{T\bar{\eta}}\right) \arccos\left(\frac{2e\_i + \left(1 + e\_i^2\right)\cos\varphi\_{T\bar{\eta}}}{1 + e\_i^2 + 2e\_i\cos\varphi\_{T\bar{\eta}}}\right) \tag{17}$$

The jth propagation path of the ith ellipse, i.e., φRij, corresponds to the random power, pij. In the first step, the powers of the delayed components, pij, are generated based on a uniform distribution:

$$f\_p\left(p\_{\vec{\imath}}\right) = \begin{cases} M\_{\vec{\imath}}/(2P\_i) & \text{for} \quad p\_{\vec{\imath}} \in \langle 0, 2P\_i/M\_i \rangle \\ 0 & \text{for} \quad p\_{\vec{\imath}} \notin \langle 0, 2P\_i/M\_i \rangle \end{cases} \tag{18}$$

where Mi is the number of the generated paths in the ith cluster, j ¼ 1, 2, …, Mi, and Pi is the cluster power read from PDP/PDS.

Then, these powers are modified by the power pattern of the receiving antenna, g<sup>2</sup> <sup>R</sup> φ<sup>R</sup> � �. Let us assume that the main lobe of this pattern is also described using the Gaussian model [39], i.e.,

$$g\_R^2(\varphi\_R) = G\_R \exp\left(-\frac{\varphi\_R^2}{\sigma\_{R\varphi}}\right) \quad \text{for} \quad \varphi\_R \in (-\pi, \pi) \tag{19}$$

where GR is the receiving antenna gain in a linear measure, <sup>σ</sup><sup>R</sup><sup>φ</sup> <sup>¼</sup> HPBWRφ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffiffi ln 2 � � <sup>p</sup> ffi 0:6HPBWRφ, and HPBWR<sup>φ</sup> is the receiving antenna HPBW in the azimuth plane.

Therefore, pij generated using by the uniform distribution, i.e., Eq. (18), should be multiplied by the value of the power pattern which corresponds to the AOA with the same indexes, i.e.,

$$p\_{Rij} = p\_{ij} g\_R^2 \left(\boldsymbol{\varphi}\_{Rij}\right) \tag{20}$$

where pij is the so-called power at the reception point and does not consider the receiving antenna pattern, while pRij is the power seen at the output of the receiving antenna or at the Rx input [41].

If the receiving antenna is isotropic or omni-directional in the azimuth plane, then it should be assumed that GR <sup>¼</sup> 1 and <sup>g</sup><sup>2</sup> <sup>R</sup> φ<sup>R</sup> � � <sup>¼</sup> 1 for <sup>φ</sup><sup>R</sup> <sup>∈</sup>h�π;πÞ.

## 4. Multi-ellipsoidal propagation model

gT φ<sup>T</sup>

124 Antennas and Wave Propagation

where CT<sup>φ</sup> <sup>¼</sup> <sup>Ð</sup>

azimuth plane.

Hence [40]:

distribution:

we have

� � <sup>¼</sup> <sup>C</sup>0exp � <sup>φ</sup><sup>2</sup>

180�

�180�

from the ellipse properties [32]:

f <sup>T</sup> φ<sup>T</sup>

f <sup>T</sup> φ<sup>T</sup> � �dφ<sup>T</sup> !�<sup>1</sup>

T 2σ<sup>T</sup><sup>φ</sup> � �

φRij ¼ sgn φTij

f <sup>p</sup> pij � �

g2 <sup>R</sup> φ<sup>R</sup>

cluster power read from PDP/PDS.

� �

(

� � <sup>¼</sup> CTφexp � <sup>φ</sup><sup>2</sup>

<sup>¼</sup> ffiffiffi

T σ<sup>T</sup><sup>φ</sup> � �

<sup>π</sup> <sup>p</sup> <sup>σ</sup><sup>T</sup>φerf <sup>π</sup>=σ<sup>T</sup><sup>φ</sup> � � � � �<sup>1</sup>

HPBWTφ, and HPBWT<sup>φ</sup> is half power beam width (HPBW) of the transmitting antenna in the

Eq. (15) is the basis for the AOD generation. A relationship between AOD and AOA results

i � � cosφTij

<sup>i</sup> þ 2ei cos φTij

<sup>2</sup>ei <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>2</sup>

<sup>¼</sup> Mi=ð Þ <sup>2</sup>Pi for pij <sup>∈</sup> h i <sup>0</sup>; <sup>2</sup>Pi=Mi 0 for pij∉h i 0; 2Pi=Mi

i � � cosφTij

!

<sup>i</sup> þ 2ei cosφTij

cosφRij <sup>¼</sup> <sup>2</sup>ei <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>2</sup>

arccos

The jth propagation path of the ith ellipse, i.e., φRij, corresponds to the random power, pij. In the first step, the powers of the delayed components, pij, are generated based on a uniform

where Mi is the number of the generated paths in the ith cluster, j ¼ 1, 2, …, Mi, and Pi is the

assume that the main lobe of this pattern is also described using the Gaussian model [39], i.e.,

R σ<sup>R</sup><sup>φ</sup> � �

where GR is the receiving antenna gain in a linear measure, <sup>σ</sup><sup>R</sup><sup>φ</sup> <sup>¼</sup> HPBWRφ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffiffi

ffi 0:6HPBWRφ, and HPBWR<sup>φ</sup> is the receiving antenna HPBW in the azimuth plane.

Then, these powers are modified by the power pattern of the receiving antenna, g<sup>2</sup>

� � <sup>¼</sup> GRexp � <sup>φ</sup><sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>e</sup><sup>2</sup>

for φ<sup>T</sup> ∈h�π;πÞ;ð Þ C<sup>0</sup> � normalizing constant (14)

for φ<sup>T</sup> ∈h�π;πÞ (15)

, <sup>σ</sup><sup>T</sup><sup>φ</sup> <sup>¼</sup> HPBWTφ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffi

ln 2 � � <sup>p</sup> ffi <sup>0</sup>:<sup>6</sup>

(16)

(17)

(18)

ln 2 � � p

<sup>R</sup> φ<sup>R</sup> � �. Let us

for φ<sup>R</sup> ∈h�π;πÞ (19)

In the case of using antennas, whose pattern width in the vertical plane exceeds several degrees, the scattering in this plane also determines the direction of reaching the propagation paths. Then, we obtain the multi-ellipsoidal propagation model by extending the multielliptical model to 3D space. If the antenna heights meet the conditions of hT, hR ≪ D, then this model is reduced to the set of the semi-ellipsoids. As in the case of the multi-elliptical model, the parameters of the individual semi-ellipsoids, i.e., their semi-axes, are defined by Eq. (1) based on PDP/PDS. The reception angle statistical properties are determined based on simulation studies. In this case, we use a procedure for the 3D modeling shown in Figure 4. This procedure also includes the elevation plane.

Similar to the multi-elliptical model, the properties of the normalized power radiation pattern of the transmit antenna, g<sup>2</sup> <sup>T</sup> θT; φ<sup>T</sup> � �, are used to generate AODs [42]:

$$f\_T(\theta\_T, \varphi\_T) = \frac{1}{4\pi} g\_T^2(\theta\_T, \varphi\_T) \sin \theta\_T \quad \text{for} \quad \theta\_T \in \langle 0, \pi/2 \rangle, \varphi\_T \in (-\pi, \pi) \tag{21}$$

In this case, we use the assumption of the independence of the scatterer position in the azimuth and elevation planes. Hence, we have

$$f\_T\left(\theta\_T, \varphi\_T\right) = f\_T(\theta\_T) \cdot f\_T\left(\varphi\_T\right) \tag{22}$$

where f <sup>T</sup>ð Þ θ<sup>T</sup> and f <sup>T</sup> φ<sup>T</sup> � � are the one-dimensional PDFs of AOD in the elevation and azimuth planes, respectively.

Using the Gaussian model [39] for gT θT;φ<sup>T</sup> � �, we can describe the PDF of AOD for the multiellipsoidal model as

$$f\_T(\theta\_T, \varphi\_T) = f\_T(\theta\_T) \cdot f\_T(\varphi\_T) = \mathbb{C}\_{T\theta} \exp\left(-\frac{(\theta\_T - \pi/2)^2}{\sigma\_{T\theta}}\right) \sin\left(\theta\_T\right) \cdot \mathbb{C}\_{T\varphi} \exp\left(-\frac{\varphi\_T^2}{\sigma\_{T\varphi}}\right) \tag{23}$$

where <sup>σ</sup><sup>T</sup><sup>θ</sup> <sup>¼</sup> HPBWTθ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffi ln 2 � � <sup>p</sup> ffi <sup>0</sup>:6HPBWTθ, HPBWT<sup>θ</sup> is the transmitting antenna HPBW in the elevation plane, and CT<sup>θ</sup> is a normalizing constant that meets the condition CT<sup>θ</sup> ¼ 90Ð� 0� f <sup>T</sup> ð Þ θ<sup>T</sup> dθ<sup>T</sup> � ��1 .

Thus, f <sup>T</sup> φ<sup>T</sup> � � is described by Eq. (15), while <sup>f</sup> <sup>T</sup>ð Þ <sup>θ</sup><sup>T</sup> is defined as

$$f\_T(\theta\_T) = \mathbb{C}\_{T\theta} \exp\left(-\frac{\left(\theta\_T - \pi/2\right)^2}{\sigma\_{T\theta}}\right) \sin \theta\_T \quad \text{for} \quad \theta\_T \in \langle 0, \pi/2\rangle \tag{24}$$

Then, pij generated using the uniform distribution, i.e., Eq. (18), should be multiplied by the value of the pattern corresponding to the elevation and azimuth angles of the same indexes.

Modeling the reception angle statistical properties for the local scattering components is based

where I0ð Þ� is the zero-order modified Bessel function and γφ ≥ 0 is a parameter describing the

For the 3D case, considering the independence of the scatterer occurrence in the azimuth and

exp γθ cos

90 ð�

0 @

0�

The values of γφ and γθ depend on surroundings of the receiving antenna in an analyzed

For γφ ¼ f g 0; 3; 30 , PDFs of AOA for the local scattering components in the azimuth plane are

It should be noted that the AOA distribution for the local scattering components is independent of the distance between Tx and Rx, D. This PDF depends only on the obstacles in the

π <sup>2</sup> � <sup>θ</sup><sup>R</sup> � � � �

for θ<sup>R</sup> ∈h0; π=2Þ,φ<sup>R</sup> ∈h�π;πÞ

� � �

π <sup>2</sup> � <sup>θ</sup><sup>R</sup> � � � � <sup>d</sup>θ<sup>R</sup>

2π I0 γθ

exp γθ cos

<sup>R</sup> <sup>θ</sup>Rij;φRij � � (29)

� � for <sup>φ</sup><sup>R</sup> <sup>∈</sup> h Þ �π;<sup>π</sup> (30)

� � <sup>¼</sup> 1 for <sup>θ</sup><sup>T</sup> <sup>∈</sup>h0;π=2Þ, and

http://dx.doi.org/10.5772/intechopen.75142

127

exp γφ cos φ<sup>R</sup> � �

> 2π I0 γφ � �

> > 1 A

�1

(31)

and γθ ≥ 0 is

<sup>R</sup> θR;φ<sup>R</sup>

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

pRij <sup>¼</sup> pijg<sup>2</sup>

5. Reception angle distribution for local scattering components

exp γφ cosφ<sup>R</sup> � �

2π I0 γφ

� � <sup>¼</sup> CM<sup>θ</sup>

¼ 2π I0ðγθÞ

For an isotropic receiving antenna, we accept GR <sup>¼</sup> 1 and <sup>g</sup><sup>2</sup>

f <sup>0</sup> φ<sup>R</sup> � � <sup>¼</sup>

reception angle dispersion in the azimuth plane.

� � <sup>¼</sup> <sup>f</sup> <sup>0</sup>ð Þ� <sup>θ</sup><sup>R</sup> <sup>f</sup> <sup>0</sup> <sup>φ</sup><sup>R</sup>

f <sup>0</sup>ð Þ θ<sup>R</sup> dθ<sup>R</sup>

1 A

�1

a parameter determining the angle dispersion in the elevation plane.

Hence

φ<sup>R</sup> ∈h�π; πÞ.

on the von Mises distribution [1, 4]:

elevation planes, we have [42]:

f <sup>0</sup> θR; φ<sup>R</sup>

90 ð�

0 @

0�

where CM<sup>θ</sup> ¼

propagation scenario.

shown in Figure 5.

A generated pair of angles, <sup>θ</sup>Rij;φRij � �, determines the direction of the ijth propagation path departing from Tx. This path intersects the ith semi-ellipsoid. The intersection point, Sij, determines the potential position of the scatterer. In the multi-ellipsoidal model, the method of determining the distance rTij between Tx and Sij requires considering the elevation plane. Hence [42]:

$$r\_{\overline{\text{T}}\overline{\text{j}}} = -\frac{1}{2a}b\_i^2 D \sin\theta\_{\overline{\text{T}}\overline{\text{j}}} \cos\varphi\_{\overline{\text{T}}\overline{\text{j}}} + \frac{1}{2a}\sqrt{\left(b\_i^2 D \sin\theta\_{\overline{\text{T}}\overline{\text{j}}} \cos\varphi\_{\overline{\text{T}}\overline{\text{j}}}\right)^2 + 4ab\_i^2 \left(a\_i^2 - \frac{D^2}{4}\right)} \ge 0\tag{25}$$

where <sup>a</sup> <sup>¼</sup> bi sin <sup>θ</sup>Tij cosφTij � �<sup>2</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> <sup>i</sup> cos <sup>2</sup>θTij <sup>þ</sup> sin <sup>θ</sup>Tij sinφTij � �<sup>2</sup> � � <sup>≥</sup> 0.

For the 3D modeling, a pair of angles <sup>θ</sup>Rij;φRij � � representing AOA in the elevation and azimuth planes is determined based on the following formula [42]:

$$\Theta\_{R\bar{\eta}} = \arctan \frac{\sqrt{\left(r\_{T\bar{\eta}}\sin\Theta\_{T\bar{\eta}}\cos\varphi\_{T\bar{\eta}} + D\right)^2 + \left(r\_{T\bar{\eta}}\sin\Theta\_{T\bar{\eta}}\sin\varphi\_{T\bar{\eta}}\right)^2}}{r\_{T\bar{\eta}}\cos\Theta\_{T\bar{\eta}}}\tag{26}$$

$$\varphi\_{\text{Rij}} = \text{sgn}\left(\varphi\_{T\text{ij}}\right) \arctan \frac{r\_{T\text{ij}}\sin\theta\_{T\text{ij}}\sin\varphi\_{T\text{ij}}}{r\_{T\text{ij}}\sin\theta\_{T\text{ij}}\cos\varphi\_{T\text{ij}} + D} \tag{27}$$

As in the 2D approach, the random power is assigned to each path that reaches the Rx and is defined by AOA. The procedure for determining this power is analogous to the multi-elliptical model and is based on the uniform distribution and the receiving antenna pattern, gR θR;φ<sup>R</sup> � �. Let us assume that the main lobe of this pattern is also described using the Gaussian model [39], i.e.,

$$\mathcal{G}\_{\mathbb{R}}^{2}\left(\theta\_{\mathbb{R}},\varphi\_{\mathbb{R}}\right) = \mathbb{G}\_{\mathbb{R}}\exp\left(-\frac{\left(\theta\_{\mathbb{R}}-\pi/2\right)^{2}}{\sigma\_{\mathbb{R}\theta}}\right)\sin\left(\theta\_{\mathbb{R}}\right)\cdot\exp\left(-\frac{\varphi\_{\mathbb{R}}^{2}}{\sigma\_{\mathbb{R}\theta}}\right) \quad \text{for} \quad \theta\tau \in (0,\pi/2), \varphi\_{\mathbb{R}}\in(-\pi,\pi) \tag{28}$$

where <sup>σ</sup><sup>R</sup><sup>θ</sup> <sup>¼</sup> HPBWRθ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffi ln 2 � � <sup>p</sup> ffi <sup>0</sup>:6HPBWR<sup>θ</sup> and HPBWR<sup>θ</sup> the receiving antenna HPBW in the elevation plane.

Then, pij generated using the uniform distribution, i.e., Eq. (18), should be multiplied by the value of the pattern corresponding to the elevation and azimuth angles of the same indexes. Hence

$$p\_{R\bar{i}\bar{j}} = p\_{\bar{i}\bar{j}} \mathbf{g}\_R^2 \left(\boldsymbol{\Theta}\_{R\bar{i}\bar{j}}, \boldsymbol{\varphi}\_{R\bar{i}\bar{j}}\right) \tag{29}$$

For an isotropic receiving antenna, we accept GR <sup>¼</sup> 1 and <sup>g</sup><sup>2</sup> <sup>R</sup> θR;φ<sup>R</sup> � � <sup>¼</sup> 1 for <sup>θ</sup><sup>T</sup> <sup>∈</sup>h0;π=2Þ, and φ<sup>R</sup> ∈h�π; πÞ.

#### 5. Reception angle distribution for local scattering components

where <sup>σ</sup><sup>T</sup><sup>θ</sup> <sup>¼</sup> HPBWTθ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffi

�1 .

A generated pair of angles, θRij;φRij

90Ð� 0�

f <sup>T</sup> ð Þ θ<sup>T</sup> dθ<sup>T</sup> � �

126 Antennas and Wave Propagation

Thus, f <sup>T</sup> φ<sup>T</sup>

Hence [42]:

g2 <sup>R</sup> θR;φ<sup>R</sup>

rTij ¼ � <sup>1</sup> 2a b2

where a ¼ bi sin θTij cosφTij

� �<sup>2</sup>

θRij ¼ arctan

� � <sup>¼</sup> GRexp � ð Þ <sup>θ</sup><sup>R</sup> � <sup>π</sup>=<sup>2</sup> <sup>2</sup>

where <sup>σ</sup><sup>R</sup><sup>θ</sup> <sup>¼</sup> HPBWRθ<sup>=</sup> <sup>2</sup> ffiffiffiffiffiffiffi

the elevation plane.

For the 3D modeling, a pair of angles θRij;φRij

ln 2

� � is described by Eq. (15), while <sup>f</sup> <sup>T</sup>ð Þ <sup>θ</sup><sup>T</sup> is defined as

<sup>f</sup> <sup>T</sup>ð Þ¼ <sup>θ</sup><sup>T</sup> CTθexp � ð Þ <sup>θ</sup><sup>T</sup> � <sup>π</sup>=<sup>2</sup> <sup>2</sup>

� �

<sup>i</sup> D sin θTij cosφTij þ

<sup>þ</sup> <sup>a</sup><sup>2</sup>

azimuth planes is determined based on the following formula [42]:

φRij ¼ sgn φTij

σ<sup>R</sup><sup>θ</sup> !

ln 2

� � <sup>p</sup> ffi <sup>0</sup>:6HPBWTθ, HPBWT<sup>θ</sup> is the transmitting antenna HPBW

sin θ<sup>T</sup> for θ<sup>T</sup> ∈h0;π=2Þ (24)

, determines the direction of the ijth propagation path

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>s</sup> � �

<sup>þ</sup> <sup>4</sup>ab<sup>2</sup> <sup>i</sup> a<sup>2</sup> <sup>i</sup> � <sup>D</sup><sup>2</sup> 4

≥ 0.

þ rTij sin θTij sinφTij

rTij sin <sup>θ</sup>Tij cosφTij <sup>þ</sup> <sup>D</sup> (27)

for θ<sup>T</sup> ∈h0; π=2Þ,φ<sup>R</sup> ∈h�π;πÞ (28)

representing AOA in the elevation and

≥ 0 (25)

(26)

� �. Let

in the elevation plane, and CT<sup>θ</sup> is a normalizing constant that meets the condition CT<sup>θ</sup> ¼

σ<sup>T</sup><sup>θ</sup> !

> 1 2a

departing from Tx. This path intersects the ith semi-ellipsoid. The intersection point, Sij, determines the potential position of the scatterer. In the multi-ellipsoidal model, the method of determining the distance rTij between Tx and Sij requires considering the elevation plane.

b2

<sup>i</sup> cos <sup>2</sup>θTij <sup>þ</sup> sin <sup>θ</sup>Tij sinφTij � �<sup>2</sup> � �

� �

rTij sin θTij cosφTij þ D � �<sup>2</sup>

arctan

As in the 2D approach, the random power is assigned to each path that reaches the Rx and is defined by AOA. The procedure for determining this power is analogous to the multi-elliptical model and is based on the uniform distribution and the receiving antenna pattern, gR θR;φ<sup>R</sup>

us assume that the main lobe of this pattern is also described using the Gaussian model [39], i.e.,

sin ð Þ� <sup>θ</sup><sup>R</sup> exp � <sup>φ</sup><sup>2</sup>

� �

<sup>i</sup> D sin θTij cosφTij � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> <sup>r</sup>

rTij cos θTij

R σ<sup>R</sup><sup>φ</sup> � �

� � <sup>p</sup> ffi <sup>0</sup>:6HPBWR<sup>θ</sup> and HPBWR<sup>θ</sup> the receiving antenna HPBW in

rTij sin θTij sin φTij

Modeling the reception angle statistical properties for the local scattering components is based on the von Mises distribution [1, 4]:

$$f\_0(\varphi\_\mathbb{R}) = \frac{\exp\left(\mathcal{V}\_\psi \cos \varphi\_\mathbb{R}\right)}{2\pi \operatorname{I}\_0\left(\mathcal{V}\_\psi\right)} \quad \text{for} \quad \varphi\_\mathbb{R} \in (-\pi, \pi) \tag{30}$$

where I0ð Þ� is the zero-order modified Bessel function and γφ ≥ 0 is a parameter describing the reception angle dispersion in the azimuth plane.

For the 3D case, considering the independence of the scatterer occurrence in the azimuth and elevation planes, we have [42]:

$$f\_0\left(\theta\_R, \varphi\_R\right) = f\_0(\theta\_R) \cdot f\_0\left(\varphi\_R\right) = \mathbb{C}\_{M\theta} \cdot \frac{\exp\left(\mathcal{V}\_\theta \cos\left(\frac{\pi}{2} - \theta\_R\right)\right)}{2\pi \operatorname{I}\_0(\mathcal{V}\_\theta)} \cdot \frac{\exp\left(\mathcal{V}\_\varphi \cos\varphi\_R\right)}{2\pi \operatorname{I}\_0\left(\mathcal{V}\_\varphi\right)}\tag{31}$$
 
$$\text{for} \quad \theta\_R \in \left<0, \pi/2\right>, \varphi\_R \in \left<-\pi, \pi\right>$$

$$\text{where } \mathsf{C}\_{\mathsf{M}\theta} = \left(\bigcap\_{\theta^{\circ}}^{\mathsf{q}\mathsf{q}^{\circ}} f\_{\theta}(\theta\_{\mathsf{R}}) \mathsf{d}\theta\_{\mathsf{R}}\right)^{-1} = 2\pi \,\mathsf{I}\_{\mathsf{I}}(\mathsf{y}\_{\theta}) \left(\int\limits\_{0^{\circ}}^{\mathsf{q}\mathsf{q}^{\circ}} \exp\left(\mathsf{y}\_{\theta} \cos\left(\frac{\pi}{2} - \theta\_{\mathsf{R}}\right)\right) \mathsf{d}\theta\_{\mathsf{R}}\right)^{-1} \text{ and } \mathsf{y}\_{\theta} \geq 0 \text{ is} $$

a parameter determining the angle dispersion in the elevation plane.

The values of γφ and γθ depend on surroundings of the receiving antenna in an analyzed propagation scenario.

For γφ ¼ f g 0; 3; 30 , PDFs of AOA for the local scattering components in the azimuth plane are shown in Figure 5.

It should be noted that the AOA distribution for the local scattering components is independent of the distance between Tx and Rx, D. This PDF depends only on the obstacles in the

Let O θR; φ<sup>R</sup>

as [41]:

lim<sup>ε</sup><sup>θ</sup> ! <sup>0</sup> ε<sup>φ</sup> ! 0 90 ð � ð�180�

�180�

0

� � <sup>¼</sup> ð Þ <sup>i</sup>; <sup>j</sup> : <sup>θ</sup>Rij <sup>∈</sup>ð Þ <sup>θ</sup><sup>R</sup> � <sup>ε</sup><sup>θ</sup> <sup>∧</sup>φRij <sup>∈</sup> <sup>φ</sup><sup>R</sup> � <sup>ε</sup><sup>φ</sup>

the signal that arrives at the input of the receiver from θ<sup>R</sup> � εθ;φ<sup>R</sup> � ε<sup>φ</sup>

<sup>~</sup><sup>f</sup> <sup>R</sup> <sup>θ</sup>R;φ<sup>R</sup> � � <sup>¼</sup> C0

hoods of <sup>θ</sup><sup>R</sup> and <sup>φ</sup>R, respectively. Thus, <sup>P</sup>

<sup>~</sup><sup>f</sup> <sup>R</sup>ðθR,φRÞdθ<sup>R</sup> <sup>d</sup>φ<sup>R</sup> <sup>¼</sup> 1.

case, marginal distribution properties are applied.

X Kð Þ θ<sup>R</sup>

X N

X Mi

j¼1

� � and <sup>L</sup> <sup>φ</sup><sup>R</sup>

i¼0

90 ð�

0

• "Corner reflector" (CR): GR�CR <sup>¼</sup> 23 dBi, HPBW<sup>θ</sup>�CR <sup>¼</sup> <sup>18</sup>�

• "Horn antenna" (HA): GR�HA <sup>¼</sup> 23 dBi, HPBW<sup>θ</sup>�HA <sup>¼</sup> <sup>40</sup>�

• "Parabolic grid" (PG): GR�PG <sup>¼</sup> 46 dBi, HPBW<sup>θ</sup>�PG <sup>¼</sup> <sup>14</sup>�

• "Other antenna" (OA): GR�OA ¼ 23 dBi and HPBW<sup>θ</sup>�OA ¼ 6

pRij <sup>θ</sup>Rij � �

pRij <sup>θ</sup>Rij � �

<sup>~</sup><sup>f</sup> <sup>R</sup>ðθRÞdθ<sup>R</sup> <sup>¼</sup> 1 and lim<sup>ε</sup>φ!<sup>0</sup>

For the simulation results presented below, we adopted the following assumptions: the PDP as shown in Figure 2, carrier frequency, f <sup>0</sup> ¼ 2:4 GHz, D ¼ 300 m, and parameters for four

<sup>~</sup><sup>f</sup> <sup>R</sup>ð Þ¼ <sup>θ</sup><sup>R</sup> <sup>C</sup><sup>R</sup><sup>θ</sup>

where Kð Þ¼ θ<sup>R</sup> ð Þ i; j : θRij ∈ð Þ θ<sup>R</sup> � ε<sup>θ</sup>

and C<sup>R</sup><sup>φ</sup> meet conditions lim<sup>ε</sup>θ!<sup>0</sup>

antenna types [43]:

n o � � , where <sup>ε</sup><sup>θ</sup> and <sup>ε</sup><sup>φ</sup> are the neighbor-

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

pRij <sup>θ</sup>Rij; <sup>φ</sup>Rij � �

pRij <sup>θ</sup>Rij; <sup>φ</sup>Rij � �

pRij <sup>θ</sup>Rij;φRij � � represents the total power of

http://dx.doi.org/10.5772/intechopen.75142

(33)

129

(34)

� � sector.

<sup>O</sup>ð Þ <sup>θ</sup>R;φ<sup>R</sup>

An estimator of a joint PDF of AOA for the delayed and local scattering components is defined

P <sup>O</sup>ð Þ <sup>θ</sup>R;φ<sup>R</sup>

> P N i¼0 P Mi j¼1

where C0 is a normalizing constant that is associated with ε<sup>θ</sup> and ε<sup>φ</sup> and provides a condition

Eq. (33) is the basis for determining PDFs of AOA in the elevation and azimuth planes. In this

and <sup>~</sup><sup>f</sup> <sup>R</sup> <sup>φ</sup><sup>R</sup>

� � <sup>¼</sup> <sup>C</sup><sup>R</sup><sup>φ</sup>

180 ð �

�180�

X <sup>L</sup>ð Þ <sup>φ</sup><sup>R</sup>

X N

X Mi

j¼1

i¼0

� � <sup>¼</sup> ð Þ <sup>i</sup>; <sup>j</sup> : <sup>φ</sup>Rij <sup>∈</sup> <sup>φ</sup><sup>R</sup> � <sup>ε</sup><sup>φ</sup>

pRij <sup>φ</sup>Rij � �

pRij <sup>φ</sup>Rij � �

n o � � , whereas C<sup>R</sup><sup>θ</sup>

<sup>~</sup><sup>f</sup> <sup>R</sup>ðφRÞdφ<sup>R</sup> <sup>¼</sup> 1, respectively.

, and HPBW<sup>φ</sup>�CR <sup>¼</sup> <sup>58</sup>�

, and HPBW<sup>φ</sup>�HA <sup>¼</sup> <sup>44</sup>�

and HPBW<sup>φ</sup>�OA ! <sup>360</sup>�

, and HPBW<sup>φ</sup>�PG <sup>¼</sup> <sup>10</sup>�

�

Thus, the PDFs of AOA in the elevation and azimuth planes have forms, respectively [41]:

Figure 5. PDFs of AOA for local scattering components and selected γφ.

immediate vicinity of Rx and the direction of Rx-Tx. For the local scattering components, the random power can also be assigned to each AOA. The generation method of such power is similar to that for the delayed scattering components. However, the power at the reception point, p0<sup>j</sup> , is determined on the basis of another uniform distribution [42]:

$$f\_p\left(p\_{0\rangle}\right) = \begin{cases} (\kappa + 1)M\_0/(2P\_0) & \text{for} \quad p\_{0\rangle} \in \langle 0, 2P\_0/(M\_0(\kappa + 1))\rangle\\ 0 & \text{for} \quad p\_{0\rangle} \notin (0, 2P\_0/(M\_0(\kappa + 1))\rangle \end{cases} \tag{32}$$

where M<sup>0</sup> is the number of the generated paths, j ¼ 1, 2, …, M0, and P<sup>0</sup> is the value of power read from PDP/PDS for τ ffi 0.

The power at the Rx input, pR0<sup>j</sup> , is determined as for the delayed scattering components, based on Eq. (20) or Eq. (28) for the 2D or 3D modeling, respectively.

#### 6. Sample results of simulation studies

#### 6.1. Influence of antenna parameters on reception angle distribution

A result of simulation studies is sets of the propagation path parameters reaching the Rx. In the multi-elliptical model, input data for the analysis of the result PDF of AOA for the delayed and local scattering components are two sets <sup>Φ</sup> <sup>¼</sup> <sup>φ</sup>Rij n o and <sup>P</sup> <sup>¼</sup> pRij n o for <sup>j</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, Mi and i ¼ 0, 1, …, N. For the multi-ellipsoidal model, a set of elevations is additionally considered, i.e., <sup>Θ</sup> <sup>¼</sup> <sup>θ</sup>Rij � �. Thus, the ijth propagation path is defined by two or three parameters, for 2D or 3D modeling, respectively. The method of determining the estimated PDFs of AOA based on these sets is presented in [41, 42].

Let O θR; φ<sup>R</sup> � � <sup>¼</sup> ð Þ <sup>i</sup>; <sup>j</sup> : <sup>θ</sup>Rij <sup>∈</sup>ð Þ <sup>θ</sup><sup>R</sup> � <sup>ε</sup><sup>θ</sup> <sup>∧</sup>φRij <sup>∈</sup> <sup>φ</sup><sup>R</sup> � <sup>ε</sup><sup>φ</sup> n o � � , where <sup>ε</sup><sup>θ</sup> and <sup>ε</sup><sup>φ</sup> are the neighborhoods of <sup>θ</sup><sup>R</sup> and <sup>φ</sup>R, respectively. Thus, <sup>P</sup> <sup>O</sup>ð Þ <sup>θ</sup>R;φ<sup>R</sup> pRij <sup>θ</sup>Rij;φRij � � represents the total power of the signal that arrives at the input of the receiver from θ<sup>R</sup> � εθ;φ<sup>R</sup> � ε<sup>φ</sup> � � sector.

An estimator of a joint PDF of AOA for the delayed and local scattering components is defined as [41]:

$$\tilde{f}\_R(\theta\_R, \varphi\_R) = \mathbf{C}\_0 \frac{\mathbf{o}(\theta\_{R, \varphi\_R})}{\sum\_{i=0}^N \sum\_{j=1}^{M\_i} p\_{Rj}(\theta\_{Rij}, \varphi\_{Rj})} \tag{33}$$

where C0 is a normalizing constant that is associated with ε<sup>θ</sup> and ε<sup>φ</sup> and provides a condition 90 �

$$\lim\_{\substack{\iota\_0 \to 0\\ \iota\_\psi \to 0}} \int \int \int\_{R} \tilde{f}\_R(\theta\_{R\prime}\,\rho\_R) \mathbf{d}\theta\_R \,\mathbf{d}\rho\_R = 1.$$

immediate vicinity of Rx and the direction of Rx-Tx. For the local scattering components, the random power can also be assigned to each AOA. The generation method of such power is similar to that for the delayed scattering components. However, the power at the reception

where M<sup>0</sup> is the number of the generated paths, j ¼ 1, 2, …, M0, and P<sup>0</sup> is the value of power

A result of simulation studies is sets of the propagation path parameters reaching the Rx. In the multi-elliptical model, input data for the analysis of the result PDF of AOA for the delayed and

i ¼ 0, 1, …, N. For the multi-ellipsoidal model, a set of elevations is additionally considered,

or 3D modeling, respectively. The method of determining the estimated PDFs of AOA based

n o

� �. Thus, the ijth propagation path is defined by two or three parameters, for 2D

<sup>¼</sup> ð Þ <sup>κ</sup> <sup>þ</sup> <sup>1</sup> <sup>M</sup>0=ð Þ <sup>2</sup>P<sup>0</sup> for <sup>p</sup>0<sup>j</sup> <sup>∈</sup>h i <sup>0</sup>; <sup>2</sup>P0=ð Þ <sup>M</sup>0ð Þ <sup>κ</sup> <sup>þ</sup> <sup>1</sup>

∉h i 0; 2P0=ð Þ M0ð Þ κ þ 1

, is determined as for the delayed scattering components, based

and P ¼ pRij

n o

for j ¼ 1, 2, …, Mi and

(32)

, is determined on the basis of another uniform distribution [42]:

0 for p0<sup>j</sup>

point, p0<sup>j</sup>

128 Antennas and Wave Propagation

i.e., Θ ¼ θRij

f <sup>p</sup> p0<sup>j</sup> � �

read from PDP/PDS for τ ffi 0. The power at the Rx input, pR0<sup>j</sup>

(

Figure 5. PDFs of AOA for local scattering components and selected γφ.

on Eq. (20) or Eq. (28) for the 2D or 3D modeling, respectively.

6.1. Influence of antenna parameters on reception angle distribution

6. Sample results of simulation studies

local scattering components are two sets Φ ¼ φRij

on these sets is presented in [41, 42].

Eq. (33) is the basis for determining PDFs of AOA in the elevation and azimuth planes. In this case, marginal distribution properties are applied.

Thus, the PDFs of AOA in the elevation and azimuth planes have forms, respectively [41]:

$$\begin{aligned} \sum\_{\mathbf{R}\in\mathcal{S}} p\_{R\bar{\mathbf{j}}}(\Theta\_{\mathbf{R}\bar{\mathbf{j}}}) \frac{\sum\_{\mathbf{K}\neq\mathbf{0}} (\Theta\_{\mathbf{R}\bar{\mathbf{j}}})}{\sum\_{i=0}^{N} \sum\_{\mathbf{j}=1}^{M\_i} p\_{R\bar{\mathbf{j}}}(\Theta\_{\mathbf{R}\bar{\mathbf{j}}})} \text{ and } \bar{f}\_{\mathbf{R}}\left(\boldsymbol{\varphi}\_{\mathbf{R}}\right) = \mathsf{C}\_{\mathsf{R}\boldsymbol{\varphi}} \frac{\mathbf{1}\_{\{\boldsymbol{\varphi}\_{\mathbf{R}}\}}(\boldsymbol{\varphi}\_{\mathbf{R}\bar{\mathbf{j}}})}{\sum\_{i=0}^{N} \sum\_{\mathbf{j}=1}^{M\_i} p\_{R\bar{\mathbf{j}}}(\boldsymbol{\varphi}\_{\mathbf{R}\bar{\mathbf{j}}})} \end{aligned} \tag{34}$$

where Kð Þ¼ θ<sup>R</sup> ð Þ i; j : θRij ∈ð Þ θ<sup>R</sup> � ε<sup>θ</sup> � � and <sup>L</sup> <sup>φ</sup><sup>R</sup> � � <sup>¼</sup> ð Þ <sup>i</sup>; <sup>j</sup> : <sup>φ</sup>Rij <sup>∈</sup> <sup>φ</sup><sup>R</sup> � <sup>ε</sup><sup>φ</sup> n o � � , whereas C<sup>R</sup><sup>θ</sup> and C<sup>R</sup><sup>φ</sup> meet conditions lim<sup>ε</sup>θ!<sup>0</sup> 90 ð� 0 <sup>~</sup><sup>f</sup> <sup>R</sup>ðθRÞdθ<sup>R</sup> <sup>¼</sup> 1 and lim<sup>ε</sup>φ!<sup>0</sup> 180 ð � �180� <sup>~</sup><sup>f</sup> <sup>R</sup>ðφRÞdφ<sup>R</sup> <sup>¼</sup> 1, respectively.

For the simulation results presented below, we adopted the following assumptions: the PDP as shown in Figure 2, carrier frequency, f <sup>0</sup> ¼ 2:4 GHz, D ¼ 300 m, and parameters for four antenna types [43]:


Figure 6 shows examples of the PDFs of AOA in the elevation and azimuth planes for the CRtransmitting antenna and OA-receiving antenna. The individual PDFs are presented for selected αT. Whereas, the exemplary marginal PDFs of AOA for the CR receiving antenna and OA transmitting antenna are shown in Figure 7. In this case, the PDFs are depicted for selected αR. These graphs show an influence of the directional antenna on the angular dispersion at the transmitting and receiving side, respectively.

The PDFs are the basis for assessing the AOA dispersion for different types of propagation environments. A quantitative evaluation of the dispersion is based on the rms angle spread (AS). This measure is defined in the elevation, σθ, and azimuth, σφ, planes, respectively [39]:

<sup>R</sup> � f <sup>R</sup>ð Þ θ<sup>R</sup> dθ<sup>R</sup> �

� �dφ<sup>R</sup> �

In both cases, we assume that the Tx and Rx antennas are oriented toward each other.

Figure 9 shows the relationship between AS and HPBWT<sup>θ</sup> ¼ HPBWR<sup>θ</sup> for HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup>

or HPBW<sup>θ</sup>�HA <sup>¼</sup> <sup>40</sup>�

Figure 9. ASs in (a) elevation and (b) azimuth planes versus HPBWT<sup>θ</sup> ¼ HPBWR<sup>θ</sup> for HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup> equal

Figure 10. ASs in (a) elevation and (b) azimuth planes versus HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup> for HPBWT<sup>θ</sup> ¼ HPBWR<sup>θ</sup> equal

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0 @

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

0 @

90 ð �

0

180 ð �

�180�

θ<sup>R</sup> � f <sup>R</sup>ð Þ θ<sup>R</sup> dθ<sup>R</sup>

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

φ<sup>R</sup> � f <sup>R</sup> φ<sup>R</sup>

vuuuut (36)

� �dφ<sup>R</sup>

. ASs versus HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup> for HPBWT<sup>θ</sup>

is presented in Figure 10.

vuuuut (35)

1 A

2

http://dx.doi.org/10.5772/intechopen.75142

131

1 A

2

σθ ¼

σφ ¼

<sup>¼</sup> HPBWR<sup>θ</sup> equal to HPBW<sup>θ</sup>�CR <sup>¼</sup> <sup>18</sup>�

or HPBW<sup>φ</sup>�PG <sup>¼</sup> <sup>10</sup>�

or HPBW<sup>θ</sup>�HA <sup>¼</sup> <sup>40</sup>�

.

.

equal to HPBW<sup>φ</sup>�CR <sup>¼</sup> <sup>58</sup>�

HPBW<sup>φ</sup>�CR <sup>¼</sup> <sup>58</sup>�

HPBW<sup>θ</sup>�CR <sup>¼</sup> <sup>18</sup>�

90 ð �

0 θ2

180 ð �

�180�

φ2 <sup>R</sup> � f <sup>R</sup> φ<sup>R</sup>

or HPBW<sup>φ</sup>�PG <sup>¼</sup> <sup>10</sup>�

The marginal PDFs of AOA for four types of antennas—CR, HA, PG, and OA—are shown in Figure 8. In this case, we assume that <sup>α</sup><sup>T</sup> <sup>¼</sup> <sup>180</sup>� , α<sup>R</sup> ¼ 0, and the radiation patterns of the transmitting and receiving antennas are the same.

Figure 6. PDFs of AOA in (a) elevation and (b) azimuth planes for selected αT.

Figure 7. PDFs of AOA in (a) elevation and (b) azimuth planes for selected αR.

Figure 8. PDFs of AOA in (a) elevation and (b) azimuth planes for four antenna types.

The PDFs are the basis for assessing the AOA dispersion for different types of propagation environments. A quantitative evaluation of the dispersion is based on the rms angle spread (AS). This measure is defined in the elevation, σθ, and azimuth, σφ, planes, respectively [39]:

Figure 6 shows examples of the PDFs of AOA in the elevation and azimuth planes for the CRtransmitting antenna and OA-receiving antenna. The individual PDFs are presented for selected αT. Whereas, the exemplary marginal PDFs of AOA for the CR receiving antenna and OA transmitting antenna are shown in Figure 7. In this case, the PDFs are depicted for selected αR. These graphs show an influence of the directional antenna on the angular disper-

The marginal PDFs of AOA for four types of antennas—CR, HA, PG, and OA—are shown in

, α<sup>R</sup> ¼ 0, and the radiation patterns of the

sion at the transmitting and receiving side, respectively.

Figure 6. PDFs of AOA in (a) elevation and (b) azimuth planes for selected αT.

Figure 7. PDFs of AOA in (a) elevation and (b) azimuth planes for selected αR.

Figure 8. PDFs of AOA in (a) elevation and (b) azimuth planes for four antenna types.

Figure 8. In this case, we assume that <sup>α</sup><sup>T</sup> <sup>¼</sup> <sup>180</sup>�

130 Antennas and Wave Propagation

transmitting and receiving antennas are the same.

$$\sigma\_{\theta} = \sqrt{\int\_{0}^{90^{\circ}} \theta\_{R}^{2} \cdot f\_{R}(\theta\_{R}) \mathrm{d}\theta\_{R} - \left(\int\_{0}^{90^{\circ}} \theta\_{R} \cdot f\_{R}(\theta\_{R}) \mathrm{d}\theta\_{R}\right)^{2}}\tag{35}$$

$$\sigma\_{\psi} = \sqrt{\int\_{-180^{\circ}}^{180^{\circ}} \varphi\_{R}^{2} \cdot f\_{R}(\varphi\_{R}) \mathrm{d}\varphi\_{R} - \left(\int\_{-180^{\circ}}^{180^{\circ}} \varphi\_{R} \cdot f\_{R}(\varphi\_{R}) \mathrm{d}\varphi\_{R}\right)^{2}} \tag{36}$$

Figure 9 shows the relationship between AS and HPBWT<sup>θ</sup> ¼ HPBWR<sup>θ</sup> for HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup> equal to HPBW<sup>φ</sup>�CR <sup>¼</sup> <sup>58</sup>� or HPBW<sup>φ</sup>�PG <sup>¼</sup> <sup>10</sup>� . ASs versus HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup> for HPBWT<sup>θ</sup> <sup>¼</sup> HPBWR<sup>θ</sup> equal to HPBW<sup>θ</sup>�CR <sup>¼</sup> <sup>18</sup>� or HPBW<sup>θ</sup>�HA <sup>¼</sup> <sup>40</sup>� is presented in Figure 10.

In both cases, we assume that the Tx and Rx antennas are oriented toward each other.

Figure 9. ASs in (a) elevation and (b) azimuth planes versus HPBWT<sup>θ</sup> ¼ HPBWR<sup>θ</sup> for HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup> equal HPBW<sup>φ</sup>�CR <sup>¼</sup> <sup>58</sup>� or HPBW<sup>φ</sup>�PG <sup>¼</sup> <sup>10</sup>� .

Figure 10. ASs in (a) elevation and (b) azimuth planes versus HPBWT<sup>φ</sup> ¼ HPBWR<sup>φ</sup> for HPBWT<sup>θ</sup> ¼ HPBWR<sup>θ</sup> equal HPBW<sup>θ</sup>�CR <sup>¼</sup> <sup>18</sup>� or HPBW<sup>θ</sup>�HA <sup>¼</sup> <sup>40</sup>� .

An influence of the radiation/reception direction of the transmitting/receiving antenna on the AS is illustrated in Figures 11 and <sup>12</sup> for (variable <sup>α</sup>T, <sup>α</sup><sup>R</sup> <sup>¼</sup> <sup>0</sup> <sup>¼</sup> const:) and (α<sup>T</sup> <sup>¼</sup> <sup>180</sup>� ¼ const:, variable αR), respectively. These graphs are obtained for four analyzed antenna types.

transmitted signals. Therefore, the assessment of the environment impact on the correlationspectral properties requires an accurate mapping of these phenomena. For this purpose, we propose the so-called Doppler multi-elliptical channel model (DMCM), which is depicted in [44]. This model describes a procedure for generating the propagation path parameters in simulation studies. In addition to the angular dispersion, DMCM also considers the movement of the objects (Tx/Rx). Obtained simulation results using DMCM give the opportunity to evaluate instantaneous or statistical (averaged) changes of the received signal properties. Additionally in [44], DMCM is verified on the basis of empirical data available in a literature. The influence analysis of the angular dispersion in DMCM on the correlational and spectral properties is presented in [45, 46]. In this case, the impact of the Rx motion direction on an autocorrelation function (ACF), power density spectrum (PSD), and following parameters—a coherence time, average Doppler frequency, Doppler spread, and asymmetry coefficient—is

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

http://dx.doi.org/10.5772/intechopen.75142

133

The basis for assessing the angular dispersion effects on the ACF and PSD is the relationship between the Doppler frequency shift (DFS) and AOA. DFS representing the ijth propagation

f Dij ¼ f <sup>D</sup>max cos φRij � β

where f <sup>D</sup>max ¼ f <sup>0</sup>v=c is the maximum DFS, f <sup>0</sup> is carrier frequency of the transmitting signal, v is

Typical assumptions are adopted in the presented analysis. The unmodulated carrier wave signal is used to assess the angular dispersion effects on the correlational and spectral properties. This approach gives an opportunity to simplify an analytical description and provides partial verification and comparison of obtained results with others presented in a literature. In this case, the PSD analyzed in a baseband is called the Doppler spectrum. In addition, the uniform distribution of phase and independence of the signal components are accepted.

Rx velocity, and β is Rx movement direction in relation to the Rx-Tx direction.

<sup>R</sup><sup>~</sup> ð Þ¼ <sup>τ</sup> <sup>X</sup> N

The PSD is obtained based on the Wiener-Khinchin theorem [48, 49]:

S f <sup>D</sup> � � <sup>¼</sup> i¼0

ð ∞

�∞

To this aim, we use the fast Fourier transform algorithm and smoothing filtering.

An unequivocal assessment of the influence of spatial parameters on the transmission properties of an environment requires normalization of ACF and PSD. Therefore, the results obtained

X Mi

pRijexp i2πf Dijτ

j¼1

� � (37)

� � (38)

<sup>R</sup>ð Þ<sup>τ</sup> exp �i2π<sup>f</sup> <sup>D</sup><sup>τ</sup> � �d<sup>τ</sup> (39)

path is determined based on the following formula:

Then, an ACF estimator is defined as [47]:

analyzed.

Figure 11. ASs in (a) elevation and (b) azimuth planes versus αT.

Figure 12. ASs in (a) elevation and (b) azimuth planes versus αR.

As a result of scattering phenomenon, the minimum AS occurs when the transmitting antenna radiates in the opposite direction to the Rx location. However, it should be remembered that in this case, the total power of the delayed components is reduced.

The results presented in this chapter are obtained for 2.4 GHz frequency. The evaluation of the AOA statistical properties for other frequency ranges is presented in [32, 37, 40, 42] for 1.8 GHz and in [41] for 28 GHz. The results presented in the chapter and these papers concern the angular distribution. As is shown in ([33], Figure 8) and ([2], Figure 1), AS is strictly linear correlated with the rms delay spread. Whereas, from ([7], Table 7.7.3-2), we see that the delay spread is reduced with the frequency increase. Therefore, we can conclude that the increase in the frequency brings the decrease in the angular dispersion.

#### 6.2. Effects of reception angle dispersion on correlational and spectral properties

Multipath propagation and channel dispersion phenomena as well as an object motion effect have a significant impact on deformation of spectral and correlational structures of the transmitted signals. Therefore, the assessment of the environment impact on the correlationspectral properties requires an accurate mapping of these phenomena. For this purpose, we propose the so-called Doppler multi-elliptical channel model (DMCM), which is depicted in [44]. This model describes a procedure for generating the propagation path parameters in simulation studies. In addition to the angular dispersion, DMCM also considers the movement of the objects (Tx/Rx). Obtained simulation results using DMCM give the opportunity to evaluate instantaneous or statistical (averaged) changes of the received signal properties. Additionally in [44], DMCM is verified on the basis of empirical data available in a literature.

The influence analysis of the angular dispersion in DMCM on the correlational and spectral properties is presented in [45, 46]. In this case, the impact of the Rx motion direction on an autocorrelation function (ACF), power density spectrum (PSD), and following parameters—a coherence time, average Doppler frequency, Doppler spread, and asymmetry coefficient—is analyzed.

The basis for assessing the angular dispersion effects on the ACF and PSD is the relationship between the Doppler frequency shift (DFS) and AOA. DFS representing the ijth propagation path is determined based on the following formula:

$$f\_{D\text{ij}} = f\_{D\text{max}} \cos \left(\varphi\_{R\text{ij}} - \beta \right) \tag{37}$$

where f <sup>D</sup>max ¼ f <sup>0</sup>v=c is the maximum DFS, f <sup>0</sup> is carrier frequency of the transmitting signal, v is Rx velocity, and β is Rx movement direction in relation to the Rx-Tx direction.

Typical assumptions are adopted in the presented analysis. The unmodulated carrier wave signal is used to assess the angular dispersion effects on the correlational and spectral properties. This approach gives an opportunity to simplify an analytical description and provides partial verification and comparison of obtained results with others presented in a literature. In this case, the PSD analyzed in a baseband is called the Doppler spectrum. In addition, the uniform distribution of phase and independence of the signal components are accepted.

Then, an ACF estimator is defined as [47]:

An influence of the radiation/reception direction of the transmitting/receiving antenna on the AS

As a result of scattering phenomenon, the minimum AS occurs when the transmitting antenna radiates in the opposite direction to the Rx location. However, it should be remembered that in

The results presented in this chapter are obtained for 2.4 GHz frequency. The evaluation of the AOA statistical properties for other frequency ranges is presented in [32, 37, 40, 42] for 1.8 GHz and in [41] for 28 GHz. The results presented in the chapter and these papers concern the angular distribution. As is shown in ([33], Figure 8) and ([2], Figure 1), AS is strictly linear correlated with the rms delay spread. Whereas, from ([7], Table 7.7.3-2), we see that the delay spread is reduced with the frequency increase. Therefore, we can conclude that the increase in

Multipath propagation and channel dispersion phenomena as well as an object motion effect have a significant impact on deformation of spectral and correlational structures of the

6.2. Effects of reception angle dispersion on correlational and spectral properties

this case, the total power of the delayed components is reduced.

Figure 11. ASs in (a) elevation and (b) azimuth planes versus αT.

132 Antennas and Wave Propagation

Figure 12. ASs in (a) elevation and (b) azimuth planes versus αR.

the frequency brings the decrease in the angular dispersion.

¼ const:,

is illustrated in Figures 11 and <sup>12</sup> for (variable <sup>α</sup>T, <sup>α</sup><sup>R</sup> <sup>¼</sup> <sup>0</sup> <sup>¼</sup> const:) and (α<sup>T</sup> <sup>¼</sup> <sup>180</sup>�

variable αR), respectively. These graphs are obtained for four analyzed antenna types.

$$\tilde{R}(\tau) = \sum\_{i=0}^{N} \sum\_{j=1}^{M\_i} p\_{R\dot{j}} \exp\left(\mathrm{i}2\pi f\_{D\dot{j}}\tau\right) \tag{38}$$

The PSD is obtained based on the Wiener-Khinchin theorem [48, 49]:

$$S(f\_D) = \int\_{-\infty}^{\infty} R(\tau) \exp\left(-\mathrm{i}2\pi f\_D \tau\right) d\tau \tag{39}$$

To this aim, we use the fast Fourier transform algorithm and smoothing filtering.

An unequivocal assessment of the influence of spatial parameters on the transmission properties of an environment requires normalization of ACF and PSD. Therefore, the results obtained in the simulation studies are normalized with respect to the average power, Rxð Þ0 , of the received signal. So, for the ACF and PSD, we have, respectively [45]:

$$r(\tau) = \frac{R(\tau)}{R(0)} \quad \text{and} \quad s(f\_D) = 2f\_{D\text{max}} \frac{S(f\_D)}{R(0)} \tag{40}$$

Based on these definitions, the properties of the normalized ACF and PSD are rð Þ¼ 0 1 and

$$\frac{1}{2f\_{D\text{max}}} \int\_{-f\_{D\text{max}}}^{f\_{D\text{max}}} s(f\_D) \,\mathbf{d}f\_D = 1.$$

In the assessment of the AOA dispersion effects on the correlational and spectral properties, a spatial scenario presented in Figure 13 is analyzed.

In this case, we assume that the Rx moves along a straight road at a constant speed, v ¼ 50km=h. The distance from the route to the Tx is Dy ¼ 500 m. The Tx emits the harmonic signal at f <sup>0</sup> ¼ 2:4 GHz, so f <sup>D</sup>max ffi 111 Hz. The antenna patterns are omni-directional. The ACF and PSD evaluations are carried out at four points of the Rx route for specific directions: (A) <sup>β</sup> <sup>¼</sup> <sup>30</sup>� , (B) <sup>β</sup> <sup>¼</sup> <sup>60</sup>� , (C) <sup>β</sup> <sup>¼</sup> <sup>90</sup>� , and (D) <sup>β</sup> <sup>¼</sup> <sup>120</sup>� . For four analyzed cases, modules of the normalized ACFs and PSDs are presented in Figure 14.

A quantitative assessment of the channel dispersion impact on the spectral and correlational properties can be based on the parameter analysis. For the ACF, the coherence time, TC, is determined based on the following condition:

$$|r(T\_{\mathcal{C}})| = \left|\frac{R(T\_{\mathcal{C}})}{R(\mathbf{0})}\right| = \frac{1}{2} \tag{41}$$

FD <sup>¼</sup> <sup>2</sup><sup>f</sup> <sup>D</sup>max <sup>ð</sup>

<sup>2</sup><sup>f</sup> <sup>D</sup>max <sup>ð</sup>

σ<sup>D</sup> ¼

<sup>μ</sup><sup>D</sup> <sup>¼</sup> <sup>1</sup>

vuuuut

Figure 14. Module of normalized ACFs and PSDs for four analyzed points along Rx route.

ffiffiffiffiffiffi σ<sup>D</sup> p

of these parameters on the analyzed route are shown in Figure 15.

simulation scenario. We can conclude that TC, σD, j j FD , and μ<sup>D</sup>

changes versus the Rx position.

3 vuuuut f <sup>D</sup>max

f <sup>D</sup>s f <sup>D</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

f <sup>D</sup> � FD � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

f <sup>D</sup> � FD � �<sup>3</sup>

s f <sup>D</sup> � �d<sup>f</sup> <sup>D</sup>

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

s f <sup>D</sup> � �d<sup>f</sup> <sup>D</sup>

� � �

� �d<sup>f</sup> <sup>D</sup> (42)

http://dx.doi.org/10.5772/intechopen.75142

� are decreasing with reduction

(43)

135

(44)

�f <sup>D</sup>max

f <sup>D</sup>max

�f <sup>D</sup>max

f <sup>D</sup>max

�f <sup>D</sup>max

To obtain independence of the above metrics from f <sup>D</sup>max, i.e., from f <sup>0</sup> and v, we introduce normalized parameters in the following forms: TCf <sup>D</sup>max, FD=f <sup>D</sup>max, σD=f <sup>D</sup>max, and μD. Changes

A characteristic feature of the presented PSD graphs is an occurrence of maxima for extreme DFSs and DFS resulted from Eq. (37) for the direct path component. The last frequency is closely related to the angle, β, between the velocity vector and the Rx-Tx direction. Based on the obtained results presented in Figures 14 and 15, we can conclude that close relations exist between the parameters of the analyzed characteristics and the spatial parameters of the

of the Rx-Tx distance. The results prove that PSDs and ACFs significantly depend not only on the time domain dispersion but also on the angular dispersion and mutual position of the Rx and Tx. A graphical illustration of this fact is shown in Figure 16, which shows the PSD

Based on Figure 16, it follows that channels in mobile radiocommunications are spatially anisotropic due to their correlation-spectral properties. This means that the analysis of the ACF and PSD requires not only knowledge of the transmission properties of an environment but also knowledge about the Rx-Tx mutual position and movement parameters of the

<sup>2</sup><sup>f</sup> <sup>D</sup>max <sup>ð</sup>

However, for the PSD, the average DFS, FD, rms Doppler spread, σD, and asymmetry coefficient, μD, [50] are defined, respectively:

Figure 13. Spatial scenario of simulation studies.

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver http://dx.doi.org/10.5772/intechopen.75142 135

Figure 14. Module of normalized ACFs and PSDs for four analyzed points along Rx route.

in the simulation studies are normalized with respect to the average power, Rxð Þ0 , of the

Based on these definitions, the properties of the normalized ACF and PSD are rð Þ¼ 0 1 and

In the assessment of the AOA dispersion effects on the correlational and spectral properties, a

In this case, we assume that the Rx moves along a straight road at a constant speed, v ¼ 50km=h. The distance from the route to the Tx is Dy ¼ 500 m. The Tx emits the harmonic signal at f <sup>0</sup> ¼ 2:4 GHz, so f <sup>D</sup>max ffi 111 Hz. The antenna patterns are omni-directional. The ACF and PSD evaluations are carried out at four points of the Rx route for specific directions:

A quantitative assessment of the channel dispersion impact on the spectral and correlational properties can be based on the parameter analysis. For the ACF, the coherence time, TC, is

, and (D) <sup>β</sup> <sup>¼</sup> <sup>120</sup>�

j j r Tð Þ <sup>C</sup> <sup>¼</sup> R Tð Þ <sup>C</sup>

� � � �

However, for the PSD, the average DFS, FD, rms Doppler spread, σD, and asymmetry coeffi-

Rð Þ0

� � � � ¼ 1

� � <sup>¼</sup> <sup>2</sup><sup>f</sup> <sup>D</sup>max

S f <sup>D</sup> � �

<sup>R</sup>ð Þ<sup>0</sup> (40)

. For four analyzed cases, modules of the

<sup>2</sup> (41)

received signal. So, for the ACF and PSD, we have, respectively [45]:

Rð Þτ

<sup>R</sup>ð Þ<sup>0</sup> and s f <sup>D</sup>

rð Þ¼ τ

spatial scenario presented in Figure 13 is analyzed.

, (C) <sup>β</sup> <sup>¼</sup> <sup>90</sup>�

normalized ACFs and PSDs are presented in Figure 14.

determined based on the following condition:

cient, μD, [50] are defined, respectively:

Figure 13. Spatial scenario of simulation studies.

1 2f <sup>D</sup>max

Ð f <sup>D</sup>max

s f <sup>D</sup>

134 Antennas and Wave Propagation

� �d<sup>f</sup> <sup>D</sup> <sup>¼</sup> 1.

, (B) <sup>β</sup> <sup>¼</sup> <sup>60</sup>�

�f <sup>D</sup>max

(A) <sup>β</sup> <sup>¼</sup> <sup>30</sup>�

$$F\_D = 2f\_{D\text{max}} \int\_{-f\_{D\text{max}}}^{f\_{D\text{max}}} f\_D s(f\_D) \,\text{d}f\_D \tag{42}$$

$$\sigma\_{D} = \sqrt{2f\_{D\text{max}} \int\_{-f\_{D\text{max}}}^{f\_{D\text{max}}} \left(f\_{D} - F\_{D}\right)^{2} s(f\_{D}) \mathbf{d}f\_{D}} \tag{43}$$

$$\mu\_D = \frac{1}{\sqrt{\sigma\_D}} \sqrt[3]{2f\_{D\text{max}} \int\_{-f\_{D\text{max}}}^{f\_{D\text{max}}} \left(f\_D - F\_D\right)^3 s(f\_D) \mathbf{d}f\_D} \tag{44}$$

To obtain independence of the above metrics from f <sup>D</sup>max, i.e., from f <sup>0</sup> and v, we introduce normalized parameters in the following forms: TCf <sup>D</sup>max, FD=f <sup>D</sup>max, σD=f <sup>D</sup>max, and μD. Changes of these parameters on the analyzed route are shown in Figure 15.

A characteristic feature of the presented PSD graphs is an occurrence of maxima for extreme DFSs and DFS resulted from Eq. (37) for the direct path component. The last frequency is closely related to the angle, β, between the velocity vector and the Rx-Tx direction. Based on the obtained results presented in Figures 14 and 15, we can conclude that close relations exist between the parameters of the analyzed characteristics and the spatial parameters of the simulation scenario. We can conclude that TC, σD, j j FD , and μ<sup>D</sup> � � � � are decreasing with reduction of the Rx-Tx distance. The results prove that PSDs and ACFs significantly depend not only on the time domain dispersion but also on the angular dispersion and mutual position of the Rx and Tx. A graphical illustration of this fact is shown in Figure 16, which shows the PSD changes versus the Rx position.

Based on Figure 16, it follows that channels in mobile radiocommunications are spatially anisotropic due to their correlation-spectral properties. This means that the analysis of the ACF and PSD requires not only knowledge of the transmission properties of an environment but also knowledge about the Rx-Tx mutual position and movement parameters of the

channel transmission characteristics (PDP/PDS) condition the nature of the angular dispersion

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

http://dx.doi.org/10.5772/intechopen.75142

137

The chapter is devoted to the problem of the multipath propagation phenomenon modeling and its impact on changes of the received signal properties. The main focus is on the method of determining the PDF of AOA, which has the significant impact on changes in the correlational and spectral properties. In contrast to the empirical models, using the geometry-based models gives the opportunity to consider in the modeling process the spatial structure of a propagation environment. Linking the environment transmission properties with the position geometry of the scatterers is the main problem of using the geometric models. The use of the multi-elliptical or multi-ellipsoidal models is a solution to this problem. In this case, the geometrical structure of the scatterer locations in the form of the set of the confocal ellipses or semi-ellipsoids is created on the basis of the transmission characteristics of a propagation environment. This ensures adapting the geometrical structure to the analyzed or modeled propagation environment. The use of the multi-ellipsoidal or multi-elliptical models gives the possibility to consider the effect of the antenna radiation patterns on the PDF of AOA. This plays a significant role in the analysis of the compatible operating of coexistent wireless systems. PDF of AOA is the basis for assessing the impact of a propagation environment on the correlational and spectral properties of the transmitted signals. The use of the geometric models presented in this chapter provides mapping of the motion effects, which cause changes in the ACFs and PSDs versus changes of the object positions. The ability to adapt to the environment transmission properties and the assessment of changes in the correlation-spectral characteristics as a function of the object locations significantly distinguishes the models described in this chapter, among those presented so far in a

of the received signal in multipath propagation environments.

7. Conclusion

literature.

Author details

Warsaw, Poland

References

Jan M. Kelner\* and Cezary Ziółkowski

\*Address all correspondence to: jan.kelner@wat.edu.pl

Military University of Technology, Faculty of Electronics, Institute of Telecommunications,

[1] Ziółkowski C, Kelner JM. Empirical models of the azimuthal reception angle—Part I: Comparative analysis of empirical models for different propagation environments. Wireless Personal Communications. 2016;91(2):771-791. DOI: 10.1007/s11277-016-3496-1

Figure 15. Changes of normalized parameters on Rx route: (a) average Doppler, (b) Doppler spread, (c) asymmetry coefficient, and (d) coherence time.

Figure 16. Averaged PSD versus Rx position.

elements of communication system. It follows that the set of parameters qualifying correctness of the PSD and ACF analysis, in addition to the characteristics related to the channel impulse response (PDP/PDS), should include the Tx and Rx positions, spatial location of movement trajectory, direction, and value of the velocity vector of the moving object. Therefore, the channel transmission characteristics (PDP/PDS) condition the nature of the angular dispersion of the received signal in multipath propagation environments.

## 7. Conclusion

The chapter is devoted to the problem of the multipath propagation phenomenon modeling and its impact on changes of the received signal properties. The main focus is on the method of determining the PDF of AOA, which has the significant impact on changes in the correlational and spectral properties. In contrast to the empirical models, using the geometry-based models gives the opportunity to consider in the modeling process the spatial structure of a propagation environment. Linking the environment transmission properties with the position geometry of the scatterers is the main problem of using the geometric models. The use of the multi-elliptical or multi-ellipsoidal models is a solution to this problem. In this case, the geometrical structure of the scatterer locations in the form of the set of the confocal ellipses or semi-ellipsoids is created on the basis of the transmission characteristics of a propagation environment. This ensures adapting the geometrical structure to the analyzed or modeled propagation environment. The use of the multi-ellipsoidal or multi-elliptical models gives the possibility to consider the effect of the antenna radiation patterns on the PDF of AOA. This plays a significant role in the analysis of the compatible operating of coexistent wireless systems. PDF of AOA is the basis for assessing the impact of a propagation environment on the correlational and spectral properties of the transmitted signals. The use of the geometric models presented in this chapter provides mapping of the motion effects, which cause changes in the ACFs and PSDs versus changes of the object positions. The ability to adapt to the environment transmission properties and the assessment of changes in the correlation-spectral characteristics as a function of the object locations significantly distinguishes the models described in this chapter, among those presented so far in a literature.

## Author details

Jan M. Kelner\* and Cezary Ziółkowski

\*Address all correspondence to: jan.kelner@wat.edu.pl

Military University of Technology, Faculty of Electronics, Institute of Telecommunications, Warsaw, Poland

## References

elements of communication system. It follows that the set of parameters qualifying correctness of the PSD and ACF analysis, in addition to the characteristics related to the channel impulse response (PDP/PDS), should include the Tx and Rx positions, spatial location of movement trajectory, direction, and value of the velocity vector of the moving object. Therefore, the

Figure 15. Changes of normalized parameters on Rx route: (a) average Doppler, (b) Doppler spread, (c) asymmetry

Figure 16. Averaged PSD versus Rx position.

coefficient, and (d) coherence time.

136 Antennas and Wave Propagation

[1] Ziółkowski C, Kelner JM. Empirical models of the azimuthal reception angle—Part I: Comparative analysis of empirical models for different propagation environments. Wireless Personal Communications. 2016;91(2):771-791. DOI: 10.1007/s11277-016-3496-1

[2] Ziółkowski C, Kelner JM. Empirical models of the azimuthal reception angle—Part II: Adaptation of the empirical models in analytical and simulation studies. Wireless Personal Communications. 2016;91(3):1285-1296. DOI: 10.1007/s11277-016-3528-x

[14] Janaswamy R. Angle and time of arrival statistics for the Gaussian scatter density model. IEEE Transactions on Wireless Communications. 2002;1(3):488-497. DOI: 10.1109/TWC.2002.

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

http://dx.doi.org/10.5772/intechopen.75142

139

[15] Jiang L, Tan SY. Geometrically Based Statistical Channel Models for Outdoor and Indoor Propagation Environments. IEEE Transactions on Vehicular Technology. 2007;56(6):3587-

[16] Le KN. On angle-of-arrival and time-of-arrival statistics of geometric scattering channels. IEEE Transactions on Vehicular Technology. 2009;58(8):4257-4264. DOI: 10.1109/TVT.2009.

[17] Olenko AY, Wong KT, EH-O N. Analytically derived TOA-DOA statistics of uplink/ downlink wireless multipaths arisen from scatterers on a hollow-disc around the mobile. IEEE Antennas and Wireless Propagation Letters. 2003;2(1):345-348. DOI: 10.1109/LAWP.

[18] Baltzis KB. A generalized elliptical scattering model for the spatial characteristics of mobile channels. Wireless Personal Communications. 2011;67(4):971-984. DOI: 10.1007/

[19] Zhang J, Pan C, Pei F, Liu G, Cheng X. Three-dimensional fading channel models: A survey of elevation angle research. IEEE Communications Magazine. 2014;52(6):218-226.

[20] Janaswamy R. Angle of arrival statistics for a 3-D spheroid model. IEEE Transactions on

[21] Olenko AY, Wong KT, Qasmi SA, Ahmadi-Shokouh J. Analytically derived uplink/downlink TOA and 2-D-DOA distributions with scatterers in a 3-D hemispheroid surrounding the mobile. IEEE Transactions on Antennas and Propagation. 2006;54(9):2446-2454. DOI:

[22] Nawaz SJ, Khan NM, Patwary MN, Moniri M. Effect of directional antenna on the Doppler spectrum in 3-D mobile radio propagation environment. IEEE Transactions on

[23] Nawaz SJ, Riaz M, Khan NM, Wyne S. Temporal analysis of a 3D ellipsoid channel model for the vehicle-to-vehicle communication environments. Wireless Personal Communica-

[24] Zajic AG, Stüber GL. Three-dimensional modeling, simulation, and capacity analysis of space-time correlated mobile-to-mobile channels. IEEE Transactions on Vehicular Tech-

[25] Olenko AY, Wong KT, Qasmi SA. Distribution of the uplink multipaths' arrival delay and azimuth-elevation arrival angle because of 'bad urban' scatterers distributed cylindrically above the mobile. Transactions on Emerging Telecommunications Technologies. 2013;

Vehicular Technology. 2011;60(7):2895-2903. DOI: 10.1109/TVT.2011.2161788

tions. 2015;82(3):1337-1350. DOI: 10.1007/s11277-015-2286-5

nology. 2008;57(4):2042-2054. DOI: 10.1109/TVT.2007.912150

Vehicular Technology. 2002;51(5):1242-1247. DOI: 10.1109/TVT.2002.801756

800547

2023255

2004.824174

s11277-011-0434-0

3593. DOI: 10.1109/TVT.2007.901055

DOI: 10.1109/MCOM.2014.6829967

24(2):113-132. DOI: 10.1002/ett.2530

10.1109/TAP.2006.880661


[14] Janaswamy R. Angle and time of arrival statistics for the Gaussian scatter density model. IEEE Transactions on Wireless Communications. 2002;1(3):488-497. DOI: 10.1109/TWC.2002. 800547

[2] Ziółkowski C, Kelner JM. Empirical models of the azimuthal reception angle—Part II: Adaptation of the empirical models in analytical and simulation studies. Wireless Per-

[3] Sieskul BT, Kupferschmidt C, Kaiser T. Spatial fading correlation for local scattering: A condition of angular distribution. IEEE Transactions on Vehicular Technology. 2011;60(3):

[4] Abdi A, Barger JA, Kaveh M. A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station. IEEE Transactions on Vehicular Technology. 2002;51(3):425-434. DOI: 10.1109/TVT.2002.

[5] WINNER II Channel Models. IST-WINNER II. Tech. Rep. Deliverable 1.1.2 v.1.2. 2007

[6] 3GPP TR 25.996 v13.1.0 (2016-12). Spatial Channel Model for Multiple Input Multiple Output (MIMO) Simulations (Release 13). Tech. Rep. 3GPP TR 25.996 v13.1.0. Valbonne, France. 3rd Generation Partnership Project (3GPP), Technical Specification Group Radio

[7] 3GPP TR 38.901 V14.2.0 (2017-09). Study on channel model for frequencies from 0.5 to 100 GHz (Release 14). Tech. Rep. 3GPP TR 38.901 V14.2.0. Valbonne, France. 3rd Generation Partnership Project (3GPP), Technical Specification Group Radio Access Network; 2017 [8] Yun Z, Iskander MF. Ray tracing for radio propagation modeling: Principles and applica-

[9] Fuschini F, Vitucci EM, Barbiroli M, Falciasecca G, Degli-Esposti V. Ray tracing propagation modeling for future small-cell and indoor applications: A review of current tech-

[10] Corucci A, Usai P, Monorchio A, Manara G. Wireless propagation modeling by using raytracing. In: Mittra R. editor. Computational Electromagnetics. Recent Advances and Engineering Applications. New York, NY, USA: Springer; 2014. pp. 575-618. DOI: 10.1007/978-1-

[11] Lim SY, Yun Z, Iskander MF. Radio propagation modeling: A unified view of the raytracing image method across emerging indoor and outdoor environments. In: Lakhtakia A, Furse CM, editors. The World of Applied Electromagnetics. In Appreciation of Magdy Fahmy Iskander. Cham, Switzerland: Springer; 2018. pp. 301-328. DOI: 10.1007/978-3-319-

[12] Yun Z, Iskander MF. Radio propagation modeling and simulation using ray tracing. In: Lakhtakia A, Furse CM, editors. The World of Applied Electromagnetics. In Appreciation of Magdy Fahmy Iskander. Cham, Switzerland: Springer; 2018. pp. 275-299. DOI: 10.1007/

[13] Lee WCY. Estimate of local average power of a mobile radio signal. IEEE Transactions on

Vehicular Technology. 1985;34(1):22-27. DOI: 10.1109/T-VT.1985.24030

tions. IEEE Access. 2015;3:1089-1100. DOI: 10.1109/ACCESS.2015.2453991

niques. Radio Science. 2015;50(6):2015RS005659. DOI: 10.1002/2015RS005659

sonal Communications. 2016;91(3):1285-1296. DOI: 10.1007/s11277-016-3528-x

1271-1278. DOI: 10.1109/TVT.2010.2103370

1002493

138 Antennas and Wave Propagation

Access Network; 2016

4614-4382-7\_17

58403-4\_13

978-3-319-58403-4\_12


[26] Ahmed A, Nawaz SJ, Gulfam SM. A 3-D propagation model for emerging land mobile radio cellular environments. PLoS One. 2015;10(8):e0132555. DOI: 10.1371/journal.pone.0132555

[38] Parsons JD, Bajwa AS. Wideband characterisation of fading mobile radio channels. IEE Proceedings F Communications, Radar and Signal Processing. 1982;129(2):95-101. DOI:

Multi-Elliptical Geometry of Scatterers in Modeling Propagation Effect at Receiver

http://dx.doi.org/10.5772/intechopen.75142

141

[39] Vaughan R, Bach Andersen J. Channels, Propagation and Antennas for Mobile Commu-

[40] Ziółkowski C, Kelner JM, Nowosielski L, Wnuk M. Modeling the distribution of the arrival angle based on transmitter antenna pattern. In: Proceedings of the 2017 11th European Conference on Antennas and Propagation (EUCAP); 19-24 March 2017; Paris,

[41] Ziółkowski C, Kelner JM. Statistical evaluation of the azimuth and elevation angles seen at the output of the receiving antenna. IEEE Transactions on Antennas and Propagation.

[42] Ziółkowski C, Kelner JM. Antenna pattern in three-dimensional modelling of the arrival angle in simulation studies of wireless channels. IET Microwaves, Antennas & Propaga-

[43] Azevedo JA, Santos FE, Sousa TA, Agrela JM. Impact of the antenna directivity on path loss for different propagation environments. IET Microwaves, Antennas & Propagation.

[44] Ziółkowski C, Kelner JM. Geometry-based statistical model for the temporal, spectral, and spatial characteristics of the land mobile channel. Wireless Personal Communica-

[45] Ziółkowski C, Kelner JM. Influence of receiver/transmitter motion direction on the correlational and spectral signal properties. In: Proceedings of the 2016 10th European Conference on Antennas and Propagation (EuCAP); 10–15 April 2016; Davos, Switzerland:

[46] Kelner JM, Ziółkowski C. Influence of receiver/transmitter motion direction on the correlational and spectral characteristics – Simulation analysis. In: Proceedings of the 2016 10th International Conference on Signal Processing and Communication Systems (ICSPCS); 19- 21 December 2016; Gold Coast, Australia: IEEE; 2016. p. 1-6. DOI: 10.1109/ICSPCS.2016.

[47] Stüber GL. Principles of Mobile Communication. 3rd ed. New York, NY, USA: Springer;

[48] Wiener N. Generalized harmonic analysis. Acta Math. 1930;55(1):117-258. DOI: 10.1007/

[49] Khintchine AY. Korrelationstheorie der stationären stochastischen Prozesse. Mathematische

[50] Beckmann P. Probability in Communication Engineering. New York, NY, USA: Harcourt,

nications. London, UK: Institution of Engineering and Technology; 2003

France: IEEE; 2017. p. 1582-1586. DOI: 10.23919/EuCAP.2017.7928823

10.1049/ip-f-1:19820016

7843381

BF02546511

Brace & World; 1967

2011

2018;66. DOI: 10.1109/TAP.2018.2796719

tion. 2017;11(6):898-906. DOI: 10.1049/iet-map.2016.0591

2015;9(13):1392-1398. DOI: 10.1049/iet-map.2015.0194

tions. 2015;83(1):631-652. DOI: 10.1007/s11277-015-2413-3

IEEE; 2016. pp. 1-4. DOI: 10.1109/EuCAP.2016.7481225

Annalen. 1934;109(1):604-615. DOI: 10.1007/BF01449156


[38] Parsons JD, Bajwa AS. Wideband characterisation of fading mobile radio channels. IEE Proceedings F Communications, Radar and Signal Processing. 1982;129(2):95-101. DOI: 10.1049/ip-f-1:19820016

[26] Ahmed A, Nawaz SJ, Gulfam SM. A 3-D propagation model for emerging land mobile radio cellular environments. PLoS One. 2015;10(8):e0132555. DOI: 10.1371/journal.pone.0132555

[27] Nawaz SJ, Wyne S, Baltzis KB, Gulfam SM, Cumanan K. A tunable 3-D statistical channel model for spatio-temporal characteristics of wireless communication networks. Transactions on Emerging Telecommunications Technologies. 2017;28(12):e3213. DOI:

[28] Jiang L, Tan SY. Simple geometrical-based AOA model for mobile communication sys-

[29] Petrus P, Reed JH, Rappaport TS. Geometrical-based statistical macrocell channel model for mobile environments. IEEE Transactions on Communications. 2002;50(3):495-502.

[30] Ertel RB, Reed JH. Angle and time of arrival statistics for circular and elliptical scattering models. IEEE Journal on Selected Areas in Communications. 1999;17(11):1829-1840. DOI:

[31] Khan NM, Simsim MT, Rapajic PB. A generalized model for the spatial characteristics of the cellular mobile channel. IEEE Transactions on Vehicular Technology. 2008;57(1):22-37.

[32] Ziółkowski C. Statistical model of the angular power distribution for wireless multipath environments. IET Microwaves, Antennas & Propagation. 2015;9(3):281-289. DOI: 10.

[33] Pedersen KI, Mogensen PE, Fleury BH. A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments. IEEE Trans-

[34] Pedersen KI, Mogensen PE, Fleury BH. Spatial channel characteristics in outdoor environments and their impact on BS antenna system performance. In: Proceedings of the 1998 48th IEEE Vehicular Technology Conference (VTC); 18-21 may 1998; Ottawa, Can-

[35] Mogensen PE, Pedersen KI, Leth-Espensen P, Fleury BH, Frederiksen F, Olesen K, Larsen SL .Preliminary measurement results from an adaptive antenna array testbed for GSM/ UMTS. In: Proceedings of the 1997 47th IEEE Vehicular Technology Conference (VTC); 4- 7 May 1997; Phoenix, AZ, USA: IEEE; 1997. vol. 3; p. 1592-1596. DOI: 10.1109/VETEC.

[36] Fleury BH, Tschudin M, Heddergott R, Dahlhaus D, Pedersen KI. Channel parameter estimation in mobile radio environments using the SAGE algorithm. IEEE Journal on

[37] Ziółkowski C, Kelner JM. Estimation of the reception angle distribution based on the power delay spectrum or profile. International Journal of Antennas and Propagation.

Selected Areas in Communications. 1999;17(3):434-450. DOI: 10.1109/49.753729

2015;2015:e936406. DOI: 10.1155/2015/936406

actions on Vehicular Technology. 2000;49(2):437-447. DOI: 10.1109/25.832975

ada: IEEE; 1998. vol. 2; p. 719-723. DOI: 10.1109/VETEC.1998.683676

tems. Electronics Letters. 2004;40(19):1203-1205. DOI: 10.1049/el:20045599

10.1002/ett.3213

140 Antennas and Wave Propagation

DOI: 10.1109/26.990911

DOI: 10.1109/TVT.2007.904532

1049/iet-map.2014.0099

1997.605826

10.1109/49.806814


**Chapter 7**

Provisional chapter

**Ultra Wideband Transient Scattering and Its**

Ultra Wideband Transient Scattering and Its

Hoi-Shun Lui, Faisal Aldhubaib, Stuart Crozier and

Hoi-Shun Lui, Faisal Aldhubaib, Stuart Crozier and

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

lights of the ongoing challenges in the field.

scattering, singularity expansion method

http://dx.doi.org/10.5772/intechopen.75059

Nicholas V. Shuley

Abstract

1. Introduction

Nicholas V. Shuley

**Applications to Automated Target Recognition**

DOI: 10.5772/intechopen.75059

Reliable radar target recognition has long been the holy grail of electromagnetic sensors. Target recognition based on the singularity expansion method (SEM) uses a time-domain electromagnetic signature and has been well studied over the last few decades. The SEM describes the late time period of the transient target signature as a sum of damped exponentials with natural resonant frequencies (NRFs). The aspect-independent and purely target geometry and material-dependent nature of the NRF set make it an excellent feature set for target characterization. In this chapter, we aim to review the background and the state of the art of resonance-based target recognition. The theoretical framework of SEM is introduced, followed by signal processing techniques that retrieve the targetdependent NRFs embedded in the transient electromagnetic target signatures. The extinction pulse, a well-known target recognition technique, is discussed. This chapter covers recent developments in using a polarimetric signature for target recognition, as well as using NRFs for subsurface sensing applications. The chapter concludes with some high-

Keywords: radar target recognition, ultra wideband radar, transient electromagnetic

The need to quickly and accurately identify enemies in confrontational situations is essential to most defense applications. Such decisions often rely upon radar target recognition. The two primary functions of radar are inherent in the acronym, whose letters stand for radio detection and ranging. There are two main categories of radar target recognition techniques: imaging

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Applications to Automated Target Recognition

#### **Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition** Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

DOI: 10.5772/intechopen.75059

Hoi-Shun Lui, Faisal Aldhubaib, Stuart Crozier and Nicholas V. Shuley Hoi-Shun Lui, Faisal Aldhubaib, Stuart Crozier and Nicholas V. Shuley

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75059

#### Abstract

Reliable radar target recognition has long been the holy grail of electromagnetic sensors. Target recognition based on the singularity expansion method (SEM) uses a time-domain electromagnetic signature and has been well studied over the last few decades. The SEM describes the late time period of the transient target signature as a sum of damped exponentials with natural resonant frequencies (NRFs). The aspect-independent and purely target geometry and material-dependent nature of the NRF set make it an excellent feature set for target characterization. In this chapter, we aim to review the background and the state of the art of resonance-based target recognition. The theoretical framework of SEM is introduced, followed by signal processing techniques that retrieve the targetdependent NRFs embedded in the transient electromagnetic target signatures. The extinction pulse, a well-known target recognition technique, is discussed. This chapter covers recent developments in using a polarimetric signature for target recognition, as well as using NRFs for subsurface sensing applications. The chapter concludes with some highlights of the ongoing challenges in the field.

Keywords: radar target recognition, ultra wideband radar, transient electromagnetic scattering, singularity expansion method

#### 1. Introduction

The need to quickly and accurately identify enemies in confrontational situations is essential to most defense applications. Such decisions often rely upon radar target recognition. The two primary functions of radar are inherent in the acronym, whose letters stand for radio detection and ranging. There are two main categories of radar target recognition techniques: imaging

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and what is termed signature recognition. Imaging radars provide a visualization of the target using techniques such as focus spot scanning and inverse synthetic aperture [1]. Signature recognition radar extracts some characteristics or a feature set that characterizes the target. Some of the techniques, such as radar cross section (RCS) [2], polarization techniques [3], highresolution range profiles (HRRP) [4], scattering centers [5], and multiple frequency measurements [6], are all under this category. The main drawback of these techniques is that the extracted parameters usually vary with incident aspect. For most radar target recognition problems, usually, the incident aspect angles of the target are not known a priori. It is, therefore, preferable to implement a technique that is purely dependent on the target itself and independent of its aspect to the radar.

respectively. The scattered field and thus the RCS of a definitive target, in general, vary as a function of incident aspects, receiving aspects, and excitation frequency. In the resonant regime which the wavelength of illumination (λ) and the target size (L) are comparable (0:4λ ≤ L ≤ 10λ), every part of the target affects every other parts. The total field of any part of the target is the vectorial sum of the incident and scattered field due to every part of the scattering body. This collective interaction determines the overall electromagnetic current induced on the target, which explains why the induced current at resonant frequencies is

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

145

Resonance-based target recognition is based on the resonating electromagnetic current induced on the target such that the ratio of excitation wavelength (λ) and the target size (L) has to be within the resonance scattering regime. The induced current serves as a secondary source and reradiates such that the resonant modes are embedded in the scattered target response. Rather than a frequency-domain characterization using RCS, one can also illuminate the target at a particular aspect in the time domain through a short-pulse excitation, measure the corresponding transient response at the same position (monostatic) or any another aspect (bistatic), and obtain the impulse response of the target. If one applies a Fourier transform to the target impulse response, this is the same as obtaining the frequency-domain "transfer function" of the target, thus evaluating the frequency dependency of the RCS at a particular transmit-receive configuration. Indeed, we assume that the bandwidth of the pulse is wide enough such that it covers at least the first few dominant resonant frequencies of the target.

In the context of linear time-invariant (LTI) systems, the impulse response characterizes the behavior of the system and circuit. In the mid-1960s, Kennaugh and Moffatt [14] extended the concept of the impulse response and applied it to transient electromagnetic scattering from definitive targets. In the early 1970s, Baum [7] introduced the SEM that describes transient scattering phenomena. According to SEM, the entire transient target signature can be divided into two parts, namely, the early time and the late time period. Conceptually, the early time period is defined as the period from when the electromagnetic pulse initially strikes the target until the target is fully illuminated, while the commencement of the late time period is the time when global resonance has been fully established and the excitation is no longer present.

In the early time, the target is partly illuminated such that the majority of the scattering events originate locally from different parts of the target: specular reflections, diffraction from nonplanar surfaces, edges, and corners. The occurrence of these events depends on when the pulse strikes on these edges and corners, and thus these early events are local, aspect dependent, and polarization dependent. They can be individually treated as scattering centers [5] or more generally using a wavefront description [15]. In the time domain, these scattering events correspond to impulse-like components, and they are time-varying (nonstationary) and can

In contrast, the commencement of the late time period is the time when the target is fully illuminated such that resonant modes at distinct frequencies are fully established. As previously mentioned the total field at any part of the target is a collective interaction of the incident

be modeled in a circuit context using an entire function [7, 16, 17].

dependent on the physical attributes of the target.

2.1. Singularity expansion method

One of the methods that overcome the aspet-dependent limitation is termed resonance-based target recognition [7]. As the name states, resonance-based target recognition essentially characterizes the radar target based on the natural resonant frequencies (NRFs) embedded in the target response. These NRFs are purely dependent on the physical attributes of the radar target, i.e., its dielectric properties and physical geometry, and these parameters are independent on the incident aspect [7] and incident polarization states [8]. Provided of course the resonances are well excited, the feasibility of using the NRFs for target recognition has been successfully demonstrated in the literature [9–11].

This chapter aims to provide an overview of the fundamentals and the development of resonance-based target recognition. We commence the chapter with a short discussion on RCS—a well-known frequency-domain method on how electromagnetic scattering is characterized at microwave frequencies. The singularity expansion method (SEM), which is the theoretical framework for resonance-based target recognition, provides the physical description of the transient scattering phenomena in time and frequency domains. Technical solutions for retrieving the target-dependent NRFs from the transient target signatures and automated target recognition (ATR) algorithms will be covered. The recent development including the prospect of using full polarimetric signatures, as well as the potential use of the techniques in other sensing applications, is also discussed. Comments and suggestions for future development are considered in conclusion.

#### 2. Resonance-based target recognition

RCS is a well-known technique to characterize target from 3 MHz up to 300 MHz (HF to mm) [2]. Attempts have also been made to perform RCS measurement in the terahertz frequency region [12, 13]. RCS is a measure of the power that is returned or scattered in a given direction, normalized with respect to the power of the incident field. Mathematically, the RCS (σ) of a target is defined as [2]

$$\sigma = 4\pi \lim\_{R \to \infty} R^2 \frac{|\mathbf{E\_s}|}{|\mathbf{E\_i}|},\tag{1}$$

where R is the range from the radar to the target and Ei and Es are the amplitudes of the incident electric field from the radar transmitter and the scattered electric field from the target, respectively. The scattered field and thus the RCS of a definitive target, in general, vary as a function of incident aspects, receiving aspects, and excitation frequency. In the resonant regime which the wavelength of illumination (λ) and the target size (L) are comparable (0:4λ ≤ L ≤ 10λ), every part of the target affects every other parts. The total field of any part of the target is the vectorial sum of the incident and scattered field due to every part of the scattering body. This collective interaction determines the overall electromagnetic current induced on the target, which explains why the induced current at resonant frequencies is dependent on the physical attributes of the target.

Resonance-based target recognition is based on the resonating electromagnetic current induced on the target such that the ratio of excitation wavelength (λ) and the target size (L) has to be within the resonance scattering regime. The induced current serves as a secondary source and reradiates such that the resonant modes are embedded in the scattered target response. Rather than a frequency-domain characterization using RCS, one can also illuminate the target at a particular aspect in the time domain through a short-pulse excitation, measure the corresponding transient response at the same position (monostatic) or any another aspect (bistatic), and obtain the impulse response of the target. If one applies a Fourier transform to the target impulse response, this is the same as obtaining the frequency-domain "transfer function" of the target, thus evaluating the frequency dependency of the RCS at a particular transmit-receive configuration. Indeed, we assume that the bandwidth of the pulse is wide enough such that it covers at least the first few dominant resonant frequencies of the target.

#### 2.1. Singularity expansion method

and what is termed signature recognition. Imaging radars provide a visualization of the target using techniques such as focus spot scanning and inverse synthetic aperture [1]. Signature recognition radar extracts some characteristics or a feature set that characterizes the target. Some of the techniques, such as radar cross section (RCS) [2], polarization techniques [3], highresolution range profiles (HRRP) [4], scattering centers [5], and multiple frequency measurements [6], are all under this category. The main drawback of these techniques is that the extracted parameters usually vary with incident aspect. For most radar target recognition problems, usually, the incident aspect angles of the target are not known a priori. It is, therefore, preferable to implement a technique that is purely dependent on the target itself

One of the methods that overcome the aspet-dependent limitation is termed resonance-based target recognition [7]. As the name states, resonance-based target recognition essentially characterizes the radar target based on the natural resonant frequencies (NRFs) embedded in the target response. These NRFs are purely dependent on the physical attributes of the radar target, i.e., its dielectric properties and physical geometry, and these parameters are independent on the incident aspect [7] and incident polarization states [8]. Provided of course the resonances are well excited, the feasibility of using the NRFs for target recognition has been

This chapter aims to provide an overview of the fundamentals and the development of resonance-based target recognition. We commence the chapter with a short discussion on RCS—a well-known frequency-domain method on how electromagnetic scattering is characterized at microwave frequencies. The singularity expansion method (SEM), which is the theoretical framework for resonance-based target recognition, provides the physical description of the transient scattering phenomena in time and frequency domains. Technical solutions for retrieving the target-dependent NRFs from the transient target signatures and automated target recognition (ATR) algorithms will be covered. The recent development including the prospect of using full polarimetric signatures, as well as the potential use of the techniques in other sensing applications, is also discussed. Comments and suggestions for future develop-

RCS is a well-known technique to characterize target from 3 MHz up to 300 MHz (HF to mm) [2]. Attempts have also been made to perform RCS measurement in the terahertz frequency region [12, 13]. RCS is a measure of the power that is returned or scattered in a given direction, normalized with respect to the power of the incident field. Mathematically, the RCS (σ) of a

where R is the range from the radar to the target and Ei and Es are the amplitudes of the incident electric field from the radar transmitter and the scattered electric field from the target,

<sup>R</sup><sup>2</sup> j j Es

Ei j j , (1)

σ ¼ 4π lim R!∞

and independent of its aspect to the radar.

144 Antennas and Wave Propagation

successfully demonstrated in the literature [9–11].

ment are considered in conclusion.

target is defined as [2]

2. Resonance-based target recognition

In the context of linear time-invariant (LTI) systems, the impulse response characterizes the behavior of the system and circuit. In the mid-1960s, Kennaugh and Moffatt [14] extended the concept of the impulse response and applied it to transient electromagnetic scattering from definitive targets. In the early 1970s, Baum [7] introduced the SEM that describes transient scattering phenomena. According to SEM, the entire transient target signature can be divided into two parts, namely, the early time and the late time period. Conceptually, the early time period is defined as the period from when the electromagnetic pulse initially strikes the target until the target is fully illuminated, while the commencement of the late time period is the time when global resonance has been fully established and the excitation is no longer present.

In the early time, the target is partly illuminated such that the majority of the scattering events originate locally from different parts of the target: specular reflections, diffraction from nonplanar surfaces, edges, and corners. The occurrence of these events depends on when the pulse strikes on these edges and corners, and thus these early events are local, aspect dependent, and polarization dependent. They can be individually treated as scattering centers [5] or more generally using a wavefront description [15]. In the time domain, these scattering events correspond to impulse-like components, and they are time-varying (nonstationary) and can be modeled in a circuit context using an entire function [7, 16, 17].

In contrast, the commencement of the late time period is the time when the target is fully illuminated such that resonant modes at distinct frequencies are fully established. As previously mentioned the total field at any part of the target is a collective interaction of the incident and scattered field from every single part of the target. At particular frequencies, the induced current on the target is freely resonating, and resonant modes are established. These resonant modes, known as natural resonance frequencies (NRFs), are theoretically dependent only on the physical attributes of the target and are independent of the aspect [7] and polarization [8]. This allows them to be an excellent candidate to be used as a feature set for target classification. Mathematically, the late time target signature can be modeled as a sum of damped exponentials with constant residues, i.e.,

$$r(t) = \sum\_{n=1}^{N} \left[ A\_n e^{s\_n t} + A\_n^\* e^{s\_n^\* t} \right], \quad t > T\_l \tag{2}$$

a computationally affordable manner. Prony's method addresses the above limitations, and it has drawn significant attention in the field. However, the primary drawback of Prony's method is that the accuracies of the extracted parameters are highly sensitive to noise and the estimated modal order of the signal. Since then, many techniques have been proposed which can accurately retrieve the modal order, NRFs, and residues with noisy target signatures. To date, direct extraction of NRFs from late time target signature has become the principal approach in this context, and matrix pencil methods (MPM) developed by Sarkar and Pereira

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

To illustrate how NRFs can be used for target classification, as well as the aspect dependency nature of the resides, an example of a 1 m wire target illuminated under different excitation aspect angles θ shown in Figure 1 is considered [21]. The wire target of length (ℓ) and radius (a) ratio <sup>ℓ</sup>=<sup>a</sup> <sup>¼</sup> 200 is excited by a plane wave with the electric field in the plane of the wire. The transient target responses are obtained using the indirect time-domain method [22]—the scattering problem is first solved in the frequency domain at a large number of discrete frequency points, followed by Gaussian windowing to model a Gaussian pulse excitation [23] and an inverse Fourier transform. Here, the electromagnetic problem is solved using the moment method solver FEKO [24] from 4.39 MHz to 9 GHz with 2048 equally spaced samples. Given the target and incident aspect angles to the target, the late time can be approximated by

where Ttr is the maximal transit time of the target, Tp is the effective pulse duration, and Tb is the estimated edge when the pulse strikes the leading edge of the target [11]. The Gaussian pulse commences at Tb ¼ 10ns with Tp ¼ 0:22ns. According to the geometry of Figure 1,

considered, resulting in Tl ≈ 16:6ns, 14:92ns and 11:9ns, respectively. The NRFs are extracted using the MPM [20] with late time samples from 17 to 140 ns. The first ten dominant extracted NRFs are listed in Table 1 and are compared with the ground truth—NRFs extracted using a

Figure 1. The wire scatterer with plane wave incidence (reprinted from [21] with permission from IEEE).

<sup>m</sup>=s, and the excitation angles of <sup>θ</sup> <sup>¼</sup> <sup>15</sup>�

Tl ¼ Tb þ Tp þ 2Ttr, (4)

, 45�

and 75�

are

147

[20] have become the main tool for this purpose.

Ttr <sup>¼</sup> <sup>ℓ</sup>cosθ=<sup>c</sup> where <sup>c</sup> <sup>¼</sup> <sup>3</sup> � 108

root searching procedure of the ½ � Z matrix [25].

where N is the modal order of the signal—the number of modes embedded in the late time response. It is assumed that only N modes are excited given that the pulse excitation is band-limited. Tl is the onset of the late time and \* denotes the complex conjugate. An is the aspect-dependent residues. sn ¼ σ<sup>n</sup> � jω<sup>n</sup> is the complex NRF. σnð Þ < 0 and ω<sup>n</sup> are the damping coefficients and resonant frequencies of the nth mode, respectively.

#### 2.2. Resonance extractions for target classification

Target classification and recognition rely on accurate extraction of the target-dependent NRFs. There are two main approaches to extract these NRFs. The first one is based on the mathematical formulation of the scattering problem. Through a Fourier transform, the short-pulse excitation corresponds to a broadband of frequencies in the frequency domain. If we formulate the entire scattering problem at a particular frequency via an integral equation formulation, the integral equation relationship can be written in a matrix form and solved via moment methods [18]. In general, this can be written as

$$[\mathbf{Z}][\mathbf{I}] = [\mathbf{V}]\_{\prime} \tag{3}$$

where ½ � Z is an impedance-like matrix corresponding to the target geometry and ½ �I and ½ � V are column vectors corresponding to the unknown current induced on the target and the excitation, respectively. The natural resonant modes of the target, sn ¼ σ<sup>n</sup> � jωn, are those such that the homogenous version of Eq. (3) has a nontrivial solution. This implies that the solution exists when the determinant of the ½ � Z matrix equals zero, thus establishing the singularities of the ½ � Z matrix. These singularities can be extracted using a typical (complex) root searching method such as Muller's method [7]. The ½ � Z matrix is constructed based solely on the geometry and dielectric properties of the target. The NRFs are therefore totally independent of the incident aspect angle. A major limitation is that the entire physical problem needs to be modeled using moment method, and each ½ � Z corresponds to only one frequency. This means that one needs to repeat the entire root searching process for all the frequency points, which requires extensive computation. Another limitation is that this method cannot be applied directly to measured data.

Shortly after the proposition of SEM by Baum, Van Blaricum and Mittra [19] proposed using Prony's method which directly retrieves the NRFs and residues from the late time response in a computationally affordable manner. Prony's method addresses the above limitations, and it has drawn significant attention in the field. However, the primary drawback of Prony's method is that the accuracies of the extracted parameters are highly sensitive to noise and the estimated modal order of the signal. Since then, many techniques have been proposed which can accurately retrieve the modal order, NRFs, and residues with noisy target signatures. To date, direct extraction of NRFs from late time target signature has become the principal approach in this context, and matrix pencil methods (MPM) developed by Sarkar and Pereira [20] have become the main tool for this purpose.

and scattered field from every single part of the target. At particular frequencies, the induced current on the target is freely resonating, and resonant modes are established. These resonant modes, known as natural resonance frequencies (NRFs), are theoretically dependent only on the physical attributes of the target and are independent of the aspect [7] and polarization [8]. This allows them to be an excellent candidate to be used as a feature set for target classification. Mathematically, the late time target signature can be modeled as a sum of damped exponen-

> r tðÞ¼ <sup>X</sup> N

2.2. Resonance extractions for target classification

[18]. In general, this can be written as

directly to measured data.

n¼1

damping coefficients and resonant frequencies of the nth mode, respectively.

Ane

snt <sup>þ</sup> <sup>A</sup><sup>∗</sup> ne s∗

where N is the modal order of the signal—the number of modes embedded in the late time response. It is assumed that only N modes are excited given that the pulse excitation is band-limited. Tl is the onset of the late time and \* denotes the complex conjugate. An is the aspect-dependent residues. sn ¼ σ<sup>n</sup> � jω<sup>n</sup> is the complex NRF. σnð Þ < 0 and ω<sup>n</sup> are the

Target classification and recognition rely on accurate extraction of the target-dependent NRFs. There are two main approaches to extract these NRFs. The first one is based on the mathematical formulation of the scattering problem. Through a Fourier transform, the short-pulse excitation corresponds to a broadband of frequencies in the frequency domain. If we formulate the entire scattering problem at a particular frequency via an integral equation formulation, the integral equation relationship can be written in a matrix form and solved via moment methods

where ½ � Z is an impedance-like matrix corresponding to the target geometry and ½ �I and ½ � V are column vectors corresponding to the unknown current induced on the target and the excitation, respectively. The natural resonant modes of the target, sn ¼ σ<sup>n</sup> � jωn, are those such that the homogenous version of Eq. (3) has a nontrivial solution. This implies that the solution exists when the determinant of the ½ � Z matrix equals zero, thus establishing the singularities of the ½ � Z matrix. These singularities can be extracted using a typical (complex) root searching method such as Muller's method [7]. The ½ � Z matrix is constructed based solely on the geometry and dielectric properties of the target. The NRFs are therefore totally independent of the incident aspect angle. A major limitation is that the entire physical problem needs to be modeled using moment method, and each ½ � Z corresponds to only one frequency. This means that one needs to repeat the entire root searching process for all the frequency points, which requires extensive computation. Another limitation is that this method cannot be applied

Shortly after the proposition of SEM by Baum, Van Blaricum and Mittra [19] proposed using Prony's method which directly retrieves the NRFs and residues from the late time response in

<sup>n</sup><sup>t</sup> � �, t > Tl (2)

½ � Z ½�¼ I ½ � V , (3)

tials with constant residues, i.e.,

146 Antennas and Wave Propagation

To illustrate how NRFs can be used for target classification, as well as the aspect dependency nature of the resides, an example of a 1 m wire target illuminated under different excitation aspect angles θ shown in Figure 1 is considered [21]. The wire target of length (ℓ) and radius (a) ratio <sup>ℓ</sup>=<sup>a</sup> <sup>¼</sup> 200 is excited by a plane wave with the electric field in the plane of the wire. The transient target responses are obtained using the indirect time-domain method [22]—the scattering problem is first solved in the frequency domain at a large number of discrete frequency points, followed by Gaussian windowing to model a Gaussian pulse excitation [23] and an inverse Fourier transform. Here, the electromagnetic problem is solved using the moment method solver FEKO [24] from 4.39 MHz to 9 GHz with 2048 equally spaced samples. Given the target and incident aspect angles to the target, the late time can be approximated by

$$T\_l = T\_b + T\_p + 2T\_{tr} \tag{4}$$

where Ttr is the maximal transit time of the target, Tp is the effective pulse duration, and Tb is the estimated edge when the pulse strikes the leading edge of the target [11]. The Gaussian pulse commences at Tb ¼ 10ns with Tp ¼ 0:22ns. According to the geometry of Figure 1, Ttr <sup>¼</sup> <sup>ℓ</sup>cosθ=<sup>c</sup> where <sup>c</sup> <sup>¼</sup> <sup>3</sup> � 108 <sup>m</sup>=s, and the excitation angles of <sup>θ</sup> <sup>¼</sup> <sup>15</sup>� , 45� and 75� are considered, resulting in Tl ≈ 16:6ns, 14:92ns and 11:9ns, respectively. The NRFs are extracted using the MPM [20] with late time samples from 17 to 140 ns. The first ten dominant extracted NRFs are listed in Table 1 and are compared with the ground truth—NRFs extracted using a root searching procedure of the ½ � Z matrix [25].

Figure 1. The wire scatterer with plane wave incidence (reprinted from [21] with permission from IEEE).

The results show that all the first ten dominant resonant modes can be extracted only at <sup>θ</sup> <sup>¼</sup> <sup>15</sup>� , while some modes cannot be retrieved at <sup>θ</sup> <sup>¼</sup> <sup>45</sup>� and <sup>θ</sup> <sup>¼</sup> <sup>75</sup>� . To investigate of what is happening as the incidence angle varies, we transform the transient signatures to the joint timefrequency (TF) domain such that the existence and occurrences of the NRFs and scattering phenomena are clearly observed. Of all the time-frequency distributions (TFDs) in the Cohen class and the reassigned TFDs [26, 27], the Smooth Pseudo Wigner-Ville Distribution (SPWVD) gives reasonable TF resolutions without introducing uninterpretable artifacts [28–30].

The SPWVD, together with the corresponding time and frequency responses of the three signals, is shown in Figure 2(a)–(c). At <sup>θ</sup> <sup>¼</sup> <sup>15</sup>� , all the ten modes are identified in the joint TF domain. In Figure 2(b), modes 1 to 5 and mode 9 are observed when the incident angle is changed to 45� . Mode 6 is not observed due to its small residue. Two resonant modes at about 1.2 and 1.5 GHz are also found in the figure. They have similar frequencies to modes 8 and 10 but with higher damping factors (marked with ^ in Table 1), which could probably be the higher layer NRFs [31]. As the incident angle changes to 75� , only modes 1 to 4, 6, 7, and 9 are excited, and they are be found correspondingly to those frequencies in Figure 2(c). In addition to the TF results, the NRFs also appear as peaks in the frequency response. As shown in both TF and frequency domains, it is apparent that the strength of the resonant modes, i.e., the residues, changes as the incident aspect varies, which validates the aspect dependency nature of the residues.

#### 2.3. Extinction pulse technique

The target-dependent nature of the NRFs implies that the NRF patterns appearing in the Splane are unique for a given target. Target recognition can thus be easily achieved by visually inspecting the patterns of the NRFs in the S-plane [32]. To automate the identification


procedure, Rothwell [9–11] proposed that target recognition can be performed by convolving the target signatures with a special type of filter, known as the "extinction pulse" or the "E-pulse," in the time domain. The E-pulse is specially designed such that it will annul all the NRFs embedded in the late time response only if it is convolved with the response from the

eð Þτ r tð Þ � τ dτ ¼ 0 for t > TL ¼ Tl þ TE, (5)

, (b) <sup>θ</sup> <sup>¼</sup> <sup>45</sup>�

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

149

, and (c)<sup>θ</sup> <sup>¼</sup> <sup>75</sup>�

(reprinted

"true target." Mathematically, the E-pulse, e tð Þ, can be defined as [9–11]

Figure 2. SPWVD of the wire scatterer with plane wave incidence at (a) <sup>θ</sup> <sup>¼</sup> <sup>15</sup>�

c tðÞ¼ <sup>ð</sup> TE

from [21] with permission from IEEE).

0

Table 1. Comparison between the extracted NRFs (snL=c) using ½ � Z matrix and MPM (reprinted from [21] with permission from IEEE).

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition http://dx.doi.org/10.5772/intechopen.75059 149

The results show that all the first ten dominant resonant modes can be extracted only at <sup>θ</sup> <sup>¼</sup> <sup>15</sup>�

ing as the incidence angle varies, we transform the transient signatures to the joint timefrequency (TF) domain such that the existence and occurrences of the NRFs and scattering phenomena are clearly observed. Of all the time-frequency distributions (TFDs) in the Cohen class and the reassigned TFDs [26, 27], the Smooth Pseudo Wigner-Ville Distribution (SPWVD)

The SPWVD, together with the corresponding time and frequency responses of the three

domain. In Figure 2(b), modes 1 to 5 and mode 9 are observed when the incident angle is

1.2 and 1.5 GHz are also found in the figure. They have similar frequencies to modes 8 and 10 but with higher damping factors (marked with ^ in Table 1), which could probably be the

excited, and they are be found correspondingly to those frequencies in Figure 2(c). In addition to the TF results, the NRFs also appear as peaks in the frequency response. As shown in both TF and frequency domains, it is apparent that the strength of the resonant modes, i.e., the residues, changes as the incident aspect varies, which validates the aspect dependency nature

The target-dependent nature of the NRFs implies that the NRF patterns appearing in the Splane are unique for a given target. Target recognition can thus be easily achieved by visually inspecting the patterns of the NRFs in the S-plane [32]. To automate the identification

1 138 �0:260 � j2:91 �0:252 � j2:87 �0:253 � j2:87 �0:252 � j2:87 2 286 �0:381 � j6:01 �0:372 � j5:93 �0:370 � j5:93 �0:373 � j5:93 3 432 �0:468 � j9:06 �0:455 � j9:01 �0:458 � j9:01 �0:444 � j9:05 4 583 �0:538 � j12:2 �0:525 � j12:1 �0:512 � j12:1 �0:545 � j12:1

6 879 �0:654 � j18:4 �0:637 � j18:3 �0:833 � j18:4 �0:881 � j17:6 7 1027 �0:704 � j21:5 �0:692 � j21:4 �0:850 � j21:6

9 1323 �0:792 � j27:7 �0:785 � j27:7 �0:732 � j27:9 �1:005 � j28:6

Table 1. Comparison between the extracted NRFs (snL=c) using ½ � Z matrix and MPM (reprinted from [21] with

, snL=c 45�

gives reasonable TF resolutions without introducing uninterpretable artifacts [28–30].

and <sup>θ</sup> <sup>¼</sup> <sup>75</sup>�

. Mode 6 is not observed due to its small residue. Two resonant modes at about

while some modes cannot be retrieved at <sup>θ</sup> <sup>¼</sup> <sup>45</sup>�

signals, is shown in Figure 2(a)–(c). At <sup>θ</sup> <sup>¼</sup> <sup>15</sup>�

higher layer NRFs [31]. As the incident angle changes to 75�

n f (MHz) Z½ � [25] Matrix pencil method [20] snL=c 15�

5 730 �0:600 � j15:3 �0:585 � j15:2 �0:609 � j15:2

8 1175 �0:749 � j24:6 �0:733 � j24:6 �1:043 � j24:5^

10 1471 �0:832 � j30:8 �0:817 � j30:8 �1:294 � j30:9^

changed to 45�

148 Antennas and Wave Propagation

of the residues.

permission from IEEE).

2.3. Extinction pulse technique

,

. To investigate of what is happen-

, only modes 1 to 4, 6, 7, and 9 are

, snL=c 75�

, snL=c

, all the ten modes are identified in the joint TF

Figure 2. SPWVD of the wire scatterer with plane wave incidence at (a) <sup>θ</sup> <sup>¼</sup> <sup>15</sup>� , (b) <sup>θ</sup> <sup>¼</sup> <sup>45</sup>� , and (c)<sup>θ</sup> <sup>¼</sup> <sup>75</sup>� (reprinted from [21] with permission from IEEE).

procedure, Rothwell [9–11] proposed that target recognition can be performed by convolving the target signatures with a special type of filter, known as the "extinction pulse" or the "E-pulse," in the time domain. The E-pulse is specially designed such that it will annul all the NRFs embedded in the late time response only if it is convolved with the response from the "true target." Mathematically, the E-pulse, e tð Þ, can be defined as [9–11]

$$c(t) = \int\_{0}^{T\_E} c(\tau)r(t - \tau)d\tau = 0 \quad \text{for} \quad t > T\_L = T\_l + T\_{E'} \tag{5}$$

where r tð Þ is the late time target signature defined in Eq. (2), Tl is the commencement of the late time period, and TE is the duration of e tð Þ.

To illustrate how E-pulse operates, an ATR scenario is shown in Figure 3 [11]. The goal here is to identify the target given a measured target signature v tð Þ from an unknown target [11]. In the target library, a number of E-pulses corresponding to different targets are stored. To perform ATR, v tð Þ is convolved with each of the E-pulses independently. According to Eq. (5), the one with a null response or smallest signal strength corresponds to the "true" target. Target identification is thus performed by monitoring the output of the convolution and picks up the one with the small energy level. To automate the process, the output is quantified using E-pulse discrimination number, EDN, and the E-pulse discrimination ratio, EDR, which are defined as follows [11]:

$$EDN = \left[\int\_{T\_L}^{T\_L+W} \left(e(t)^\* v(t)\right)^2 dt\right] \left[\int\_0^{T\_f} e^2(t) dt\right]^{-1} \tag{6}$$

$$EDR = 10\log\_{10}\left[\frac{EDN}{\min\{EDN\}}\right] \tag{7}$$

EDNp, <sup>q</sup> ¼

formulation in [10] together with the NRFs extracted from MPM.

3. Recent developments

Ð TLþW TL

Figure 4. Validation of the E-pulse technique for new applications with known targets: (a) to (c) illustrate how the EDNp, <sup>q</sup> and DRp, <sup>q</sup> values are computed for the case of three known targets with target signatures (r1ð Þt , r2ð Þt , r3ð Þt ) and E-pulses (e1ð Þt ,e2ð Þt ,e3ð Þt ). As we know a priori of the "true" target, our goal is to test if we could obtain positive DRp, <sup>q</sup> (p 6¼ q) values.

> Ð Te 0 e2 <sup>q</sup> ð Þ<sup>t</sup> dt " # <sup>Ð</sup>

DRp, <sup>q</sup> ¼ 10log10

.EDNp, <sup>p</sup> is the case when the E-pulse and target response are from the same target, while EDNp, <sup>q</sup> is the case which the E-pulse and target returns are from different targets. A positive value of DRp, <sup>q</sup> implies that EDNp, <sup>q</sup> is greater than EDNp, <sup>p</sup> and thus a successful target recognition. As we have a priori knowledge of correct targets, positive DRp, <sup>q</sup> values are simply a validation of the success of E-pulse ATR or detection in the aforementioned new applications. In practice, the E-pulse of a target can be obtained by numerically solving the convolution integral in Eq. (5) given the target signature [36] or using the formulation in [10] given the NRFs of the target. The E-pulses used in our studies (e.g., [33–35]) are constructed using the

Upon the introduction of SEM in the 1970s until the mid-1990s, research has mainly concentrated on three directions: theoretical studies with better modeling and description of early

eqð Þ<sup>t</sup> <sup>∗</sup> rpð Þ<sup>t</sup> � �<sup>2</sup> dt " #

> TLþW TL r2

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

151

EDNp, <sup>q</sup> EDNp, <sup>p</sup>

<sup>p</sup>ð Þ<sup>t</sup> dt " # (8)

� � (9)

Therefore, the target signature yielding the smallest EDN and 0 dB EDR is the one corresponding to the target of interest. This forms the basis for E-pulse ATR.

In most of our studies, we know a priori which one is the "right" target, and our goal is to determine if E-pulse is capable of discriminating the targets in new applications; for instance, the "banded" E-pulse technique that better discriminates between similar targets [33], a novel technique for subsurface target detection [34], and ATR using polarimetric signatures (Section 3.1). Figure 4 shows the flowchart of how we validate the E-pulse technique. For the case of three targets, there are three target signatures and three E-pulses, resulting in nine convolutions. Instead of using EDN and EDR in Eqs. (6) and (7), we modify them and introduce EDNp, <sup>q</sup> and discrimination ratio, DRp, <sup>q</sup>, to quantify and convolution outcome and discrimination performance. They are given as follows [33, 35]:

Figure 3. Automated target recognition using the E-pulse technique [11]. The goal is to determine which target does the unknown target signature v tð Þ correspond to. v tð Þ is convolved with all the E-pulses in the target library, and the corresponding EDN and EDR values are computed. the E-pulse that results in minimum value of EDN or equivalently 0 dB of EDR indicates the E-pulse "matches" with v tð Þ—The true target is thus the one that generate this E-pulse.

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition http://dx.doi.org/10.5772/intechopen.75059 151

Figure 4. Validation of the E-pulse technique for new applications with known targets: (a) to (c) illustrate how the EDNp, <sup>q</sup> and DRp, <sup>q</sup> values are computed for the case of three known targets with target signatures (r1ð Þt , r2ð Þt , r3ð Þt ) and E-pulses (e1ð Þt ,e2ð Þt ,e3ð Þt ). As we know a priori of the "true" target, our goal is to test if we could obtain positive DRp, <sup>q</sup> (p 6¼ q) values.

$$EDN\_{p,q} = \frac{\begin{bmatrix} \int\_{\tau\_L}^{T\_L+W} \left( e\_q(t)^\* r\_p(t) \right)^2 dt \\ \int\_{\tau\_L}^{T\_\epsilon} e\_q^2(t) dt \end{bmatrix}}{\begin{bmatrix} \int\_0^{T\_\epsilon} e\_q^2(t) dt \\ \int\_{\tau\_L}^{T\_\epsilon} r\_p^2(t) dt \end{bmatrix}} \tag{8}$$

$$DR\_{p,q} = 10 \log\_{10} \left[ \frac{EDN\_{p,q}}{EDN\_{p,p}} \right] \tag{9}$$

.EDNp, <sup>p</sup> is the case when the E-pulse and target response are from the same target, while EDNp, <sup>q</sup> is the case which the E-pulse and target returns are from different targets. A positive value of DRp, <sup>q</sup> implies that EDNp, <sup>q</sup> is greater than EDNp, <sup>p</sup> and thus a successful target recognition. As we have a priori knowledge of correct targets, positive DRp, <sup>q</sup> values are simply a validation of the success of E-pulse ATR or detection in the aforementioned new applications.

In practice, the E-pulse of a target can be obtained by numerically solving the convolution integral in Eq. (5) given the target signature [36] or using the formulation in [10] given the NRFs of the target. The E-pulses used in our studies (e.g., [33–35]) are constructed using the formulation in [10] together with the NRFs extracted from MPM.

#### 3. Recent developments

where r tð Þ is the late time target signature defined in Eq. (2), Tl is the commencement of the late

To illustrate how E-pulse operates, an ATR scenario is shown in Figure 3 [11]. The goal here is to identify the target given a measured target signature v tð Þ from an unknown target [11]. In the target library, a number of E-pulses corresponding to different targets are stored. To perform ATR, v tð Þ is convolved with each of the E-pulses independently. According to Eq. (5), the one with a null response or smallest signal strength corresponds to the "true" target. Target identification is thus performed by monitoring the output of the convolution and picks up the one with the small energy level. To automate the process, the output is quantified using E-pulse discrimination number, EDN, and the E-pulse discrimination ratio,

e tð Þ<sup>∗</sup> ð Þ v tð Þ <sup>2</sup>

Therefore, the target signature yielding the smallest EDN and 0 dB EDR is the one

In most of our studies, we know a priori which one is the "right" target, and our goal is to determine if E-pulse is capable of discriminating the targets in new applications; for instance, the "banded" E-pulse technique that better discriminates between similar targets [33], a novel technique for subsurface target detection [34], and ATR using polarimetric signatures (Section 3.1). Figure 4 shows the flowchart of how we validate the E-pulse technique. For the case of three targets, there are three target signatures and three E-pulses, resulting in nine convolutions. Instead of using EDN and EDR in Eqs. (6) and (7), we modify them and introduce EDNp, <sup>q</sup> and discrimination ratio, DRp, <sup>q</sup>, to quantify and convolution outcome and discrimina-

Figure 3. Automated target recognition using the E-pulse technique [11]. The goal is to determine which target does the unknown target signature v tð Þ correspond to. v tð Þ is convolved with all the E-pulses in the target library, and the corresponding EDN and EDR values are computed. the E-pulse that results in minimum value of EDN or equivalently 0 dB of EDR indicates the E-pulse "matches" with v tð Þ—The true target is thus the one that generate this E-pulse.

dt

3 7 5 ð Te

2 4

EDN minf g EDN � �

0 e 2 ð Þt dt

3 5

�1

(6)

(7)

time period, and TE is the duration of e tð Þ.

150 Antennas and Wave Propagation

EDR, which are defined as follows [11]:

EDN ¼

tion performance. They are given as follows [33, 35]:

TLð þW

2 6 4

TL

corresponding to the target of interest. This forms the basis for E-pulse ATR.

EDR ¼ 10 log <sup>10</sup>

Upon the introduction of SEM in the 1970s until the mid-1990s, research has mainly concentrated on three directions: theoretical studies with better modeling and description of early time scattering phenomena [15–16], signal processing solutions to better retrieve SEM parameters for target characterization [17, 19–20, 36–39], and development of E-pulse and other ATR solutions for target recognition [11, 33, 40–44]. Most studies focused on ATR for targets in free space. The E-pulse proposed by Rothwell et al. [9–11, 36, 42] and the MPM algorithm by Sarkar and Pereira [20] have become the benchmark for filter-based ATR techniques and NRF extraction in this context. Since the 2000s, research activities have shifted toward applying the techniques developed in resonance-based target recognition for different applications. These include subsurface target detection, nondestructive evaluation, and medical diagnosis. Instead of using the linearly polarized electromagnetic wave to excite the target, the prospect of using fully polarimetric target signatures for ATR has also been investigated.

In polarimetry, the Sinclair scattering matrices [46] in the linear and circular polarization bases

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

SVH SVV and ½ �¼ S Lð Þ ;<sup>R</sup>

Here, the subscripts denote the polarization channels (sets of transmit-receive polarization), where V, H, L, and R correspond to vertical, horizontal, left-hand, and right-hand circular polarization, respectively. The scattering matrices relate the incident and scattered electric field of the target under different polarization bases at a particular frequency. The first and second subscripts in each of the term inside the Sinclair scattering matrix correspond to the polarization state from the transmitting and receiving antennas. For instance, SVH corresponds to the case where the target is illuminated using vertically polarized plane wave and the horizontally polarized scattered field is measured. Once the Sinclair scattering matrix is measured in an orthogonal basis, all the polarization states (linear, circular, and elliptical) can be synthesized such that it is not necessary to illuminate the target with a large number of linear polarization angles from the same aspect [9, 45]. Only four target signatures (three for monostatic) are required which considerably reduces the amount of data to be

To investigate the impact of the excitation and receiving states to the performance of ATR, numerical examples of three wire targets shown in Figure 5 were studied [47]. The targets are made up of two wire segments—a vertical wire segment (main body) of 1 m and a horizontal wire segment of 0.3 m (Target 1 and Target 3) and 0.2 m (Target 2) located at the center (Target 1 and Target 2) and at 0.2 m away from the end of the main body (Target 3), respectively. The scattering problems of these targets are solved in the frequency domain using FEKO [24] with 512 equally spaced frequency samples from 3.9 MHz to 2 GHz. The polarimetric transient signatures, or equivalently the scattering matrices in the time

Figure 5. Wire targets and corresponding excitation polarization references. (a) Target 1: r ¼ 0:3m, target 2: r ¼ 0:2m, and

(b) target 3 (reprinted from [47] with permission from IEEE).

SLL SLR

SRL SRR : (10)

http://dx.doi.org/10.5772/intechopen.75059

153

SHH SHV

are given by

processed.

domain, i.e.,

½ �¼ S Hð Þ ; V

In this section, we first discuss the use of full polarimetric measurement and its impact on resonance-based target recognition. The merit of using polarimetric transient signatures is demonstrated through numerical examples. Then, different strategies for handling the multiple-aspect multiple-polarization data sets in multi-static scenarios are evaluated. Toward the end of the section, research activities on applying the E-pulse technique for different applications will be covered.

#### 3.1. ATR using polarimetric signatures

The aspect dependency, as demonstrated in the above wire scattering example, as well as the polarization dependency of the residues, would limit the reliability and performance of ATR as it is uncertain whether all the dominant NRFs are well excited because the target orientation is usually not known a priori. In the mid-2000s, Shuley et al. [9] studied the residues retrieved from a target at 18 linear polarization angles at the same aspect and found that the residues of some NRFs could be so small at some polarization angles such that these NRFs are not retrieved. This was the first instance where polarization dependency of the residues is demonstrated. If we evaluate the RCS of the target at the resonant frequencies, the amplitudes of the scattered field, in general, vary as the observation aspects and polarization. This explains the aspect and polarization dependencies of the residues.

To accurately characterize a target using NRFs extracted from measured target signatures, it is important to incorporate more than one target signature obtained from different aspects and polarization states. Lui and Shuley [45] investigated different ways to process target signatures obtained from a number of polarization angles at a single aspect. Our results show that using the modified MPM [39] that allows extraction of one set of NRFs from multiple target signatures is preferred as it does not corrupt the original time-domain information and the risk of ignoring dominant modes is eliminated [45]. The major drawback is that the computational load grows as the number of target signatures (aspects and polarization angles) is increased. Also, when linear polarization is used, it is not trivial to decide how many polarization angles are sufficient. To better understand the impact of using the different polarimetric response for ATR, examples of some simple wire targets will first be presented. Then, different ways to handle data set obtained from multiple-aspect multiple polarization measurements for target classification will be presented.

In polarimetry, the Sinclair scattering matrices [46] in the linear and circular polarization bases are given by

time scattering phenomena [15–16], signal processing solutions to better retrieve SEM parameters for target characterization [17, 19–20, 36–39], and development of E-pulse and other ATR solutions for target recognition [11, 33, 40–44]. Most studies focused on ATR for targets in free space. The E-pulse proposed by Rothwell et al. [9–11, 36, 42] and the MPM algorithm by Sarkar and Pereira [20] have become the benchmark for filter-based ATR techniques and NRF extraction in this context. Since the 2000s, research activities have shifted toward applying the techniques developed in resonance-based target recognition for different applications. These include subsurface target detection, nondestructive evaluation, and medical diagnosis. Instead of using the linearly polarized electromagnetic wave to excite the target, the prospect of using

In this section, we first discuss the use of full polarimetric measurement and its impact on resonance-based target recognition. The merit of using polarimetric transient signatures is demonstrated through numerical examples. Then, different strategies for handling the multiple-aspect multiple-polarization data sets in multi-static scenarios are evaluated. Toward the end of the section, research activities on applying the E-pulse technique for different

The aspect dependency, as demonstrated in the above wire scattering example, as well as the polarization dependency of the residues, would limit the reliability and performance of ATR as it is uncertain whether all the dominant NRFs are well excited because the target orientation is usually not known a priori. In the mid-2000s, Shuley et al. [9] studied the residues retrieved from a target at 18 linear polarization angles at the same aspect and found that the residues of some NRFs could be so small at some polarization angles such that these NRFs are not retrieved. This was the first instance where polarization dependency of the residues is demonstrated. If we evaluate the RCS of the target at the resonant frequencies, the amplitudes of the scattered field, in general, vary as the observation aspects and polarization. This explains the

To accurately characterize a target using NRFs extracted from measured target signatures, it is important to incorporate more than one target signature obtained from different aspects and polarization states. Lui and Shuley [45] investigated different ways to process target signatures obtained from a number of polarization angles at a single aspect. Our results show that using the modified MPM [39] that allows extraction of one set of NRFs from multiple target signatures is preferred as it does not corrupt the original time-domain information and the risk of ignoring dominant modes is eliminated [45]. The major drawback is that the computational load grows as the number of target signatures (aspects and polarization angles) is increased. Also, when linear polarization is used, it is not trivial to decide how many polarization angles are sufficient. To better understand the impact of using the different polarimetric response for ATR, examples of some simple wire targets will first be presented. Then, different ways to handle data set obtained from multiple-aspect multiple polarization measurements for target

fully polarimetric target signatures for ATR has also been investigated.

applications will be covered.

152 Antennas and Wave Propagation

classification will be presented.

3.1. ATR using polarimetric signatures

aspect and polarization dependencies of the residues.

$$\begin{bmatrix} \mathbf{S}(H, V) \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{HH} & \mathbf{S}\_{HV} \\ \mathbf{S}\_{VH} & \mathbf{S}\_{VV} \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} \mathbf{S}(L, R) \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{LL} & \mathbf{S}\_{LR} \\ \mathbf{S}\_{RL} & \mathbf{S}\_{RR} \end{bmatrix}. \tag{10}$$

Here, the subscripts denote the polarization channels (sets of transmit-receive polarization), where V, H, L, and R correspond to vertical, horizontal, left-hand, and right-hand circular polarization, respectively. The scattering matrices relate the incident and scattered electric field of the target under different polarization bases at a particular frequency. The first and second subscripts in each of the term inside the Sinclair scattering matrix correspond to the polarization state from the transmitting and receiving antennas. For instance, SVH corresponds to the case where the target is illuminated using vertically polarized plane wave and the horizontally polarized scattered field is measured. Once the Sinclair scattering matrix is measured in an orthogonal basis, all the polarization states (linear, circular, and elliptical) can be synthesized such that it is not necessary to illuminate the target with a large number of linear polarization angles from the same aspect [9, 45]. Only four target signatures (three for monostatic) are required which considerably reduces the amount of data to be processed.

To investigate the impact of the excitation and receiving states to the performance of ATR, numerical examples of three wire targets shown in Figure 5 were studied [47]. The targets are made up of two wire segments—a vertical wire segment (main body) of 1 m and a horizontal wire segment of 0.3 m (Target 1 and Target 3) and 0.2 m (Target 2) located at the center (Target 1 and Target 2) and at 0.2 m away from the end of the main body (Target 3), respectively. The scattering problems of these targets are solved in the frequency domain using FEKO [24] with 512 equally spaced frequency samples from 3.9 MHz to 2 GHz. The polarimetric transient signatures, or equivalently the scattering matrices in the time domain, i.e.,

Figure 5. Wire targets and corresponding excitation polarization references. (a) Target 1: r ¼ 0:3m, target 2: r ¼ 0:2m, and (b) target 3 (reprinted from [47] with permission from IEEE).

$$\begin{bmatrix} \mathbf{S}\_{(H,V)}(t) \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{HH}(t) & \mathbf{S}\_{HV}(t) \\ \mathbf{S}\_{VH}(t) & \mathbf{S}\_{VV}(t) \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} \mathbf{S}\_{(L,R)}(t) \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{LL}(t) & \mathbf{S}\_{LR}(t) \\ \mathbf{S}\_{RL}(t) & \mathbf{S}\_{RR}(t) \end{bmatrix} \tag{11}$$

result which indicate the E-pulses successfully distinguish between Target 1 and Target 3. With different positions and length of the horizontal wire segments, the E-pulse is capable of distinguishing between Target 2 and Target 3 for both SVVð Þt and SHHð Þt . Under monostatic configuration and circularly polarized illumination, SLLðÞ¼ t SRRð Þt and SLRðÞ¼ t SRLð Þt , and thus we only need to consider SLLð Þt and SLRð Þt . DRp, <sup>q</sup>values (p 6¼ q) of at least 26 dB result in all cases. Such results indicate that using circularly polarized target signatures can successfully

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

155

In this example, ATR performance under different polarization states is studied. When the target is illuminated under linear polarization, only the main body or the horizontal wire segment is excited (details of the extracted NRFs can be found in [47]). The E-pulses are constructed using the incomplete set of NRFs. The target is poorly characterized and is not fully illuminated—these are the two main causes of the inconsistency in ATR performance. An example of the inconsistency in ATR performance due to aspect dependencies of the NRFs is reported in [48]. When the target is illuminated under circular polarization, the NRFs of both wire segments are adequately excited. The constructed E-pulse contains the domain NRFs of the entire target. The target is well characterized and well illuminated under circular polarization. The consistent ATR performance originates from the fact that the NRFs of both wire segments of the targets are well excited. The findings from this example demonstrated the importance of including all the dominant NRFs (including both the global and partial/substructure resonances [30]) for target classification, as well as the importance of exciting all the

dominant NRFs, especially when a library of similar targets is considered [33].

3.2. Target classification using multiple-aspect multiple-polarization data set

Owing to the aspect and polarization dependencies of the residues, it is unlikely that the entire set of dominant NRFs can be excited from only one target signature. As shown above, target signatures obtained from different aspects and polarization states excite a different subset of dominant NRFs. Certainly, the use of multiple target signatures obtained from multiple aspect and polarization states for target characterization allows us to retrieve at least a larger subset of dominant NRFs within the frequency bandwidth. The multiple-aspect multiple-polarization data set [49], a data set that consists of transient target signatures obtained with different transmit-receive configurations and polarization basis, is thus required. There are a number of possible ways to illuminate the target and post-process these target signatures, and we want to identify an efficient way to handle the data. To illustrate the different possible ways to handle such large data sets, an example of a simple human breast model shown in Figure 6 (a) and (b), a lossless dielectric hemisphere with a small different dielectric spheres embedded, that mimics the breast cancer detection scenario [50–53] is used. The radius of the lossless hemisphere is 60 mm with the relative permittivity of 5 (fat infiltrated tissue at ~3 GHz [54]). A 10 mm radius lossless dielectric sphere with a relative permittivity of 50 (taken from the Debye model [55] for <3 GHz) embedded inside the hemisphere is used to model the tumor. The target is illuminated using plane wave at six different aspects (<sup>θ</sup> <sup>¼</sup> <sup>105</sup>�

and

, where θ is measured from the positive z-axis, while

discriminate the three targets.

<sup>ϕ</sup> <sup>¼</sup> <sup>30</sup>�

, 60�

, 90�

, 120�

, 150�

and 180�

are determined using the aforementioned indirect time-domain method [22]. For each target, the NRFs from each target signature is extracted using MPM [20] resulting eight sets of NRFs and residues. At each polarization state, the E-pulses for each of the three targets (denoted as target q) are constructed using the extracted NRFs [10]. To evaluate the E-pulse ATR performance, the E-pulse validation procedures shown in Figure 4 are applied. The E-pulse of each target (denoted as target q) is convolved with the target signatures from different targets (denoted as target p) but with the same polarization state. Before the convolution, both the E-pulses and target signatures are resampled as usually the sampling rate of the E-pulse and the target signatures are not the same [35]. The EDNp, <sup>q</sup> and DRp, <sup>q</sup> are computed, resulting in nine sets of EDNp, <sup>q</sup>s and DRp, <sup>q</sup>s for each polarization state. The corresponding results are tabulated in Table 2.

Under vertical excitation, the main body is excited but not the horizontal wire segment. Theoretically, the cross polarized response should be zero in this case and vice versa for horizontal polarization excitation, and thus we only consider the co-polarized components. As tabulated in Table 2, the E-pulse technique fails to recognize between Target 1 and Target 2 for the case of SVVð Þt , with DR1, <sup>2</sup> and DR2,<sup>1</sup> values near to 0 dB as only the NRFs corresponding to the main body are excited. For the case of SHHð Þt , the horizontal wire segments of the two targets are well excited, and DR1,<sup>2</sup> and DR2,<sup>1</sup> values of 46.6 and 126.2 dB are obtained, which indicates successful target recognition. However, almost 0 dB of DR1,<sup>3</sup> and DR3,<sup>1</sup> values result. This is because the length of the horizontal wire segment of Target 1 and Target 3 is identical and the transient responses are strongly dominated by the horizontal wire segments. Under vertical polarization, the current distributions of the two targets are different due to different positions of the horizontal wire segments. The DR1,<sup>3</sup> and DR3, <sup>1</sup> values of 42.9 and 65.4 dB


Table 2. ATR using target signatures under different polarization bases using E-pulse technique with the corresponding DRp, <sup>q</sup> (dB) values (reprinted from [47] with permission from IEEE).

result which indicate the E-pulses successfully distinguish between Target 1 and Target 3. With different positions and length of the horizontal wire segments, the E-pulse is capable of distinguishing between Target 2 and Target 3 for both SVVð Þt and SHHð Þt . Under monostatic configuration and circularly polarized illumination, SLLðÞ¼ t SRRð Þt and SLRðÞ¼ t SRLð Þt , and thus we only need to consider SLLð Þt and SLRð Þt . DRp, <sup>q</sup>values (p 6¼ q) of at least 26 dB result in all cases. Such results indicate that using circularly polarized target signatures can successfully discriminate the three targets.

<sup>S</sup>ð Þ <sup>H</sup>;<sup>V</sup> ð Þ<sup>t</sup> <sup>¼</sup> SHHð Þ<sup>t</sup> SHVð Þ<sup>t</sup>

154 Antennas and Wave Propagation

corresponding results are tabulated in Table 2.

SVHð Þt SVVð Þt 

DRp,q(dB) (a) SVV ð Þt (b) SHH ð Þt

DRp,q(dB) (c) SLLð Þt (d) SLRð Þt

DRp, <sup>q</sup> (dB) values (reprinted from [47] with permission from IEEE).

are determined using the aforementioned indirect time-domain method [22]. For each target, the NRFs from each target signature is extracted using MPM [20] resulting eight sets of NRFs and residues. At each polarization state, the E-pulses for each of the three targets (denoted as target q) are constructed using the extracted NRFs [10]. To evaluate the E-pulse ATR performance, the E-pulse validation procedures shown in Figure 4 are applied. The E-pulse of each target (denoted as target q) is convolved with the target signatures from different targets (denoted as target p) but with the same polarization state. Before the convolution, both the E-pulses and target signatures are resampled as usually the sampling rate of the E-pulse and the target signatures are not the same [35]. The EDNp, <sup>q</sup> and DRp, <sup>q</sup> are computed, resulting in nine sets of EDNp, <sup>q</sup>s and DRp, <sup>q</sup>s for each polarization state. The

Under vertical excitation, the main body is excited but not the horizontal wire segment. Theoretically, the cross polarized response should be zero in this case and vice versa for horizontal polarization excitation, and thus we only consider the co-polarized components. As tabulated in Table 2, the E-pulse technique fails to recognize between Target 1 and Target 2 for the case of SVVð Þt , with DR1, <sup>2</sup> and DR2,<sup>1</sup> values near to 0 dB as only the NRFs corresponding to the main body are excited. For the case of SHHð Þt , the horizontal wire segments of the two targets are well excited, and DR1,<sup>2</sup> and DR2,<sup>1</sup> values of 46.6 and 126.2 dB are obtained, which indicates successful target recognition. However, almost 0 dB of DR1,<sup>3</sup> and DR3,<sup>1</sup> values result. This is because the length of the horizontal wire segment of Target 1 and Target 3 is identical and the transient responses are strongly dominated by the horizontal wire segments. Under vertical polarization, the current distributions of the two targets are different due to different positions of the horizontal wire segments. The DR1,<sup>3</sup> and DR3, <sup>1</sup> values of 42.9 and 65.4 dB

p ¼ 1 p ¼ 2 p ¼ 3 p ¼ 1 p ¼ 2 p ¼ 3

p ¼ 1 p ¼ 2 p ¼ 3 p ¼ 1 p ¼ 2 p ¼ 3

q ¼ 1 0 0.01 42.9 0 46.6 0.002 q ¼ 2 �0.01 0 42.9 126.2 0 126.2 q ¼ 3 65.4 65.4 0 �0.002 53.9 0

q ¼ 1 0 31.4 36.0 0 38.9 42.9 q ¼ 2 27.0 0 43.8 26.7 0 44.6 q ¼ 3 48.4 53.1 0 43.6 49.3 0

Table 2. ATR using target signatures under different polarization bases using E-pulse technique with the corresponding

and <sup>S</sup>ð Þ <sup>L</sup>;<sup>R</sup> ð Þ<sup>t</sup> <sup>¼</sup> SLLð Þ<sup>t</sup> SLRð Þ<sup>t</sup>

SRLð Þt SRRð Þt 

, (11)

In this example, ATR performance under different polarization states is studied. When the target is illuminated under linear polarization, only the main body or the horizontal wire segment is excited (details of the extracted NRFs can be found in [47]). The E-pulses are constructed using the incomplete set of NRFs. The target is poorly characterized and is not fully illuminated—these are the two main causes of the inconsistency in ATR performance. An example of the inconsistency in ATR performance due to aspect dependencies of the NRFs is reported in [48]. When the target is illuminated under circular polarization, the NRFs of both wire segments are adequately excited. The constructed E-pulse contains the domain NRFs of the entire target. The target is well characterized and well illuminated under circular polarization. The consistent ATR performance originates from the fact that the NRFs of both wire segments of the targets are well excited. The findings from this example demonstrated the importance of including all the dominant NRFs (including both the global and partial/substructure resonances [30]) for target classification, as well as the importance of exciting all the dominant NRFs, especially when a library of similar targets is considered [33].

#### 3.2. Target classification using multiple-aspect multiple-polarization data set

Owing to the aspect and polarization dependencies of the residues, it is unlikely that the entire set of dominant NRFs can be excited from only one target signature. As shown above, target signatures obtained from different aspects and polarization states excite a different subset of dominant NRFs. Certainly, the use of multiple target signatures obtained from multiple aspect and polarization states for target characterization allows us to retrieve at least a larger subset of dominant NRFs within the frequency bandwidth. The multiple-aspect multiple-polarization data set [49], a data set that consists of transient target signatures obtained with different transmit-receive configurations and polarization basis, is thus required. There are a number of possible ways to illuminate the target and post-process these target signatures, and we want to identify an efficient way to handle the data. To illustrate the different possible ways to handle such large data sets, an example of a simple human breast model shown in Figure 6 (a) and (b), a lossless dielectric hemisphere with a small different dielectric spheres embedded, that mimics the breast cancer detection scenario [50–53] is used. The radius of the lossless hemisphere is 60 mm with the relative permittivity of 5 (fat infiltrated tissue at ~3 GHz [54]). A 10 mm radius lossless dielectric sphere with a relative permittivity of 50 (taken from the Debye model [55] for <3 GHz) embedded inside the hemisphere is used to model the tumor. The target is illuminated using plane wave at six different aspects (<sup>θ</sup> <sup>¼</sup> <sup>105</sup>� and <sup>ϕ</sup> <sup>¼</sup> <sup>30</sup>� , 60� , 90� , 120� , 150� and 180� , where θ is measured from the positive z-axis, while

Figure 6. (a) Cross-sectional views (x-y plane) of the target. (b) Cross-sectional view (<sup>ϕ</sup> <sup>¼</sup> <sup>30</sup>� ,<sup>θ</sup> <sup>¼</sup> <sup>105</sup>� ) of the breast volume under plane wave illuminations from <sup>θ</sup><sup>t</sup> <sup>¼</sup> <sup>105</sup>� , <sup>ϕ</sup><sup>t</sup> <sup>¼</sup> <sup>30</sup>� , and <sup>ϕ</sup><sup>t</sup> <sup>¼</sup> <sup>210</sup>� , respectively. (reprinted from [49] with permission from IEEE).

ϕ is measured from the positive x-axis), resulted in 36 transmit-receive combinations (6 are monostatic, and 30 are bistatic) and 8 polarization states, i.e., 288 target signatures. It is apparent that the amount of data is tremendously increased when both aspect and polarization domains are considered. Efficient methods to handle the data are thus required.

in the polarization matrix is utilized (regardless single or multiple aspects), circularly polarized components are preferred over linearly polarized components as some of the dominant modes may not be retrieved in certain linear polarization states. Compared to previous studies that require a large number of target signatures from multiple linear polarization angles (6 [45] and 18 [9]), the computational load is reduced to the maximum of four or even one without deteriorating the accuracies of the extracted NRF when polarimetric signature is utilized to handle the polarization dependency of the residues. The findings presented here provide us with guidelines on how we should illuminate the targets and process the multiple-aspect multiple-polarization data set to extract a set of NRF that includes all the dominant NRFs for target characterization. With the "completed" set of NRF, the E-pulse is constructed and then used for ATR. In most ATR scenarios in which only one target signature from the target is measured, the traditional E-pulse technique using the "completed" E-pulse should be able to effectively distinguish the correct targets from the others. In situations, where more than one target signatures from a target are considered for ATR, e.g., the situation of multidirectional Epulse presented in [48], novel ATR procedures will be required to effectively utilize the

The shaded boxes indicate the NRF is properly retrieved in the extraction process (P, total number of target signatures;

Table 3. Comparison of the extracted natural resonance frequencies (NRFs) using multiple-aspect multiple-polarization

Case 3 All (6 � 6)

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

P 288 144 144 36 36 36 36 36 36 36 36

VV HH HV VH LL RR RL LR

http://dx.doi.org/10.5772/intechopen.75059

157

Other than target recognition problems with the target in free space, research activities have also focused on subsurface target detection—where the target is located at a particular depth below an interface. The motivation of subsurface target detection first originated from unexploded ordinance (UXO) detection using ground-penetrating radar [56–60] and later detection of tumors inside a breast volume [49, 51, 55, 61] as well as detection of small changes

"completed" set of NRF with the extensive data set.

3.3. Subsurface target detection

Aspects Case 1

NRFs σ�jω c

data.

All (6 � 6)

�3:<sup>48</sup> � <sup>j</sup>27:97<sup>1</sup> �4:<sup>52</sup> � <sup>j</sup>32:93<sup>2</sup> �2:<sup>13</sup> � <sup>j</sup>41:59<sup>3</sup> �3:<sup>14</sup> � <sup>j</sup>45:27<sup>4</sup> �3:<sup>27</sup> � <sup>j</sup>52:66<sup>5</sup> �4:<sup>16</sup> � <sup>j</sup>57:63<sup>6</sup>

Reprinted from [49] with permission from IEEE)

Pol. Linear + circular (4 þ 4)

Case 2 All (6 � 6)

Linear (4)

Circular (4)

in hip prostheses [62–64].

First, extraction results of the entire multiple-aspect multiple-polarization data sets with 288 target signatures (Case 1: 6 � 6aspects and eight polarizations) are tabulated in column 2 of Table 3. Six dominant resonant modes are extracted. This result is treated as a ground truth as the extraction has taken all the data into account. Next, we consider NRF extractions in linear and circular polarization bases separately with 144 target signatures (Case 2: 36 aspects and four polarization states) in each basis. All the six resonant modes are retrieved in both bases, and the corresponding boxes in columns 3 and 4 in Table 3 are shaded. The results indicate that both bases should give similar results in the NRF extraction process once all the four components in the Sinclair matrix are utilized.

Lastly, we perform NRF extraction on the multiple-aspect-only data at each of the eight polarization states (Case 3: 36 aspects and one polarization states). When HH data is used, five out of six resonant modes were extracted. When VV, HV, and VH data is used, only four modes were retrieved. When any one of the circularly polarized target signatures is used, all the six modes are accurately retrieved in all the co- and cross polarized results. The results show that we can retrieve all the dominant NRFs when only one of the four circularly polarized target signatures is used in the extraction process.

In summary, it is essential to include multiple-aspect data for target characterization because the NRF can be poorly excited at specific transmit-receive configurations [39, 48]. If all the four components in the scattering matrix are utilized for resonance extraction, both linear and circular polarization bases should give similar results. If only one out of the four components


The shaded boxes indicate the NRF is properly retrieved in the extraction process (P, total number of target signatures; Reprinted from [49] with permission from IEEE)

in the polarization matrix is utilized (regardless single or multiple aspects), circularly polarized components are preferred over linearly polarized components as some of the dominant modes may not be retrieved in certain linear polarization states. Compared to previous studies that require a large number of target signatures from multiple linear polarization angles (6 [45] and 18 [9]), the computational load is reduced to the maximum of four or even one without deteriorating the accuracies of the extracted NRF when polarimetric signature is utilized to handle the polarization dependency of the residues. The findings presented here provide us with guidelines on how we should illuminate the targets and process the multiple-aspect multiple-polarization data set to extract a set of NRF that includes all the dominant NRFs for target characterization. With the "completed" set of NRF, the E-pulse is constructed and then used for ATR. In most ATR scenarios in which only one target signature from the target is measured, the traditional E-pulse technique using the "completed" E-pulse should be able to effectively distinguish the correct targets from the others. In situations, where more than one target signatures from a target are considered for ATR, e.g., the situation of multidirectional Epulse presented in [48], novel ATR procedures will be required to effectively utilize the "completed" set of NRF with the extensive data set.

#### 3.3. Subsurface target detection

ϕ is measured from the positive x-axis), resulted in 36 transmit-receive combinations (6 are monostatic, and 30 are bistatic) and 8 polarization states, i.e., 288 target signatures. It is apparent that the amount of data is tremendously increased when both aspect and polariza-

, <sup>ϕ</sup><sup>t</sup> <sup>¼</sup> <sup>30</sup>�

, and <sup>ϕ</sup><sup>t</sup> <sup>¼</sup> <sup>210</sup>�

,<sup>θ</sup> <sup>¼</sup> <sup>105</sup>�

, respectively. (reprinted from [49] with

) of the breast

First, extraction results of the entire multiple-aspect multiple-polarization data sets with 288 target signatures (Case 1: 6 � 6aspects and eight polarizations) are tabulated in column 2 of Table 3. Six dominant resonant modes are extracted. This result is treated as a ground truth as the extraction has taken all the data into account. Next, we consider NRF extractions in linear and circular polarization bases separately with 144 target signatures (Case 2: 36 aspects and four polarization states) in each basis. All the six resonant modes are retrieved in both bases, and the corresponding boxes in columns 3 and 4 in Table 3 are shaded. The results indicate that both bases should give similar results in the NRF extraction process once all the four compo-

Lastly, we perform NRF extraction on the multiple-aspect-only data at each of the eight polarization states (Case 3: 36 aspects and one polarization states). When HH data is used, five out of six resonant modes were extracted. When VV, HV, and VH data is used, only four modes were retrieved. When any one of the circularly polarized target signatures is used, all the six modes are accurately retrieved in all the co- and cross polarized results. The results show that we can retrieve all the dominant NRFs when only one of the four circularly polarized target

In summary, it is essential to include multiple-aspect data for target characterization because the NRF can be poorly excited at specific transmit-receive configurations [39, 48]. If all the four components in the scattering matrix are utilized for resonance extraction, both linear and circular polarization bases should give similar results. If only one out of the four components

tion domains are considered. Efficient methods to handle the data are thus required.

Figure 6. (a) Cross-sectional views (x-y plane) of the target. (b) Cross-sectional view (<sup>ϕ</sup> <sup>¼</sup> <sup>30</sup>�

nents in the Sinclair matrix are utilized.

volume under plane wave illuminations from <sup>θ</sup><sup>t</sup> <sup>¼</sup> <sup>105</sup>�

permission from IEEE).

156 Antennas and Wave Propagation

signatures is used in the extraction process.

Other than target recognition problems with the target in free space, research activities have also focused on subsurface target detection—where the target is located at a particular depth below an interface. The motivation of subsurface target detection first originated from unexploded ordinance (UXO) detection using ground-penetrating radar [56–60] and later detection of tumors inside a breast volume [49, 51, 55, 61] as well as detection of small changes in hip prostheses [62–64].

Table 3. Comparison of the extracted natural resonance frequencies (NRFs) using multiple-aspect multiple-polarization data.

In the subsurface target detection problem, usually the excitation and the measured field points are located in one medium, and the target is located in another medium. The scattered response is no longer solely dependent on the target itself but also interactions between the target and the interface. These interactions vary as the dielectric contrast between the two media, the depth and the orientation of the target [56–60], as well as the interactions between the heterogeneity of the medium and the target. In [65], a transient scattering of metallic targets sited below lossless and lossy half space is studied using joint TF analysis. All the interactions of the target and the dielectric interface, as well as attenuation phenomena due to the non-zero conductivity of the half space, are clearly observed in the joint TF domain. Considering a relatively simple situation where both media are homogeneous, studies have demonstrated that there are two types of resonances associated with the entire scattering problem—a target resonance and an image resonance. A target resonance is the NRF associated with the target attributes (geometry and the dielectric properties) and the dielectric properties of the medium in which the target is embedded [56–60]. An image resonance is the NRF that corresponds to the target depth—the distance between the target and the interface of the two media [56, 57]. For target detection and recognition applications, the target NRF is of interest. Studies [56–58, 63] showed that the target NRFs forms a spiral trajectory in the S-plane as the target depth changes. The spiral trajectory surrounds the target NRF within a homogeneous environment the case where the target is fully immersed in the environment without a dielectric interface.

within a time window with a proper direction. The time window is moved by small time steps, and the extraction process is repeated until the entire signal is covered. The STMPM was first applied to chipless radio-frequency identification (RFID) application [71–74]. Data is encoded as NRFs of the RFID tagged by incorporating notches in the structure such that each RFID tag

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

159

In addition to the applications above, the concept of late time resonances has also been applied to sensing applications in other disciplines. This includes monitoring the deployment of arterial stents [75], nondestructive evaluation of the maturity of fruit [76], automated detection of objects fallen on railway tracks [77], nondestructive evaluation of layered materials [78, 79],

The fundamentals and development of resonance-based target recognition over the last 40 years have been briefly reviewed. Prospects of using polarimetric transient signatures for target classification and recognition have been demonstrated through numerical examples. Results show that the use of polarimetric signatures will undoubtedly enhance the ATR

Other than using a linear polarization basis to synthesize the circularly polarized target signature, the challenges for designing circularly polarized time-domain antennas [81] that directly generate circularly polarized pulses are an ongoing research topic for antenna engineers. In situations where the targets under test are very similar, the resonance-based target recognition would perform poorly and probably fails. Recent results in [51, 82, 83] have shown that incorporating polarimetric features, e.g., characteristic polarization states (CPS) of the resonance modes, can solve the problem. Development of novel target recognition schemes that

In addition to defense and security applications, recent studies have been looking into the potential of applying the technology to nondestructive evaluations, medical diagnosis, RFID, and agricultural applications. Compared to radar applications where the target is isolated (e.g., aircrafts in the sky) and located in the far-field region and direct signal path from transmitting and receiving antennas exists higher-order interactions between the targets and antennas as well as scattering from surrounding objects become significant especially when the target is located in the near-field region in medical diagnosis [52, 53] and aforementioned RFID applications. Novel calibration procedures [84, 85] and signal processing solutions for NRF extractions of multiple targets [86–88] are still ongoing research topics for researchers to explore. Regarding subsurface target detection, our results demonstrate that the E-pulse is capable of detecting changes of the NRFs of subsurface targets due to depth [64, 66] and geometrical changes [67]. In conjunction with target NRF, other features embedded in the target signature, such as the "turn-on" time of the resonance [65], could also be used for these applications. Rather than solely relies on NRF for ATR, the trend of having a "combined" feature set with

utilize these novel feature sets would be of practical interest to the radar community.

has different NRFs [73, 74].

and detection of concealed handguns [80].

4. Conclusions and ongoing challenges

performance while reducing the amount of data to process.

Other than the characterization of the target resonances for the subsurface target, attempts have also been made to apply the E-pulse for monitoring depth changes [64, 66] and geometrical changes [67] of a hip prosthesis model sited below a half space of tissue. Results show that the E-pulse technique is capable of detecting both depth changes and physical changes of the target. A subsurface target detection technique is proposed in which the E-pulse is constructed using the NRFs for a target inside a homogenous environment [60] to approximate the target NRFs for subsurface targets. This E-pulse is convolved with target signatures from the subsurface target and shows that this approximate technique can distinguish between different targets [34, 68, 69].

#### 3.4. Other applications

A critical issue that affects the accuracies of the extracted NRFs is the commencement of the late time or the turn-on time of the resonant modes [30]. The damped exponential model given by Eq. (2) is strictly only valid during the late time period. The early time consists of highfrequency scattering centers that are mainly local scattering events, which can be modeled using the entire functions (e.g., a Gaussian [17]). The inclusion of the early time period into the NRF extraction process will undoubtedly degrade the accuracies of the retrieved NRFs. Automated detection of the commencement of late time without a priori knowledge of the target geometry or orientation becomes crucial for ATR. Hargrave et al. [70] proposed a method to estimate the commencement of the late time based on intrinsic differences between the fullrank Hankel matrix generated from the early time data and the rank-deficient late time matrix generated by discrete resonant components. Rezaiesarlak and Manteghi [71, 72] propose the short-time matrix pencil method (STMPM), which mostly applies the MPM for NRF extraction within a time window with a proper direction. The time window is moved by small time steps, and the extraction process is repeated until the entire signal is covered. The STMPM was first applied to chipless radio-frequency identification (RFID) application [71–74]. Data is encoded as NRFs of the RFID tagged by incorporating notches in the structure such that each RFID tag has different NRFs [73, 74].

In addition to the applications above, the concept of late time resonances has also been applied to sensing applications in other disciplines. This includes monitoring the deployment of arterial stents [75], nondestructive evaluation of the maturity of fruit [76], automated detection of objects fallen on railway tracks [77], nondestructive evaluation of layered materials [78, 79], and detection of concealed handguns [80].

## 4. Conclusions and ongoing challenges

In the subsurface target detection problem, usually the excitation and the measured field points are located in one medium, and the target is located in another medium. The scattered response is no longer solely dependent on the target itself but also interactions between the target and the interface. These interactions vary as the dielectric contrast between the two media, the depth and the orientation of the target [56–60], as well as the interactions between the heterogeneity of the medium and the target. In [65], a transient scattering of metallic targets sited below lossless and lossy half space is studied using joint TF analysis. All the interactions of the target and the dielectric interface, as well as attenuation phenomena due to the non-zero conductivity of the half space, are clearly observed in the joint TF domain. Considering a relatively simple situation where both media are homogeneous, studies have demonstrated that there are two types of resonances associated with the entire scattering problem—a target resonance and an image resonance. A target resonance is the NRF associated with the target attributes (geometry and the dielectric properties) and the dielectric properties of the medium in which the target is embedded [56–60]. An image resonance is the NRF that corresponds to the target depth—the distance between the target and the interface of the two media [56, 57]. For target detection and recognition applications, the target NRF is of interest. Studies [56–58, 63] showed that the target NRFs forms a spiral trajectory in the S-plane as the target depth changes. The spiral trajectory surrounds the target NRF within a homogeneous environment the case where the target is fully immersed in the environment without a dielectric interface. Other than the characterization of the target resonances for the subsurface target, attempts have also been made to apply the E-pulse for monitoring depth changes [64, 66] and geometrical changes [67] of a hip prosthesis model sited below a half space of tissue. Results show that the E-pulse technique is capable of detecting both depth changes and physical changes of the target. A subsurface target detection technique is proposed in which the E-pulse is constructed using the NRFs for a target inside a homogenous environment [60] to approximate the target NRFs for subsurface targets. This E-pulse is convolved with target signatures from the subsurface target and shows that this approximate technique can distinguish between different

A critical issue that affects the accuracies of the extracted NRFs is the commencement of the late time or the turn-on time of the resonant modes [30]. The damped exponential model given by Eq. (2) is strictly only valid during the late time period. The early time consists of highfrequency scattering centers that are mainly local scattering events, which can be modeled using the entire functions (e.g., a Gaussian [17]). The inclusion of the early time period into the NRF extraction process will undoubtedly degrade the accuracies of the retrieved NRFs. Automated detection of the commencement of late time without a priori knowledge of the target geometry or orientation becomes crucial for ATR. Hargrave et al. [70] proposed a method to estimate the commencement of the late time based on intrinsic differences between the fullrank Hankel matrix generated from the early time data and the rank-deficient late time matrix generated by discrete resonant components. Rezaiesarlak and Manteghi [71, 72] propose the short-time matrix pencil method (STMPM), which mostly applies the MPM for NRF extraction

targets [34, 68, 69].

158 Antennas and Wave Propagation

3.4. Other applications

The fundamentals and development of resonance-based target recognition over the last 40 years have been briefly reviewed. Prospects of using polarimetric transient signatures for target classification and recognition have been demonstrated through numerical examples. Results show that the use of polarimetric signatures will undoubtedly enhance the ATR performance while reducing the amount of data to process.

Other than using a linear polarization basis to synthesize the circularly polarized target signature, the challenges for designing circularly polarized time-domain antennas [81] that directly generate circularly polarized pulses are an ongoing research topic for antenna engineers. In situations where the targets under test are very similar, the resonance-based target recognition would perform poorly and probably fails. Recent results in [51, 82, 83] have shown that incorporating polarimetric features, e.g., characteristic polarization states (CPS) of the resonance modes, can solve the problem. Development of novel target recognition schemes that utilize these novel feature sets would be of practical interest to the radar community.

In addition to defense and security applications, recent studies have been looking into the potential of applying the technology to nondestructive evaluations, medical diagnosis, RFID, and agricultural applications. Compared to radar applications where the target is isolated (e.g., aircrafts in the sky) and located in the far-field region and direct signal path from transmitting and receiving antennas exists higher-order interactions between the targets and antennas as well as scattering from surrounding objects become significant especially when the target is located in the near-field region in medical diagnosis [52, 53] and aforementioned RFID applications. Novel calibration procedures [84, 85] and signal processing solutions for NRF extractions of multiple targets [86–88] are still ongoing research topics for researchers to explore. Regarding subsurface target detection, our results demonstrate that the E-pulse is capable of detecting changes of the NRFs of subsurface targets due to depth [64, 66] and geometrical changes [67]. In conjunction with target NRF, other features embedded in the target signature, such as the "turn-on" time of the resonance [65], could also be used for these applications. Rather than solely relies on NRF for ATR, the trend of having a "combined" feature set with other parameters (e.g., CPS, turn-on time) will become the fashion for further development in this context.

[11] Ilavarasan P, Ross JE, Rothwell EJ, Chen KM, Nyquist DP. Performance of an automated radar target pulse discrimination scheme using E pulses and S pulses. IEEE Transactions

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

161

[12] Iwaszczuk K, Heiselberg H, Jepsen PU. Terahertz radar cross section measurements.

[13] Lui HS, Taimre T, Lim YL, Bertling K, Dean P, Khanna SP, Lachab M, Valavanis A, Indjin D, Linfield EH, Davies AG, Rakic AD. Terahertz radar cross section characterization using self-mixing interferometry with a quantum Cascade laser. Electronics Letters. 22nd

[14] Kennaugh EM, Moffatt DL. Transient and impulse response approximations. Proceedings

[15] Heyman E, Felsen L. A wavefront interpretation of the singularity expansion method.

[16] Felsen L. Comments on early time SEM. IEEE Transactions on Antennas and Propagation.

[17] Jang S, Choi W, Sarkar TK, Salazar-Palma M, Kyungjung K, Baum CE. Exploiting early time response using the fractional Fourier transform for analyzing transient radar returns.

[18] Harrington R. Field computation by the moment method. 2nd ed. USA: IEEE Press; 1993

[19] Van Blaricum M, Mittra R. A technique for extracting the poles and residues of a system directly from its transient response. IEEE Transactions on Antennas and Propagation.

[20] Sarkar TK, Pereira O. Using the matrix pencil method to estimate the parameters of a sum of complex exponentials. IEEE Antennas and Propagation Magazine. Feb, 1995;37(1):48-55

[21] Lui HS, Shuley N. On the analysis of electromagnetic transients from radar targets using smooth pseudo Wigner-Ville distribution (SPWVD). Proc. IEEE Antenna Propag. Society

[22] Lui HS, Shuley NV. On the modelling of transient scattering under ultra wideband sources. Asia Pacific Symp. Electromag. Compat., Beijing, China. 12–16 April 2010. pp.

[24] FEKO, 32 Techno Lane, Technopark, Stellenbosch, 7600, South Africa: EM Software and

[25] Rothwell EJ, Chen KM, Nyquist DP. Approximate natural response of an arbitrarily shaped thin wire Scatterer. IEEE Transactions on Antennas Propagation. Oct 1991;39(10):

[23] Rao SM. Time Domain Electromagnetics. San Diego: Academic Press; 1999

on Antennas and Propagation. May, 1993;41(5):582-588

Oct. 2015;51(22):1774-1776

of the IEEE. 1965;53:893-901

1985;33:118-119

1975;23:777-781

854-857

1457-1462

System S.A., (pty) Ltd

Optics Express. 2010;18(25):26399-26408. DOI: 10.1364/OE.18.026399

IEEE Transactions on Antennas and Propagation. 1985;33:706-718

IEEE Transactions on Antennas and Propagation. 2004;52:3109-3121

Int. Symp., pp. 5701-5704, Honolulu, HI, Jun. 10–15, 2007

## Author details

Hoi-Shun Lui<sup>1</sup> \*, Faisal Aldhubaib<sup>2</sup> , Stuart Crozier<sup>1</sup> and Nicholas V. Shuley<sup>1</sup>

\*Address all correspondence to: h.lui@uq.edu.au

1 School of Information Technology and Electrical Engineering, The University of Queensland, St. Lucia, Queensland, Australia

2 Electronics Department, College of Technological Studies, Public Authority for Applied Education, Kuwait

## References


[11] Ilavarasan P, Ross JE, Rothwell EJ, Chen KM, Nyquist DP. Performance of an automated radar target pulse discrimination scheme using E pulses and S pulses. IEEE Transactions on Antennas and Propagation. May, 1993;41(5):582-588

other parameters (e.g., CPS, turn-on time) will become the fashion for further development in

1 School of Information Technology and Electrical Engineering, The University of

2 Electronics Department, College of Technological Studies, Public Authority for Applied

[1] Sullivan RJ. Radar Foundations for Imaging and Advanced Concepts. Raleigh: Scitech

[2] Knott EF, Shaeffer JF, Tuley MT. Radar Cross Section: Its Predication, Measurement and

[3] Copeland JR. Radar target classification by polarization properties. Proceedings of the

[4] Li HJ, Yang SH. Using range profiles as feature vectors to identify aerospace objects. IEEE

[5] Hurst M, Mittra R. Scattering center analysis via Prony's method. IEEE Transactions on

[6] Lin H, Ksienski AA. Optimum frequencies for aircraft classification. IEEE Transactions on

[7] Baum CE. The singularity expansion method. In: Felsen LB, editor. Transient Electromagnetic Fields. 1st ed. Vol. 10. Berlin; New York: Springer-Verlag; 1976. pp. 129-176

[8] Shuley N, Longstaff D. Role of polarisation in automatic target recognition using reso-

[9] Rothwell E, Nyquist D, Chen KM, Drachman B. Radar target discrimination using the extinction-pulse technique. IEEE Transactions on Antennas and Propagation. 1985;33:929-937

[10] Rothwell E, Chen KM, Nyquist DP, Sun W. Frequency domain E-pulse synthesis and target discrimination. IEEE Transactions on Antennas and Propagation. April, 1987;35(4):

, Stuart Crozier<sup>1</sup> and Nicholas V. Shuley<sup>1</sup>

this context.

Author details

160 Antennas and Wave Propagation

Education, Kuwait

Publishing; 2004

IRE. 1960;48:1290-1296

References

426-434

\*, Faisal Aldhubaib<sup>2</sup>

\*Address all correspondence to: h.lui@uq.edu.au

Queensland, St. Lucia, Queensland, Australia

Reduction. Dedham, MA: Artech house; 1985

Antennas and Propagation. 1987;35:986-988

Aerospaces Electronic Systems. 1981;17:656-665

nance descriptions. Electronics Letters. 2004;40:268-270

Transactions on Antennas and Propagation. 1993;41:261-268

Hoi-Shun Lui<sup>1</sup>


[41] Chen KM, Nyquist D, Rothwell E, Webb L, Drachman B. Radar target discrimination by convolution of radar return with extinction-pulses and single-mode extraction signals.

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

163

[42] Rothwell EJ, Chen KM. A hybrid E-pulse/least squares technique for natural resonance

[43] Carrion MC, Gallego A, Porti J, Ruiz DP. Subsectional-polynomial E-pulse synthesis and application to radar target discrimination. IEEE Transactions on Antennas and Propaga-

[44] Blanco D, Ruiz DP, Alameda E, Carrion MC. An asymptotically unbiased E-pulse-based scheme for radar target discrimination. IEEE Transactions on Antennas and Propagation.

[45] Lui HS, Shuley N. Resonance Based Radar Target Detection with Multiple Polarisations. Proc. IEEE Antennas Propag. Soc. Int. Symp. USNC/URSI National Radio Science Meet-

[47] Lui HS, Shuley NVZ. Resonance based target recognition using ultra-wideband Polarimetric signatures. IEEE Transactions on Antennas and Propagation, New Jersey. August

[48] Zhang H, Fan Z, Ding D, Chen R. Radar target recognition based on multi-directional E-pulse technique. IEEE Transactions on Antennas and Propagation. Nov. 2013;61(11):5838-5843 [49] Lui HS. Characterization of radar target using multiple transient responses. IEEE Anten-

[50] Lui HS, Fhager A, Persson M. On the forward scattering of microwave breast imaging.

[51] Lui HS, Fhager A, Yang J, Persson M. Characterization and detection of breast cancer using ultra wideband Polarimetric transients. European Conference on Antennas and

[52] Lui HS, Fhager A, Persson M. Preliminary Investigations of Three-Dimensional Microwave Tomography Using Different Data Sets. European Conference on Antennas and

[53] Lui HS, Fhager A, Persson M. Antenna Configurations of Microwave Breast Imaging.

[54] Gabriel S, Lau RW, Gabriel C. The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. Physics in Medicine and Biology. Nov.

[55] Huo Y, Bansal R, Zhu Q. Modeling of noninvasive microwave characterization of breast

tumors. IEEE Transactions on Biomedical Engineering. 2004;51:1089-1094

ing. 9–14 July, 2006, Albuquerque, New Mexico, USA. pp. 3259-3262

[46] Mott H. Remote Sensing with Polarimetric Radar. New Jersey: Wiley; 2007

nas on Wireless Propagation Letters. Sept. 2015;14:1750-1753

International Journal of Biomedical Imaging, 15 Pages. May 2012

Propagation. Gothenburg, Sweden, 8–11 April 2013. pp. 2909-2913

Asia Pacific Microwave Conference, 5–8 Nov., 2013, Seoul, Korea

Propagation, Prague, Czech Republic. 26–30 March 2012

IEEE Transactions on Antennas and Propagation. 1986;34:896-904

extraction. Proceedings of the IEEE. 1988;76:296-298

tion. 1993;41:1204-1211

2004;52:1348-1350

2012;60(8):3985-3988

1996;41(11):2271-2293


[41] Chen KM, Nyquist D, Rothwell E, Webb L, Drachman B. Radar target discrimination by convolution of radar return with extinction-pulses and single-mode extraction signals. IEEE Transactions on Antennas and Propagation. 1986;34:896-904

[26] Cohen L. Time-Frequency Analysis. Englewood Cliffs, NJ: Prentice Hall; 1995

Signal Process. Comm. Sys., Gold Coast, Australia, 17–19 Dec. 2007

MATLAB. CNRS (France) and Rice University (USA; 1996)

Conference Centre, UK. 15–18 Oct., 2007

Society International Symposium. 1975;13:416-419

Antennas and Propagation. May 2010;58(5):1699-1710

on Signal Processing. Nov. 1995;43(11):2665-2677

Propagation. 1987;35:715-720

48(4):612-618

1981;29:327-331

53-62

162 Antennas and Wave Propagation

[27] Auger F, Flandrin P, Goncalves P, Lemoine O. Time-Frequency Toolbox – For Use with

[28] Lui HS, Shuley NV, Longstaff ID. Time-frequency analysis of late time electromagnetic transients from radar targets. Proc. IET Radar 2007, Pp. 1-5, the Edinburgh International

[29] Lui HS, Shuley NV. Joint time-frequency analysis on UWB radar signals. Proc. Int.l Conf.

[30] Lui HS, Shuley NVZ. Evolutions of partial and global resonances in transient electromagnetic scattering. IEEE Antennas on Wireless Propagation Letters. 2008;7:435-439

[31] Tesche FM. On the analysis of scattering and antenna problems using the singularity expansion technique. IEEE Transactions on Antennas and Propagation. Jan., 1973;21(1):

[32] Van Blaricum M, Pearson L, Mittra R. An efficient scheme for radar target recognition based on the complex natural resonances of the target. IEEE Antennas and Propagation

[33] Lui HS, Shuley NVZ. Radar target identification using a "banded" E-pulse technique. IEEE Transactions on Antennas and Propagation. Dec., 2006;54(12):3874-3881

[34] Lui HS, Shuley NVZ, Rakic AD. A novel, fast, approximate target detection technique for metallic target below a frequency dependent Lossy Halfspace. IEEE Transactions on

[35] Lui HS, Shuley NVZ. Sampling procedures in resonance based radar target identification.

[36] Rothwell E, Kun-Mu C, Nyquist D. Extraction of the natural frequencies of a radar target from a measured response using E-pulse techniques. IEEE Transactions on Antennas and

[37] Gallego A, Medouri A, Carmen Carrion M. Estimation of number of natural resonances of transient signal using E-pulse technique. Electronics Letters. 1991;27:2253-2256

[38] Ruiz DP, Carrion MC, Gallego A, Medouri A. Parameter estimation of exponentially damped sinusoids using a higher order correlation-based approach. IEEE Transactions

[39] Sarkar TK, Park S, Koh J, Rao SM. Application of the matrix pencil method for estimating the SEM (singularity expansion method ) poles of source-free transient responses from multiple look directions. IEEE Transactions on Antennas and Propagation. April, 2000;

[40] Kennaugh E. The K-pulse concept. IEEE Transactions on Antennas and Propagation.

IEEE Transactions on Antennas and Propagation. May 2008;56(5):1487-1491


[56] Vitebskiy S, Carin L. Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space. IEEE Transactions on Antennas and Propagation. 1995;43:1303-1312

[69] Lui HS, Shuley NV. Performance evaluation of subsurface target recognition based on ultra-wideband short pulse excitation. IEEE Int. Symp. Antennas Propag., pp. 1-4,

Ultra Wideband Transient Scattering and Its Applications to Automated Target Recognition

http://dx.doi.org/10.5772/intechopen.75059

165

[70] Hargrave CO, Clarkson IVL, Lui HS. Late-time estimation for resonance-based radar target identification. IEEE Transactions on Antennas and Propagation. Nov., 2014;62(11):

[71] Rezaiesarlak R, Manteghi M. Short-time matrix pencil method for Chipless RFID detection applications. IEEE Transactions on Antennas Propagation. May 2013;61(5):2801-2806

[72] Rezaiesarlak R, Manteghi M. On the application of short-time matrix pencil method for wideband scattering from resonant structures. IEEE Transactions on Antennas Propaga-

[73] Blischak AT, Manteghi M. Embedded singularity Chipless RFID tags. IEEE Transactions

[74] Rezaiesarlak R, Manteghi M. Complex-natural-resonance-based Design of Chipless RFID tag for high-density data. IEEE Transactions on Antennas Propagation. Feb 2014;62(2):

[75] Manteghi M, Cooperand DB, Vlachos PP. Application of singularity expansion method for monitoring the deployment of arterial stents. Microwave and Optical Technology

[76] Tantisopharak T, Moon H, Youryon P, Bunya-Athichart K, Krairiksh M, Sarkar TK. Nondestructive determination of the maturity of the durian fruit in the frequency domain using the change in the natural frequency. IEEE Transactions on Antennas Propagation.

[77] Mroué A, Heddebaut M, Elbahhar F, Rivenq A, Rouvaen J-M. Automatic radar target recognition of objects falling on railway tracks. Measurement Science and Technology.

[78] Stenholm G, Rothwell EJ, Nyquist DP, Kempel LC, Frasch LL. E-pulse diagnostics of simple layered materials. IEEE Transactions on Antennas and Propagation. Dec. 2003;

[79] Wierzba JF, Rothwell EJ. E-pulse diagnostics of curved coated conductors with varying thickness and curvature. IEEE Transactions on Antennas and Propagation. September

[80] Harmer SW, Andrews DA, Rezgui ND, Bowring NJ. Detection of handguns by their complex natural resonant frequencies. IET Microwaves, Antennas and Propagation.

[81] Shlivinski A. Time domain circularly polarized antenna. IEEE Transactions on Antennas

Toronto, Ontario, Canada, July 11–17, 2010

on Antennas Propagation. Nov 2011;59(11):3961-3968

Jan. 2012;23(2). DOI: 10.1088/0957-0233/23/2/025401

2010, 2010;4(9):1182-1190. DOI: 10.1049/iet-map.2009.0382

and Propagation. June 2009;57(6):1606-1611

5865-5871

898-904

tion. Jan. 2015;63(1):328-335

Letters. 2012;54(10):2241-2246

Feb 2016;64(5):1779-1787

51(12):3221-3227

2006;54(9):2672-2676


[69] Lui HS, Shuley NV. Performance evaluation of subsurface target recognition based on ultra-wideband short pulse excitation. IEEE Int. Symp. Antennas Propag., pp. 1-4, Toronto, Ontario, Canada, July 11–17, 2010

[56] Vitebskiy S, Carin L. Moment-method modeling of short-pulse scattering from and the resonances of a wire buried inside a lossy, dispersive half-space. IEEE Transactions on

[57] Geng N, Jackson DR, Carin L. On the resonances of a dielectric BOR buried in a dispersive layered medium. IEEE Transactions on Antennas and Propagation. 1999;47:1305-1313 [58] Wang Y, Longstaff ID, Leat CJ, Shuley NV. Complex natural resonances of conducting planar objects buried in a dielectric half-space. IEEE Transactions on Geoscience Remote

[59] Chen C-C, Peters L Jr. Buried unexploded ordnance identification via complex natural resonances. IEEE Transactions on Antennas and Propagation. 1997;45:1645-1654

[60] Baum CE. Detection and Identification of Visually Obscured Targets. Philadelphia: Taylor

[61] Lui HS, Li BK, Shuley N, Crozier S. Preliminary Investigation of Breast Tumor Detection Using the E-Pulse Technique. Proc. IEEE Antenna Propag. Soc. Int. Symp. USNC/URSI National Radio Science Meeting, pp. 283-286, 9–14 July. 2006. Albuquerque, New Mexico,

[62] Lui HS, Shuley N, Crozier S. A Concept for Hip Prosthesis Identification Using Ultra Wideband Radar. Proc. 26th Annual Int. Conf. of the IEEE Engineering in Medicine and

[63] Lui HS, Shuley N, Crozier S. Hip Prosthesis Detection Based on Complex Natural Resonances. Proc. 27th Annual Int.l Conf. of the IEEE Engineering in Medicine and Biology

[64] Lui HS, Shuley N, Padhi SK, Crozier S. Detection of hip prosthesis depth changes using an E-pulse technique. Topical Meeting on Biomedical Electromagnetics. 17th International Zurich Symposium on Electromagnetic Compatibility. 27 February to 3 march

[65] Lui HS, Shuley N, Persson M. Joint time-frequency analysis of transient electromagnetic scattering from a subsurface target. IEEE Antennas and Propagation Magazine. Oct. 2012;

[66] Lui HS, Shuley NVZ. Detection of depth changes of a metallic target buried inside a Lossy Halfspace using the E-pulse technique. IEEE Transactions on Electromagnetic Compati-

[67] Lui HS, Aldhubaib F, Shuley NVZ, Hui HT. Subsurface target recognition based on transient electromagnetic scattering. IEEE Transactions on Antennas and Propagation.

[68] Lui HS, Shuley N. Subsurface target recognition using an approximated method. Asia

Pacific Microw. Conf. Singapore, 7–10 Dec. 2009. pp. 2216-2219

Biology Society. pp. 1439-1442, 1–5 Sept. 2004, San Francisco, CA, USA

Society, 1–5 Sept. 2005. Shanghai, China. pp. 1571-1574

Antennas and Propagation. 1995;43:1303-1312

Sensing. 2001;39:1183-1189

2006, Singapore. pp. 81-84

bility. November 2007;49(4):868-875

October 2009;57(10):3398-3401

54(5):109-130

& Francis; 1999

164 Antennas and Wave Propagation

USA


[82] Aldhubaib F, Shuley NVZ, Lui HS. Characteristic polarization states in an ultra-wideband context based on singularity expansion method. IEEE Geoscience and Remote Sensing Letters. Oct. 2009;6(4):792-796

**Chapter 8**

Provisional chapter

**Anisotropic Propagation of Electromagnetic Waves**

DOI: 10.5772/intechopen.75123

This chapter will analyze the properties of electromagnetic wave propagation in anisotropic media. Of particular interest are positive index, anisotropic, and magneto-dielectric media. Engineered anisotropic media provide unique electromagnetic properties including a higher effective refractive index, high permeability with relatively low magnetic loss tangent at microwave frequencies, and lower density and weight than traditional media. This chapter presents research including plane wave solutions to propagation in anisotropic media, a mathematical derivation of birefringence in anisotropic media, modal decomposition of rectangular waveguides filled with anisotropic media, and the full derivation of anisotropic transverse resonance in a partially loaded waveguide. These are fundamental theories in the area of electromagnetic wave propagation that have been reformulated for fully anisotropic magneto-dielectric media. The ensuing results will aide interested parties in understanding wave behavior for anisotropic media to enhance

designs for radio frequency devices based on anisotropic and magnetic media.

Keywords: anisotropic, wave propagation, dispersion, birefringence, waveguides,

Recently engineered materials have come to play a dominant role in the design and implementation of electromagnetic devices and especially antennas. Metamaterials, ferrites, and magneto-dielectrics have all come to play a crucial role in advances made both in the functionality and characterization of such devices. In fact, a movement towards utilizing customized material properties to replace the functionality of traditional radio frequency (RF) components such as broadband matching circuitry, ground planes, and directive elements is apparent in the literature and not just replacement of traditional substrates and superstrates with engineered

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Anisotropic Propagation of Electromagnetic Waves

Gregory Mitchell

Gregory Mitchell

Abstract

transverse resonance

1. Introduction

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75123


#### **Anisotropic Propagation of Electromagnetic Waves** Anisotropic Propagation of Electromagnetic Waves

DOI: 10.5772/intechopen.75123

#### Gregory Mitchell Gregory Mitchell

[82] Aldhubaib F, Shuley NVZ, Lui HS. Characteristic polarization states in an ultra-wideband context based on singularity expansion method. IEEE Geoscience and Remote

[83] Lui HS, Persson M. Characterization of radar targets based on ultra wideband Polarimetric transient signatures. XXX URSI General Assembly and Scientific Symposium of Inter-

[84] Sarkar TK, Tseng FI, Rao SM, Dianat SA, Hollmann BZ. Deconvolution of impulse response from time-limited input and output: Theory and experiment. IEEE Transactions

[85] Bannawata L, Boonpoongaa A, Burintramartb S, Akkaraekthalina P. On the resolution improvement of radar target identification with filtering antenna effects. International

[86] Rezaiesarlak R, Manteghi M. A space–time–frequency Anticollision algorithm for identifying chipless RFID. IEEE Transactions on Antennas Propagation. Mar 2014;62(3):1425-

[87] Lee W, Sarkar TK, Moon H, Salazar-Palma M. Identification of multiple objects using their natural resonant frequencies. IEEE Antennas and Wireless Propagation Letters.

[88] Lee JH, Jeong SH, Park GS, Lee YC, Cho SW. Performance analysis of natural frequencybased multiple radar target recognition for multiple-input–multiple-output radar appli-

national Union of Radio Science. pp. 1-4. Istanbul, Turkey. 13–20 Aug. 2011

Journal of Antennas Propagation. accepted for publication, Dec 2017

cation. IET Radar, Sonar and Navigation. June 2014;8(5):457-464

on Instrumentation and Measurement. 1985;34:541-546

Sensing Letters. Oct. 2009;6(4):792-796

1432

2013;12:54-57

166 Antennas and Wave Propagation

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.75123

#### Abstract

This chapter will analyze the properties of electromagnetic wave propagation in anisotropic media. Of particular interest are positive index, anisotropic, and magneto-dielectric media. Engineered anisotropic media provide unique electromagnetic properties including a higher effective refractive index, high permeability with relatively low magnetic loss tangent at microwave frequencies, and lower density and weight than traditional media. This chapter presents research including plane wave solutions to propagation in anisotropic media, a mathematical derivation of birefringence in anisotropic media, modal decomposition of rectangular waveguides filled with anisotropic media, and the full derivation of anisotropic transverse resonance in a partially loaded waveguide. These are fundamental theories in the area of electromagnetic wave propagation that have been reformulated for fully anisotropic magneto-dielectric media. The ensuing results will aide interested parties in understanding wave behavior for anisotropic media to enhance designs for radio frequency devices based on anisotropic and magnetic media.

Keywords: anisotropic, wave propagation, dispersion, birefringence, waveguides, transverse resonance

#### 1. Introduction

Recently engineered materials have come to play a dominant role in the design and implementation of electromagnetic devices and especially antennas. Metamaterials, ferrites, and magneto-dielectrics have all come to play a crucial role in advances made both in the functionality and characterization of such devices. In fact, a movement towards utilizing customized material properties to replace the functionality of traditional radio frequency (RF) components such as broadband matching circuitry, ground planes, and directive elements is apparent in the literature and not just replacement of traditional substrates and superstrates with engineered

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

structures. A firm theoretical understanding of the electromagnetic properties of these materials is necessary for both design and simulation of new and improved RF devices.

The concept of artificial media is also exemplified by the proliferation of metamaterials research over the last few decades. Metamaterials incorporate the use of artificial microstructures made of subwavelength inclusions that are usually implemented with periodic and/or multilayered structures known as unit cells [6]. These devices operate where the wavelength is much larger than the characteristic dimensions of the unit cell elements. One characteristic feature of some types of metamaterials is wave propagation anisotropy [7]. Anisotropic metamaterials are used in applications such as directive lensing [8, 9], cloaking [10], electronic

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123 169

Finally, a class of engineered materials exists that exhibits positive refractive index, anisotropy, and magneto-dielectric properties with reduced propagation loss at microwave frequencies compared to traditional ferrites. These materials show the unique ability to provide broadband impedance matches for very low profile antennas by exploiting the inherent anisotropy to redirect surface waves thus improving the impedance match of the antenna when very close to a ground plane. Antenna profile on the orders of a twentieth and a fortieth of a wavelength have been demonstrated using these materials with over an octave of bandwidth and positive

The recent development of low loss anisotropic magneto-dielectrics greatly expands the current antenna design space. Here we present a rigorous derivation of the wave equation and dispersion relationships for anisotropic magneto-dielectric media. All results agree with those presented by Meng et al. [15, 16]. Furthermore, setting μ<sup>r</sup> = I, where I is the identity matrix, yields results that agree with those presented by Pozar and Graham for anisotropic dielectric media [4, 5]. This section and the following section expand on the results presented by Meng et al., Pozar and Graham. Incorporating a fully developed derivation of anisotropic properties of both ε<sup>r</sup> and μ<sup>r</sup> expands upon the simplification imposed by both Pozar and Graham that uses an isotropic value of μ<sup>r</sup> = 1. An expansion on the results of Meng et al. given in Section 4 develops the waveguide theory including a full modal decomposition utilizing the biaxial definition of anisotropy versus their simplified uniaxial definition. The derivation of anisotropic cavity resonance in Section 4 differs from that of Meng et al. by addressing the separate issue of how the direct relationship of an arbitrary volume of anisotropic material will distort the geometry of a cavity to maintain resonance at a given frequency. This property is especially important for the design of conformal cavity backed antennas for ground and air-based vehicle mobile vehicular platforms. Furthermore, the analysis of anisotropic properties is not restricted to double negative (DNG) materials, which is the case for both of the Meng et al. studies.

In order to solve for the propagation constants, we will need to formulate the dispersion relationship from the anisotropic wave equation. This allows us to solve for the propagation constant in the normal direction of the anisotropic medium. We start with the anisotropic, time harmonic form of Maxwell's source free equations for the electric and magnetic fields E and H

beam steering [11], and metasurfaces [12] among others.

3. Plane wave solutions in an anisotropic medium

3.1. Source free anisotropic wave equation

realized gain [13, 14].

Inherently, many of these engineered materials have anisotropic properties. Previously, the study of anisotropy had been limited mostly to the realm of optical frequencies where the phenomenon occurs naturally in substances such as liquid crystals and plasmas. However, the recent development of the aforementioned engineered materials has encouraged the study of electromagnetic anisotropy for applications at megahertz (MHz) and gigahertz (GHz) frequencies.

For the purposes of this chapter, an anisotropic electromagnetic medium defines permittivity (ε<sup>r</sup> ) and permeability μ<sup>r</sup> as separate tensors where the values differ in all three Cartesian directions (εx6¼εy6¼ε<sup>z</sup> and μx6¼μy6¼μz). This is known as the biaxial definition of anisotropic material which is more encompassing than the uniaxial definition which makes the simplifying assumption that ε<sup>x</sup> = ε<sup>y</sup> = ε<sup>t</sup> and μ<sup>x</sup> = μ<sup>y</sup> = μt. The anisotropic definition also differs from the traditional isotropic definition where ε<sup>r</sup> and μ<sup>r</sup> are the same in all three Cartesian directions defining each by a single value. For the definition of the tensor equations see Section 3.1. Anisotropic media yield characteristics such as conformal surfaces, focusing and refraction of electromagnetic waves as they propagate through a material, high impedance surfaces for artificial magnetic conductors as well as high index, low loss, and lightweight ferrite materials. The following sections aim to discuss in more detail some RF applications directly impacted by the incorporation of anisotropic media and also give a firm understanding of electromagnetic wave propagation as it applies to anisotropic media for different RF applications.

## 2. Applications of anisotropy in radio frequency devices

Traditionally, the study of anisotropic properties was limited to a narrow application space where traditional ferrites, which exhibit natural anisotropy were the enabling technology. These types of applications included isolators, absorbers, circulators and phase shifters [1]. Traditional ferrites are generally very heavy and very lossy at microwave frequencies which are the two main limiting factors narrowing their use in RF devices; however, propagation loss is an important asset to devices such as absorbers. Anisotropy itself leads to propagation of an RF signal in different directions, which is important in devices such as circulators and isolators [1]. For phase shifters and other control devices the microwave signal is controlled by changing the bias field across the ferrite [1, 2]. However, newer versions of some of these devices, utilizing FETs and diodes in the case of phase shifters, rely on isotropic media to enable higher efficiency devices.

As early as 1958, Collin showed that at microwave frequencies, where the wavelength is larger, it is possible to fabricate artificial dielectric media having anisotropic properties [3]. This has led some to investigate known theoretical solutions to typical RF problems, such as a microstrip patch antenna, and extend them utilizing anisotropic wave propagation in dielectric media [4, 5]. The anisotropic dielectric antenna shows interesting features of basic antenna applications featuring anisotropic substrates. While these solutions establish a framework for electromagnetic wave propagation in anisotropic media, they simplify the problem by necessarily setting μ<sup>r</sup> to 1 and only focusing on dielectric phenomena of anisotropy.

The concept of artificial media is also exemplified by the proliferation of metamaterials research over the last few decades. Metamaterials incorporate the use of artificial microstructures made of subwavelength inclusions that are usually implemented with periodic and/or multilayered structures known as unit cells [6]. These devices operate where the wavelength is much larger than the characteristic dimensions of the unit cell elements. One characteristic feature of some types of metamaterials is wave propagation anisotropy [7]. Anisotropic metamaterials are used in applications such as directive lensing [8, 9], cloaking [10], electronic beam steering [11], and metasurfaces [12] among others.

Finally, a class of engineered materials exists that exhibits positive refractive index, anisotropy, and magneto-dielectric properties with reduced propagation loss at microwave frequencies compared to traditional ferrites. These materials show the unique ability to provide broadband impedance matches for very low profile antennas by exploiting the inherent anisotropy to redirect surface waves thus improving the impedance match of the antenna when very close to a ground plane. Antenna profile on the orders of a twentieth and a fortieth of a wavelength have been demonstrated using these materials with over an octave of bandwidth and positive realized gain [13, 14].

## 3. Plane wave solutions in an anisotropic medium

structures. A firm theoretical understanding of the electromagnetic properties of these mate-

Inherently, many of these engineered materials have anisotropic properties. Previously, the study of anisotropy had been limited mostly to the realm of optical frequencies where the phenomenon occurs naturally in substances such as liquid crystals and plasmas. However, the recent development of the aforementioned engineered materials has encouraged the study of electromagnetic anisotropy for applications at megahertz (MHz) and gigahertz (GHz) frequencies.

For the purposes of this chapter, an anisotropic electromagnetic medium defines permittivity (ε<sup>r</sup> ) and permeability μ<sup>r</sup> as separate tensors where the values differ in all three Cartesian directions (εx6¼εy6¼ε<sup>z</sup> and μx6¼μy6¼μz). This is known as the biaxial definition of anisotropic material which is more encompassing than the uniaxial definition which makes the simplifying assumption that ε<sup>x</sup> = ε<sup>y</sup> = ε<sup>t</sup> and μ<sup>x</sup> = μ<sup>y</sup> = μt. The anisotropic definition also differs from the traditional isotropic definition where ε<sup>r</sup> and μ<sup>r</sup> are the same in all three Cartesian directions defining each by a single value. For the definition of the tensor equations see Section 3.1. Anisotropic media yield characteristics such as conformal surfaces, focusing and refraction of electromagnetic waves as they propagate through a material, high impedance surfaces for artificial magnetic conductors as well as high index, low loss, and lightweight ferrite materials. The following sections aim to discuss in more detail some RF applications directly impacted by the incorporation of anisotropic media and also give a firm understanding of electromagnetic

rials is necessary for both design and simulation of new and improved RF devices.

168 Antennas and Wave Propagation

wave propagation as it applies to anisotropic media for different RF applications.

Traditionally, the study of anisotropic properties was limited to a narrow application space where traditional ferrites, which exhibit natural anisotropy were the enabling technology. These types of applications included isolators, absorbers, circulators and phase shifters [1]. Traditional ferrites are generally very heavy and very lossy at microwave frequencies which are the two main limiting factors narrowing their use in RF devices; however, propagation loss is an important asset to devices such as absorbers. Anisotropy itself leads to propagation of an RF signal in different directions, which is important in devices such as circulators and isolators [1]. For phase shifters and other control devices the microwave signal is controlled by changing the bias field across the ferrite [1, 2]. However, newer versions of some of these devices, utilizing FETs and diodes in the case of phase shifters, rely on isotropic media to enable higher efficiency devices. As early as 1958, Collin showed that at microwave frequencies, where the wavelength is larger, it is possible to fabricate artificial dielectric media having anisotropic properties [3]. This has led some to investigate known theoretical solutions to typical RF problems, such as a microstrip patch antenna, and extend them utilizing anisotropic wave propagation in dielectric media [4, 5]. The anisotropic dielectric antenna shows interesting features of basic antenna applications featuring anisotropic substrates. While these solutions establish a framework for electromagnetic wave propagation in anisotropic media, they simplify the problem by neces-

2. Applications of anisotropy in radio frequency devices

sarily setting μ<sup>r</sup> to 1 and only focusing on dielectric phenomena of anisotropy.

The recent development of low loss anisotropic magneto-dielectrics greatly expands the current antenna design space. Here we present a rigorous derivation of the wave equation and dispersion relationships for anisotropic magneto-dielectric media. All results agree with those presented by Meng et al. [15, 16]. Furthermore, setting μ<sup>r</sup> = I, where I is the identity matrix, yields results that agree with those presented by Pozar and Graham for anisotropic dielectric media [4, 5]. This section and the following section expand on the results presented by Meng et al., Pozar and Graham. Incorporating a fully developed derivation of anisotropic properties of both ε<sup>r</sup> and μ<sup>r</sup> expands upon the simplification imposed by both Pozar and Graham that uses an isotropic value of μ<sup>r</sup> = 1. An expansion on the results of Meng et al. given in Section 4 develops the waveguide theory including a full modal decomposition utilizing the biaxial definition of anisotropy versus their simplified uniaxial definition. The derivation of anisotropic cavity resonance in Section 4 differs from that of Meng et al. by addressing the separate issue of how the direct relationship of an arbitrary volume of anisotropic material will distort the geometry of a cavity to maintain resonance at a given frequency. This property is especially important for the design of conformal cavity backed antennas for ground and air-based vehicle mobile vehicular platforms. Furthermore, the analysis of anisotropic properties is not restricted to double negative (DNG) materials, which is the case for both of the Meng et al. studies.

#### 3.1. Source free anisotropic wave equation

In order to solve for the propagation constants, we will need to formulate the dispersion relationship from the anisotropic wave equation. This allows us to solve for the propagation constant in the normal direction of the anisotropic medium. We start with the anisotropic, time harmonic form of Maxwell's source free equations for the electric and magnetic fields E and H

$$\nabla \mathbf{x} \underline{\mathbf{E}} = j \omega \mu\_o \underline{\mu}\_r \cdot \underline{\mathbf{H}} \tag{1}$$

Ex ¼ � <sup>j</sup> k 2 <sup>o</sup>μyε<sup>x</sup> � <sup>k</sup><sup>2</sup> z

Ey <sup>¼</sup> <sup>j</sup> k 2 <sup>o</sup>μxε<sup>y</sup> � <sup>k</sup><sup>2</sup> z

Hx <sup>¼</sup> <sup>j</sup> k 2 <sup>o</sup>μxε<sup>y</sup> � k

Hy ¼ � <sup>j</sup> k 2 <sup>o</sup>μyε<sup>x</sup> � k

ing solutions for H and E, respectively

(17) and (18) yields

<sup>∇</sup><sup>x</sup> xo

equation

3.2. Dispersion equation for Hz

We expand (19) in terms of (13)–(16)

<sup>ε</sup><sup>x</sup> ð Þ d=dy HZ � ð Þ d=dz Hy � � <sup>þ</sup> yo

<sup>Π</sup><sup>x</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

<sup>Π</sup><sup>y</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

<sup>Π</sup><sup>z</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

2 z

> 2 z

H ¼ � μ<sup>r</sup>

E ¼ ε<sup>r</sup>

∇xε<sup>r</sup>

∇xμ<sup>r</sup>

xoΠ<sup>x</sup> þ y o

=dy<sup>2</sup> � �Hx � �=ε<sup>z</sup> � <sup>d</sup><sup>2</sup>

=dz<sup>2</sup> � �Hy � �=ε<sup>x</sup> � <sup>d</sup><sup>2</sup>

=dx<sup>2</sup> � �Hz � �=ε<sup>y</sup> � <sup>d</sup><sup>2</sup>

<sup>=</sup>dxdy � �Hy � <sup>d</sup><sup>2</sup>

<sup>=</sup>dydz � �Hz � <sup>d</sup><sup>2</sup>

<sup>=</sup>dxdz � �Hx � <sup>d</sup><sup>2</sup>

The relationships for the transverse field components, applied to (1) and (2), yield the follow-

Taking the cross product of both sides and substituting (1) and (2) for the right hand side of

�<sup>1</sup> � ð Þ¼ <sup>∇</sup>xH <sup>k</sup><sup>2</sup>

�<sup>1</sup> � ð Þ¼� <sup>∇</sup>xE <sup>k</sup><sup>2</sup>

Equations (19) and (20) represent the vector wave equations in an anisotropic medium [12].

<sup>ε</sup><sup>y</sup> ½ð Þ <sup>d</sup>=dz Hx � ð Þ <sup>d</sup>=dx Hz�þ zo

<sup>o</sup>μ<sup>r</sup> � H, �

Evaluating the remaining cross product of (21) yields the final form of the expanded wave

Π<sup>y</sup> þ zoΠ<sup>z</sup> ¼ k

2

<sup>=</sup>dz<sup>2</sup> � �Hx � <sup>d</sup><sup>2</sup>

<sup>=</sup>dx<sup>2</sup> � �Hy � <sup>d</sup><sup>2</sup>

<sup>=</sup>dy<sup>2</sup> � �Hz � <sup>d</sup><sup>2</sup>

�1 =jωε<sup>o</sup> � �

> �1 =jωε<sup>o</sup>

ωμoμyð Þ d=dy Hz þ kzð Þ d=dx Ez

ωμoμxð Þ d=dx Hz � kzð Þ d=dy Ez

ωεoεyð Þ d=dy Ez � kzð Þ d=dx Hz

� �, (13)

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123 171

� �, (14)

� �, (15)

ð Þ ωεoεxð Þ d=dx Ez þ kzð Þ d=dy Hz : (16)

� � � ð Þ <sup>∇</sup>xH : (18)

<sup>ε</sup><sup>z</sup> ð Þ d=dx Hy � ð Þ d=dy Hx � �o

� ð Þ ∇xE , (17)

<sup>o</sup>μ<sup>r</sup> � H, (19)

<sup>o</sup>ε<sup>r</sup> � E: (20)

<sup>o</sup>μ<sup>r</sup> � H, (22)

=dxdz � �Hz � �=εy, (23)

=dxdy � �Hx � �=εz, (24)

=dxdy � �Hx � �=εx: (25)

<sup>¼</sup> <sup>k</sup><sup>2</sup>

(21)

$$\nabla \mathbf{x} \underline{\mathbf{H}} = -\mathbf{j} \omega \varepsilon\_o \underline{\varepsilon\_r} \cdot \underline{\mathbf{E}} \tag{2}$$

where ω is the frequency in radians, ε<sup>o</sup> is the permittivity of free space, μ<sup>o</sup> is the permeability of free space, E = xoEx + yoEy + zoEz and H = xoHx + yoHy + zoHz. We define μ<sup>r</sup> and ε<sup>r</sup> as

$$\underline{\varepsilon\_{\tau}} = \begin{bmatrix} \varepsilon\_{\tau} & 0 & 0\\ 0 & \varepsilon\_{y} & 0\\ 0 & 0 & \varepsilon\_{z} \end{bmatrix},\tag{3}$$

$$\underline{\mu\_{r}} = \begin{bmatrix} \mu\_{x} & 0 & 0\\ 0 & \mu\_{y} & 0\\ 0 & 0 & \mu\_{z} \end{bmatrix}.\tag{4}$$

Applying Eqs. (3) and (4) to Eqs. (1) and (2) yields the following

$$\mathbf{x}\_o \left(\frac{d\mathbf{E}\_Z}{dy} - \frac{d\mathbf{E}\_Y}{dz}\right) + \underline{\mathbf{y}}\_o \left(\frac{d\mathbf{E}\_X}{dz} - \frac{d\mathbf{E}\_Z}{dx}\right) + \underline{\mathbf{z}}\_o \left(\frac{d\mathbf{E}\_Y}{dx} - \frac{d\mathbf{E}\_X}{dy}\right) = -j\omega\mu\_o \left(\mu\_x H\_X \underline{\mathbf{x}}\_o + \mu\_y H\_Y \underline{\mathbf{y}}\_o + \mu\_z H\_Z \underline{\mathbf{z}}\_o\right),\tag{5}$$

$$\mathbb{E}\_o \left( \frac{dH\_Z}{dy} - \frac{dH\_Y}{dz} \right) + \underline{y}\_o \left( \frac{dH\_X}{dz} - \frac{dH\_Z}{dx} \right) + \underline{z}\_o \left( \frac{dH\_Y}{dx} - \frac{dH\_X}{dy} \right) = j\omega \varepsilon\_o \left( \varepsilon\_x E\_X \underline{x}\_o + \varepsilon\_y E\_Y \underline{y}\_o + \varepsilon\_z E\_Z \underline{z}\_o \right). \tag{6}$$

Using the radiation condition, we assume a solution of E(x, y, z) = E(x, y)e�jkzz [17]. Now isolate the individual components of (5) by taking the dot product with xo, yo, and zo respectively. This operation yields the following equations

$$(d/dy)E\_z - j\mathbf{k}\_z E\_y = -j\omega\mu\_o\mu\_x H\_{x\nu} \tag{7}$$

$$\text{ijk}\_z \mathbf{E}\_x - (\mathbf{d}/d\mathbf{x})\mathbf{E}\_z = -\mathbf{j}\omega\mu\_o\mu\_y\mathbf{H}\_{y\prime} \tag{8}$$

$$(d/dx)\mathcal{E}\_y - (d/dy)\mathcal{E}\_x = -j\omega\mu\_o\mu\_zH\_z. \tag{9}$$

Assuming a solution of H(x, y, z) = H(x, y)e�jkzz for (6), the same procedure yields [17]

$$(\mathbf{d}/d\mathbf{y})H\_z-\mathbf{j}k\_zH\_y=\mathbf{j}\omega\varepsilon\_o\varepsilon\_xE\_{\mathbf{x}\prime}\tag{10}$$

$$\text{ijk}\_z H\_x - (d/d\mathbf{x})H\_z = \text{j}\omega \varepsilon\_o \varepsilon\_y E\_{y\prime} \tag{11}$$

$$(\mathbf{d}/d\mathbf{x})\mathbf{H}\_y - (\mathbf{d}/d\mathbf{y})\mathbf{H}\_x = -j\omega\mu\_o\mu\_z\mathbf{H}\_z.\tag{12}$$

Using (7)–(12) allows for the transverse field components of the electric and magnetic fields in terms of the derivatives of Hz and Ez as

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123 171

$$E\_x = -\frac{\dot{j}}{k\_o^2 \mu\_y \varepsilon\_x - k\_z^2} \left( \omega \mu\_o \mu\_y (d/dy) H\_z + k\_z (d/dx) E\_z \right), \tag{13}$$

$$E\_y = \frac{j}{k\_o^2 \mu\_x \varepsilon\_y - k\_z^2} \left( \omega \mu\_o \mu\_x (d/dx) H\_z - k\_z (d/dy) E\_z \right), \tag{14}$$

$$H\_{\rm x} = \frac{\dot{j}}{k\_o^2 \mu\_x \varepsilon\_y - k\_z^2} \left(\omega \varepsilon\_o \varepsilon\_y (d/dy) E\_z - k\_z (d/dx) H\_z\right), \tag{15}$$

$$H\_y = -\frac{j}{k\_o^2 \mu\_y \varepsilon\_x - k\_z^2} (\omega \varepsilon\_o \varepsilon\_x (d/dx) E\_z + k\_z (d/dy) H\_z). \tag{16}$$

The relationships for the transverse field components, applied to (1) and (2), yield the following solutions for H and E, respectively

$$\underline{\mathbf{H}} = -\left(\underline{\underline{\mu\_r}}^{-1} / \mathrm{j}\omega\varepsilon\_o \right) \cdot (\nabla x \underline{\mathbf{E}}) , \tag{17}$$

$$
\underline{E} = \begin{pmatrix} \underline{\varepsilon\_r}^{-1} / \mathrm{j}\omega \varepsilon\_o \\ \underline{\varepsilon\_r} \end{pmatrix} \cdot (\nabla \underline{\mathrm{x}} \underline{\mathrm{H}}).\tag{18}
$$

Taking the cross product of both sides and substituting (1) and (2) for the right hand side of (17) and (18) yields

$$\underline{\nabla \mathbf{x} \varepsilon\_r}^{-1} \cdot (\nabla \mathbf{x} \underline{\mathbf{H}}) = k\_o^2 \underline{\mu}\_r \cdot \underline{\mathbf{H}} \tag{19}$$

$$
\begin{array}{c}
\nabla \underline{\mathbf{x}} \underline{\mu\_r}^{-1} \cdot (\nabla \underline{\mathbf{x}} \underline{\mathbf{E}}) = -k\_o^2 \underline{\underline{\varepsilon\_r}} \cdot \underline{\mathbf{E}}.\end{array} \tag{20}
$$

Equations (19) and (20) represent the vector wave equations in an anisotropic medium [12].

#### 3.2. Dispersion equation for Hz

∇xE ¼ jωμoμ<sup>r</sup> � H, (1)

∇xH ¼ �jωεoε<sup>r</sup> � E, (2)

<sup>5</sup>, (3)

<sup>5</sup>: (4)

� �

� �

<sup>o</sup> <sup>þ</sup> <sup>μ</sup>zHZzo

<sup>o</sup> <sup>þ</sup> <sup>ε</sup>zEZzo

,

(5)

:

(6)

¼ �jωμ<sup>o</sup> μxHXxo þ μyHYy

¼ jωε<sup>o</sup> εxEXxo þ εyEYy

ð Þ d=dy Ez � jkzEy ¼ �jωμoμxHx, (7)

jkzEx � ð Þ d=dx Ez ¼ �jωμoμyHy, (8)

ð Þ d=dx Ey � ð Þ d=dy Ex ¼ �jωμoμzHz: (9)

ð Þ d=dy Hz � jkzHy ¼ jωεoεxEx, (10)

jkzHx � ð Þ d=dx Hz ¼ jωεoεyEy, (11)

ð Þ d=dx Hy � ð Þ d=dy Hx ¼ �jωμoμzHz: (12)

where ω is the frequency in radians, ε<sup>o</sup> is the permittivity of free space, μ<sup>o</sup> is the permeability of

ε<sup>x</sup> 0 0 0 ε<sup>y</sup> 0 0 0 ε<sup>z</sup>

μ<sup>x</sup> 0 0 0 μ<sup>y</sup> 0 0 0 μ<sup>z</sup>

3 7

> 3 7

free space, E = xoEx + yoEy + zoEz and H = xoHx + yoHy + zoHz. We define μ<sup>r</sup> and ε<sup>r</sup> as

2 6 4

2 6 4

dEY dx � dEX dy

� �

dHY dx � dHX dy

Using the radiation condition, we assume a solution of E(x, y, z) = E(x, y)e�jkzz [17]. Now isolate the individual components of (5) by taking the dot product with xo, yo, and zo respectively. This

Assuming a solution of H(x, y, z) = H(x, y)e�jkzz for (6), the same procedure yields [17]

Using (7)–(12) allows for the transverse field components of the electric and magnetic fields in

� �

ε<sup>r</sup> ¼

μ<sup>r</sup> ¼

þ zo

þ zo

Applying Eqs. (3) and (4) to Eqs. (1) and (2) yields the following

xo dEZ dy � dEY dz

xo

� �

170 Antennas and Wave Propagation

� �

dHZ dy � dHY dz

þ y o

> þ y o

operation yields the following equations

terms of the derivatives of Hz and Ez as

dEX dz � dEZ dx

� �

dHX dz � dHZ dx

� �

We expand (19) in terms of (13)–(16)

$$\nabla \mathbf{x} \left\{ \frac{\mathbf{x}\_{\rho}}{\varepsilon\_{\varepsilon}} \left[ (d/dy)\mathcal{H}\_{\mathbf{Z}} - (d/dz)\mathcal{H}\_{y} \right] + \frac{\mathbf{y}\_{\rho}}{\varepsilon\_{\varepsilon}} \left[ (d/dz)\mathcal{H}\_{\mathbf{x}} - (d/dx)\mathcal{H}\_{z} \right] + \frac{\mathbf{z}\_{\rho}}{\varepsilon\_{\varepsilon}} \left[ (d/dx)\mathcal{H}\_{y} - (d/dy)\mathcal{H}\_{x} \right] \right\} = k\_{o}^{2} \underline{\mu\_{r}} \cdot \underline{\mathbf{H}}. \tag{21}$$

Evaluating the remaining cross product of (21) yields the final form of the expanded wave equation

$$
\underline{\mathbf{x}}\_o \Pi\_\mathbf{x} + \underline{\mathbf{y}}\_o \Pi\_\mathbf{y} + \underline{\mathbf{z}}\_o \Pi\_\mathbf{z} = k\_o^2 \underline{\mu\_r} \cdot \underline{\mathbf{H}}.\tag{22}
$$

$$\Pi\_{\mathbf{x}} = \left[ \left( d^2 / d\mathbf{x} dy \right) \mathcal{H}\_y - \left( d^2 / dy^2 \right) \mathcal{H}\_{\mathbf{x}} \right] / \varepsilon\_z - \left[ \left( d^2 / dz^2 \right) \mathcal{H}\_{\mathbf{x}} - \left( d^2 / dx dz \right) \mathcal{H}\_z \right] / \varepsilon\_y \tag{23}$$

$$\Pi\_y = \left[ \left( d^2 / dydz \right) H\_z - \left( d^2 / dz^2 \right) H\_y \right] / \varepsilon\_x - \left[ \left( d^2 / dx^2 \right) H\_y - \left( d^2 / dxdy \right) H\_x \right] / \varepsilon\_z \tag{24}$$

$$
\Pi\_z = \left[ \left( d^2 / d\mathbf{x} d\mathbf{z} \right) \mathcal{H}\_x - \left( d^2 / d\mathbf{x}^2 \right) \mathcal{H}\_z \right] / \varepsilon\_y - \left[ \left( d^2 / dy^2 \right) \mathcal{H}\_z - \left( d^2 / d\mathbf{x} dy \right) \mathcal{H}\_x \right] / \varepsilon\_x. \tag{25}
$$

Taking the dot product of (22) with zo allows the isolation of Hz on the right hand side of the equation in terms of (265 on the left hand side

$$\left[\left(d^2/dydz\right)H\_y - \left(d^2/dy^2\right)H\_z\right]/\varepsilon\_x - \left[\left(d^2/d\mathbf{x}^2\right)H\_z + \left(d^2/dxdz\right)H\_x\right]/\varepsilon\_y = k\_v^2\mu\_zH\_z,\tag{26}$$

k2 z

dispersion equation for Ez

for kz in the medium.

μy

2 z d2 <sup>=</sup>dx<sup>2</sup> � �Ez <sup>þ</sup>

3.4. Transmission and reflection from an anisotropic half-space

restricted by the radiation condition in all three dimensions

Figure 1. A plane wave incident from free space on an anisotropic boundary.

Ez/dx2 and d<sup>2</sup>

k2 <sup>o</sup>ε<sup>x</sup>

k 2 <sup>o</sup>μyε<sup>x</sup> � k 3 <sup>5</sup> <sup>d</sup><sup>2</sup>

<sup>=</sup>dx<sup>2</sup> � �Ez <sup>þ</sup>

k2 z

> 2 z d2

" #

μx

Ez/dy2 terms in (35) gives the following second order differential

<sup>=</sup>dy<sup>2</sup> � �Ez <sup>þ</sup> <sup>k</sup>

d2

2

�j k ð Þ xxþkyyþkzz , (37)

�j k ð Þ xxþkyyþkzz : (38)

<sup>=</sup>dy<sup>2</sup> � �Ez <sup>¼</sup> <sup>k</sup>

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123

2

<sup>o</sup>εzEz ¼ 0: (36)

<sup>o</sup>εzEz: (35)

173

<sup>μ</sup><sup>x</sup> <sup>k</sup><sup>2</sup> <sup>z</sup> � k 2 <sup>o</sup>μxε<sup>y</sup> � � � <sup>1</sup>

k2 <sup>o</sup>ε<sup>y</sup>

Birefringence is a characteristic of anisotropic media where a single incident wave entering the boundary of an anisotropic medium gives rise to two refracted waves as shown in Figure 1 or a single incident wave leaving gives rise to two reflected waves as shown in Figure 2. We call these two waves the ordinary wave and the extraordinary wave. To see how the anisotropy of a medium gives rise to the birefringence phenomenon, Eqs. (28) and (36) will yield a solution

Equations (28) and (36) yield the following solutions in unbounded anisotropic media

Ezð Þ¼ x; y; z Eoe

Hzð Þ¼ x; y; z Hoe

k 2 <sup>o</sup>μxε<sup>y</sup> � k

<sup>μ</sup><sup>y</sup> <sup>k</sup><sup>2</sup> <sup>z</sup> � k 2 <sup>o</sup>μyε<sup>x</sup> � � � <sup>1</sup>

Combining the d2

2 4

By keeping in mind that <sup>d</sup>/dz <sup>=</sup> �jkz, setting Ez = 0, and differentiating (15) and (16) by d2 /dxdz and d2 /dydz, produces the following result

$$\left[\frac{k\_z^2}{\varepsilon\_y(k\_z^2 - k\_o^2 \varepsilon\_y \mu\_x)} - \frac{1}{\varepsilon\_y}\right] (d^2/dx^2)H\_z + \left[\frac{k\_z^2}{\varepsilon\_x\left(k\_z^2 - k\_o^2 \varepsilon\_x \mu\_y\right)} - \frac{1}{\varepsilon\_x}\right] (d^2/dy^2)H\_z = k\_o^2 \mu\_z H\_z. \tag{27}$$

Combining the d2 Hz/dx2 and d2 Hz/dy2 terms in (27) gives the following second order differential dispersion equation for Hz

$$\frac{k\_o^2 \mu\_x}{k\_o^2 \mu\_x \varepsilon\_y - k\_z^2} \left( d^2 / d\mathbf{x}^2 \right) H\_z + \frac{k\_o^2 \mu\_y}{k\_o^2 \mu\_y \varepsilon\_x - k\_z^2} \left( d^2 / dy^2 \right) H\_z + k\_o^2 \mu\_z H\_z = 0. \tag{28}$$

#### 3.3. Dispersion equation for Ez

Expanding the ∇xE term of (18) in terms of (13)–(16) yields

$$\begin{split} \nabla \mathbf{x} \left\{ \underline{\mathbf{x}}\_o [(d/dy)\mathbf{E}\_Z - (d/dz)\mathbf{E}\_Y]/\mu\_x + \underline{\mathbf{y}}\_o [(d/dz)\mathbf{E}\_X - (d/dx)\mathbf{E}\_Z]/\mu\_y \right\} \\ + \underline{\mathbf{z}}\_o [(d/dx)\mathbf{E}\_Y - (d/dy)\mathbf{E}\_I]/\mu\_z \right\} = k\_o^2 \underline{\mathbf{z}}\_r \cdot \underline{\mathbf{z}}. \end{split} \tag{29}$$

Evaluating the remaining cross product of (29) gives the final form of the expanded wave equation

$$
\underline{\mathbf{x}}\_o \underline{\xi}\_x + \underline{y}\_o \underline{\xi}\_y + \underline{z}\_o \underline{\xi}\_z = k\_o^2 \underline{\xi}\_r \cdot \underline{\mathbf{E}} \tag{30}
$$

$$\mathcal{L}\_{\mathbf{x}} = \left[ \left( d^2 / d\mathbf{x} dy \right) \mathcal{E}\_Y - \left( d^2 / dy^2 \right) \mathcal{E}\_{\mathbf{x}} \right] / \mu\_z - \left[ \left( d^2 / dz^2 \right) \mathcal{E}\_{\mathbf{x}} - \left( d^2 / d\mathbf{x} dz \right) \mathcal{E}\_z \right] / \mu\_{y^\*} \tag{31}$$

$$\mathcal{E}\_y = \left[ \left( d^2 / dydz \right) \mathcal{E}\_z - \left( d^2 / dz^2 \right) \mathcal{E}\_y \right] / \mu\_x - \left[ \left( d^2 / dx^2 \right) \mathcal{E}\_y - \left( d^2 / dx dy \right) \mathcal{E}\_x \right] / \mu\_{z'} \tag{32}$$

$$\xi\_z = \left[ \left( d^2 / d\mathbf{x} d\mathbf{z} \right) \mathbb{E}\_\mathbf{x} - \left( d^2 / d\mathbf{x}^2 \right) \mathbb{E}\_z \right] / \mu\_y - \left[ \left( d^2 / dy^2 \right) \mathbb{E}\_z - \left( d^2 / dy d\mathbf{z} \right) \mathbb{E}\_y \right] / \mu\_\mathbf{x}. \tag{33}$$

Taking the dot product of (30) with zo allows isolation of the Ez component on the right hand side of the equation in terms of (33) on the left hand side

$$\left[\left(d^2/dxdz\right)\mathbf{E}\_x - \left(d^2/dx^2\right)\mathbf{E}\_z\right]/\mu\_y + \left[\left(d^2/dydz\right)\mathbf{E}\_y - \left(d^2/dy^2\right)\mathbf{E}\_z\right]/\mu\_x = \mathbf{k}\_o^2 \varepsilon\_z \mathbf{E}\_z \tag{34}$$

Keeping in mind that <sup>d</sup>/dz <sup>=</sup> �jkz, setting Hz = 0, and differentiating (15) and (16) by d2 /dxdz and d2 /dydz produces the following result

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123 173

$$
\left[\frac{k\_z^2}{\mu\_y \left(k\_z^2 - k\_o^2 \mu\_y \varepsilon\_x\right)} - \frac{1}{\mu\_y}\right] \left(d^2/dx^2\right)\mathcal{E}\_z + \left[\frac{k\_z^2}{\mu\_x \left(k\_z^2 - k\_o^2 \mu\_x \varepsilon\_y\right)} - \frac{1}{\mu\_x}\right] \left(d^2/dy^2\right)\mathcal{E}\_z = k\_o^2 \varepsilon\_z \mathcal{E}\_z. \tag{35}
$$

Combining the d2 Ez/dx2 and d<sup>2</sup> Ez/dy2 terms in (35) gives the following second order differential dispersion equation for Ez

$$\frac{k\_o^2 \varepsilon\_x}{k\_o^2 \mu\_y \varepsilon\_x - k\_z^2} \left( d^2 / d\mathbf{x}^2 \right) \mathbf{E}\_z + \frac{k\_o^2 \varepsilon\_y}{k\_o^2 \mu\_x \varepsilon\_y - k\_z^2} \left( d^2 / dy^2 \right) \mathbf{E}\_z + k\_o^2 \varepsilon\_z \mathbf{E}\_z = \mathbf{0}.\tag{36}$$

#### 3.4. Transmission and reflection from an anisotropic half-space

Taking the dot product of (22) with zo allows the isolation of Hz on the right hand side of the

By keeping in mind that <sup>d</sup>/dz <sup>=</sup> �jkz, setting Ez = 0, and differentiating (15) and (16) by d2

ε<sup>x</sup> k<sup>2</sup> <sup>z</sup> � k 2 <sup>o</sup>εxμ<sup>y</sup> � � � <sup>1</sup>

> k2 <sup>o</sup>μ<sup>y</sup>

k 2 <sup>o</sup>μyε<sup>x</sup> � k

2 4 <sup>=</sup>dx<sup>2</sup> � �Hz <sup>þ</sup> <sup>d</sup><sup>2</sup>

k 2 z

> 2 z d2

o

o <sup>¼</sup> <sup>k</sup><sup>2</sup> <sup>o</sup>ε<sup>r</sup> � E:

Evaluating the remaining cross product of (29) gives the final form of the expanded wave

<sup>ξ</sup><sup>y</sup> <sup>þ</sup> zoξ<sup>z</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

Taking the dot product of (30) with zo allows isolation of the Ez component on the right hand

Keeping in mind that <sup>d</sup>/dz <sup>=</sup> �jkz, setting Hz = 0, and differentiating (15) and (16) by d2

<sup>=</sup>dydz � �Ey � <sup>d</sup><sup>2</sup>

=dxdz � �Hx � �=ε<sup>y</sup> <sup>¼</sup> <sup>k</sup>

εx

Hz/dy2 terms in (27) gives the following second order differen-

2

½ � ð Þ d=dz EX � ð Þ d=dx EZ =μ<sup>y</sup>

<sup>o</sup>ε<sup>r</sup> � E, (30)

=dxdz � �Ez � �=μy, (31)

=dxdy � �Ex � �=μz, (32)

=dydz � �Ey � �=μx: (33)

2

<sup>o</sup>εzEz, (34)

/dxdz and

<sup>=</sup>dy<sup>2</sup> � �Hz <sup>þ</sup> <sup>k</sup>

<sup>=</sup>dz<sup>2</sup> � �Ex � <sup>d</sup><sup>2</sup>

<sup>=</sup>dx<sup>2</sup> � �Ey � <sup>d</sup><sup>2</sup>

<sup>=</sup>dy<sup>2</sup> � �Ez � <sup>d</sup><sup>2</sup>

=dy<sup>2</sup> � �Ez � �=μ<sup>x</sup> <sup>¼</sup> <sup>k</sup>

3 <sup>5</sup> <sup>d</sup><sup>2</sup>

<sup>=</sup>dy<sup>2</sup> � �Hz <sup>¼</sup> <sup>k</sup>

2

2

<sup>o</sup>μzHz ¼ 0: (28)

<sup>o</sup>μzHz, (26)

<sup>o</sup>μzHz: (27)

/dxdz

(29)

equation in terms of (265 on the left hand side

/dydz, produces the following result

εy

Hz/dx2 and d2

2 z d2 <sup>=</sup>dx<sup>2</sup> � �Hz <sup>þ</sup>

Expanding the ∇xE term of (18) in terms of (13)–(16) yields

<sup>=</sup>dxdy � �EY � <sup>d</sup><sup>2</sup>

<sup>=</sup>dydz � �Ez � <sup>d</sup><sup>2</sup>

<sup>=</sup>dxdz � �Ex � <sup>d</sup><sup>2</sup>

side of the equation in terms of (33) on the left hand side

=dx<sup>2</sup> � �Ez � �=μ<sup>y</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup>

∇x xo½ � ð Þ d=dy EZ � ð Þ d=dz EY =μ<sup>x</sup> þ y

þzo½ � ð Þ d=dx EY � ð Þ d=dy EX =μ<sup>z</sup>

xoξ<sup>x</sup> þ y o

=dy<sup>2</sup> � �Ex � �=μ<sup>z</sup> � <sup>d</sup><sup>2</sup>

=dz<sup>2</sup> � �Ey � �=μ<sup>x</sup> � <sup>d</sup><sup>2</sup>

=dx<sup>2</sup> � �Ez � �=μ<sup>y</sup> � <sup>d</sup><sup>2</sup>

k2 <sup>o</sup>μ<sup>x</sup>

n

d2 <sup>=</sup>dx<sup>2</sup> � �Hz <sup>þ</sup>

=dy<sup>2</sup> � �Hz � �=ε<sup>x</sup> � <sup>d</sup><sup>2</sup>

<sup>=</sup>dydz � �Hy � <sup>d</sup><sup>2</sup>

d2

172 Antennas and Wave Propagation

k 2 z

" #

tial dispersion equation for Hz

k2 <sup>o</sup>μxε<sup>y</sup> � k

3.3. Dispersion equation for Ez

<sup>ξ</sup><sup>x</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

<sup>ξ</sup><sup>y</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

<sup>ξ</sup><sup>z</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

<sup>=</sup>dxdz � �Ex � <sup>d</sup><sup>2</sup>

/dydz produces the following result

d2

ε<sup>y</sup> k 2 <sup>z</sup> � k 2 <sup>o</sup>εyμ<sup>x</sup> � � � <sup>1</sup>

Combining the d2

equation

d2

and d2

Birefringence is a characteristic of anisotropic media where a single incident wave entering the boundary of an anisotropic medium gives rise to two refracted waves as shown in Figure 1 or a single incident wave leaving gives rise to two reflected waves as shown in Figure 2. We call these two waves the ordinary wave and the extraordinary wave. To see how the anisotropy of a medium gives rise to the birefringence phenomenon, Eqs. (28) and (36) will yield a solution for kz in the medium.

Equations (28) and (36) yield the following solutions in unbounded anisotropic media restricted by the radiation condition in all three dimensions

$$E\_z(x, y, z) = E\_o e^{-j\left(k\_x x + k\_y y + k\_z z\right)},\tag{37}$$

$$H\_z(x,y,z) = H\_0 e^{-j\left(k\_x x + k\_y y + k\_z z\right)}.\tag{38}$$

Figure 1. A plane wave incident from free space on an anisotropic boundary.

kz is a second order polynomial, which yields only the values for the positive and negative

Electromagnetic wave behavior of waveguides is well understood in the literature. The mode within a waveguide that are based on the voltage and current distributions within the waveguide make up the basis for the electric and magnetic field calculations. This section derives similar formulations for a rectangular waveguide uniformly filled with an anisotropic medium as shown in Figure 3. Figure 3 shows propagation in the zo-direction along the length of the waveguide. Rectangular waveguides are most commonly used for material measurement and characterization, and therefore understanding how electromagnetic waves propagate in an anisotropic waveguide is important for material characterization purposes. Furthermore, this section shows how the anisotropic derivation of waveguide behavior parallels that of a typical waveguide, and therefore how anisotropy may be applied to other waveguide geometries.

Assume source free Maxwell's equations in the same form as (1) and (2). Then the transverse

<sup>υ</sup>ð Þþ x; y V<sup>00</sup>

<sup>υ</sup>ð Þþ x; y I

�jkzz

�jkzz

<sup>υ</sup>ð Þz e 00 <sup>υ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � , � (42)

<sup>υ</sup>ð Þ <sup>x</sup>; <sup>y</sup> � , � (43)

, (44)

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123 175

, (45)

00 <sup>υ</sup>ð Þz h<sup>00</sup>

propagation of the single ordinary wave.

4.1. Anisotropic mode functions

electromagnetic fields are defined

walls.

ETυð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

HTυð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>X</sup>

m

m

X n V0 <sup>υ</sup>ð Þz e 0

X n I 0 <sup>υ</sup>ð Þz h<sup>0</sup>

Vυð Þ¼ z Voe

Iυð Þ¼ z Ioe

Figure 3. Cross section of a closed rectangular waveguide filled with anisotropic metamaterial and surrounded by PEC

4. Anisotropic rectangular waveguide

Figure 2. A plane wave incident from an anisotropic medium on a free space boundary.

Plugging (37) into (36) (equivocally we could substitute (38) into (19)) allows the generation of a polynomial equation whose solutions give the values of kz in the anisotropic medium. Noting that d<sup>2</sup> <sup>=</sup>dx<sup>2</sup> ¼ �<sup>k</sup> 2 <sup>x</sup> and <sup>d</sup><sup>2</sup> <sup>=</sup>dy<sup>2</sup> ¼ �<sup>k</sup> 2 <sup>y</sup>, (36) simplifies as

$$k\_o^2 k\_x^2 \varepsilon\_x \mathbb{E}\_z / \left(k\_o^2 \mu\_y \varepsilon\_x - k\_z^2\right) + k\_o^2 \varepsilon\_y k\_y^2 \mathbb{E}\_z / \left(k\_o^2 \mu\_x \varepsilon\_y - k\_z^2\right) - k\_o^2 \varepsilon\_z \mathbb{E}\_z = 0. \tag{39}$$

Dividing out the k<sup>2</sup> oEzterm and multiplying through by both denominators gives us the following factored polynomial

$$
\left(k\_o^2 \mu\_y \varepsilon\_x - k\_z^2\right) \left(k\_o^2 \mu\_x \varepsilon\_y - k\_z^2\right) \varepsilon\_z - k\_x^2 \varepsilon\_x \left(k\_o^2 \mu\_y \varepsilon\_y - k\_z^2\right) - \varepsilon\_y k\_y^2 \left(k\_o^2 \mu\_y \varepsilon\_x - k\_z^2\right) = 0. \tag{40}
$$

Finally, multiplying out (40) yields a fourth order polynomial whose roots yield the four values of kz describing the ordinary wave and extraordinary wave in the positive and negative propagation directions

$$k\_z^4 \mu\_z + \left[k\_x^2 \mu\_x + k\_y^2 \mu\_y - \left(\varepsilon\_x \mu\_y + \varepsilon\_y \mu\_x\right)k\_o^2 \mu\_z\right]k\_z^2 + \left[k\_o^4 \varepsilon\_x \varepsilon\_y \mu\_x \mu\_y \mu\_z - k\_o^2 k\_x \varepsilon\_x \mu\_y \mu\_x - k\_o^2 \varepsilon\_y k\_y \mu\_x \mu\_y\right] = 0. \tag{41}$$

Equation (41) is directly responsible for the existence of the extraordinary wave that is characteristic of the birefringence phenomenon. In an isotropic medium, the resulting polynomial for kz is a second order polynomial, which yields only the values for the positive and negative propagation of the single ordinary wave.

#### 4. Anisotropic rectangular waveguide

Electromagnetic wave behavior of waveguides is well understood in the literature. The mode within a waveguide that are based on the voltage and current distributions within the waveguide make up the basis for the electric and magnetic field calculations. This section derives similar formulations for a rectangular waveguide uniformly filled with an anisotropic medium as shown in Figure 3. Figure 3 shows propagation in the zo-direction along the length of the waveguide. Rectangular waveguides are most commonly used for material measurement and characterization, and therefore understanding how electromagnetic waves propagate in an anisotropic waveguide is important for material characterization purposes. Furthermore, this section shows how the anisotropic derivation of waveguide behavior parallels that of a typical waveguide, and therefore how anisotropy may be applied to other waveguide geometries.

#### 4.1. Anisotropic mode functions

Plugging (37) into (36) (equivocally we could substitute (38) into (19)) allows the generation of a polynomial equation whose solutions give the values of kz in the anisotropic medium. Noting

<sup>o</sup>μxε<sup>y</sup> � k

oEzterm and multiplying through by both denominators gives us the follow-

2 z � � � <sup>ε</sup>yk<sup>2</sup>

<sup>o</sup>μxε<sup>y</sup> � k

<sup>o</sup>εxεyμxμyμ<sup>z</sup> � k

2 z � � � <sup>k</sup>

2

h i

2

<sup>y</sup> <sup>k</sup><sup>2</sup>

<sup>o</sup> kxεxμyμ<sup>x</sup> � k

<sup>o</sup>μyε<sup>x</sup> � k

2 <sup>o</sup>εykyμxμ<sup>y</sup>

� �

2 z

<sup>o</sup>εzEz ¼ 0: (39)

¼ 0: (40)

¼ 0: (41)

<sup>y</sup>, (36) simplifies as

<sup>þ</sup> <sup>k</sup><sup>2</sup> <sup>o</sup>εyk 2 <sup>y</sup>Ez<sup>=</sup> <sup>k</sup><sup>2</sup>

> 2 <sup>x</sup>ε<sup>x</sup> <sup>k</sup><sup>2</sup>

Finally, multiplying out (40) yields a fourth order polynomial whose roots yield the four values of kz describing the ordinary wave and extraordinary wave in the positive and negative

Equation (41) is directly responsible for the existence of the extraordinary wave that is characteristic of the birefringence phenomenon. In an isotropic medium, the resulting polynomial for

that d<sup>2</sup>

k4 <sup>z</sup>μ<sup>z</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

<sup>=</sup>dx<sup>2</sup> ¼ �<sup>k</sup>

174 Antennas and Wave Propagation

Dividing out the k<sup>2</sup>

ing factored polynomial

k2 <sup>o</sup>μyε<sup>x</sup> � k

propagation directions

<sup>x</sup>μ<sup>x</sup> þ k 2

2 <sup>x</sup> and <sup>d</sup><sup>2</sup>

k 2 o k2

� �

<sup>x</sup>εxEz= k

2 z k2

<sup>y</sup>μ<sup>y</sup> � εxμ<sup>y</sup> þ εyμ<sup>x</sup> � �

h i

<sup>=</sup>dy<sup>2</sup> ¼ �<sup>k</sup>

2 <sup>o</sup>μyε<sup>x</sup> � <sup>k</sup><sup>2</sup> z

2

Figure 2. A plane wave incident from an anisotropic medium on a free space boundary.

� �

<sup>o</sup>μxε<sup>y</sup> � <sup>k</sup><sup>2</sup> z � �ε<sup>z</sup> � <sup>k</sup>

> k2 <sup>o</sup>μ<sup>z</sup>

k 2 <sup>z</sup> þ k 4 Assume source free Maxwell's equations in the same form as (1) and (2). Then the transverse electromagnetic fields are defined

$$\underline{E}\_{\mathcal{V}\boldsymbol{\nu}}(\mathbf{x},\boldsymbol{y},\boldsymbol{z}) = \sum\_{m} \sum\_{n} \left[ V'\_{\boldsymbol{\nu}}(\mathbf{z}) \underline{\boldsymbol{\varepsilon}}'\_{\boldsymbol{\nu}}(\mathbf{x},\boldsymbol{y}) + V''\_{\boldsymbol{\nu}}(\mathbf{z}) \underline{\boldsymbol{\varepsilon}}''\_{\boldsymbol{\nu}}(\mathbf{x},\boldsymbol{y}) \right],\tag{42}$$

$$\underline{H}\_{\rm Tv}(\mathbf{x}, y, z) = \sum\_{m} \sum\_{n} \left[ I\_{\nu}'(z) \underline{h}\_{\nu}'(\mathbf{x}, y) + I\_{\nu}''(z) \underline{h}\_{\nu}''(\mathbf{x}, y) \right],\tag{43}$$

$$V\_{\upsilon}(z) = V\_{\upsilon} \varepsilon^{-jk\_z z},\tag{44}$$

$$I\_{\upsilon}(z) = I\_{\upsilon} \mathbf{e}^{-jk\_{z}z},\tag{45}$$

Figure 3. Cross section of a closed rectangular waveguide filled with anisotropic metamaterial and surrounded by PEC walls.

where ET and HT are the transverse electric and magnetic fields, V(z) and I(z) are the voltage and current at point z, e and h are the waveguide mode equations, and υ є [m, n] is the mode number defined by the two indices m and n.

#### 4.1.1. Incident TE mode

Assuming only a TE type mode in the waveguide sets Ez = 0. Then (13)–(16) become

$$E\_{x\nu}^{\prime} = -j\omega\mu\_o\mu\_y(\mathrm{d}/\mathrm{d}y)H\_{z\nu}^{\prime}/\left(k\_o^2\mu\_y\varepsilon\_x - k\_{z\nu}^2\right),\tag{46}$$

and substituting (48) and (49) for H<sup>00</sup>

zυ

<sup>z</sup>υ= k 2 <sup>o</sup>μxε<sup>y</sup> � k

> μxk 2 <sup>x</sup>υ= k 2 <sup>o</sup>μxε<sup>y</sup> � k

<sup>o</sup> μxε<sup>y</sup> þ μyε<sup>x</sup> � � � <sup>k</sup>

equations for the TE mode vectors in (42) and (43)

yυμxμy=μ<sup>2</sup>

�y o

> þy o

E 0

> E 0

can be solved for H<sup>00</sup>

k 2 <sup>z</sup><sup>υ</sup> <sup>¼</sup> <sup>k</sup><sup>2</sup>

4.1.2. Incident TM mode

n

�

4k<sup>2</sup> xυk<sup>2</sup>

e00

h <sup>00</sup>

=dx<sup>2</sup> � �H<sup>00</sup>

Assuming a solution of the form

k 2 <sup>o</sup>μ<sup>x</sup> <sup>d</sup><sup>2</sup> <sup>x</sup><sup>υ</sup> and H<sup>00</sup>

2 <sup>o</sup>μ<sup>y</sup> <sup>d</sup><sup>2</sup>

=dy<sup>2</sup> � �H<sup>00</sup>

<sup>z</sup><sup>υ</sup> <sup>¼</sup> Hocosð Þ kxυ<sup>x</sup> cos kyυ<sup>y</sup> � �<sup>e</sup>

which meets the boundary conditions at the PEC walls of the waveguide, then plugging (58)

Solving (62) for kz<sup>υ</sup> gives the following equation which yields four solutions to the propagation

2 <sup>y</sup>υμy=μ<sup>z</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r ),

<sup>o</sup> μxε<sup>y</sup> � μyε<sup>x</sup> � � � <sup>k</sup><sup>2</sup>

Equations (62) and (63) provide the criteria for determining the cutoff frequency for the propagation of modes inside the waveguide. Plugging (59) into (46)–(49) yields the following

2 <sup>y</sup>υ= k 2 <sup>o</sup>μyε<sup>x</sup> � k

into (53) imposes the following restriction on the values of the tensors in (3) and (4)

<sup>x</sup>υμx=μ<sup>z</sup> � k

<sup>υ</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>j</sup>ωμoHo xoμykyυcosð Þ kxυ<sup>x</sup> sin kyυ<sup>y</sup> � �<sup>=</sup> <sup>k</sup>

<sup>υ</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> jkzυHo xokxυsinð Þ kxυ<sup>x</sup> cos kyυ<sup>y</sup> � �<sup>=</sup> <sup>k</sup>

Assuming only a TM type mode in the waveguide sets Hz = 0. Then (13)–(16) become

<sup>x</sup><sup>υ</sup> ¼ �jkzυð Þ d=dx E

<sup>y</sup><sup>υ</sup> ¼ �jkzυð Þ d=dy E

<sup>μ</sup>xkxυsinð Þ kxυ<sup>x</sup> cos kyυ<sup>y</sup> � �<sup>=</sup> <sup>k</sup><sup>2</sup>

kyυcosð Þ kxυ<sup>x</sup> sin kyυ<sup>y</sup> � �<sup>=</sup> <sup>k</sup>

0 <sup>z</sup>υ= k 2 <sup>o</sup>μyε<sup>x</sup> � k

0 <sup>z</sup>υ<sup>=</sup> <sup>k</sup><sup>2</sup>

2 zυ � � <sup>þ</sup> <sup>μ</sup>y<sup>k</sup>

constant for the ordinary and extraordinary waves described in Section 3.4

2

<sup>z</sup> þ k 2 <sup>z</sup>υ= k 2 <sup>o</sup>μyε<sup>x</sup> � k

�jkzυz

2 zυ � � <sup>þ</sup> <sup>k</sup>

H<sup>00</sup>

<sup>y</sup><sup>υ</sup> yields the following differential equation that

2 oμzH<sup>00</sup>

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123

, (59)

<sup>z</sup><sup>υ</sup> ¼ 0: (58)

177

2 zυ � � <sup>þ</sup> <sup>k</sup>

kx<sup>υ</sup> ¼ mπ=a, (60)

ky<sup>υ</sup> ¼ nπ=b, (61)

� � <sup>¼</sup> <sup>μ</sup>z: (62)

<sup>y</sup>υμy=μ<sup>z</sup>

2 zυ 2:

(63)

(64)

2 zυ

<sup>x</sup>υμx=μ<sup>z</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup>

2 <sup>o</sup>μyε<sup>x</sup> � k

2 zυ � �i

> 2 zυ

,

2 zυ

� �, (66)

� �, (67)

� �i: (65)

<sup>o</sup>μxε<sup>y</sup> � k

2 <sup>o</sup>μxε<sup>y</sup> � k

2 zυ

2 zυ

h i<sup>2</sup>

h � �

� � �

2 <sup>o</sup>μyε<sup>x</sup> � k

<sup>o</sup>μxε<sup>y</sup> � k

$$E\_{yv}'' = j\omega\mu\_o\mu\_x(d/dx)H\_{zv}''/(k\_o^2\mu\_x\varepsilon\_y - k\_{zv}^2),\tag{47}$$

$$H\_{yv}^{\prime\prime} = -\text{jk}\_{zv}^2(d/dx)H\_{zv}^{\prime\prime}/(\text{k}\_o^2\mu\_x\varepsilon\_y - \text{k}\_{zv}^2),\tag{48}$$

$$\mathcal{H}''\_{y\upsilon} = -j k\_{z\upsilon}^2 (d/dy) \mathcal{H}''\_{z\upsilon} / \left(k\_o^2 \mu\_y \varepsilon\_x - k\_{z\upsilon}^2\right). \tag{49}$$

To solve for H<sup>00</sup> <sup>z</sup><sup>υ</sup> we formulate the anisotropic wave equation from (1) and (2) where (52) resembles (19)

$$\nabla \times \nabla \times \underline{\mathbf{H}} = j\omega \varepsilon\_o \underline{\varepsilon\_r} \cdot \left( -j\omega \mu\_o \underline{\mu\_r} \cdot \underline{\mathbf{H}} \right) \tag{50}$$

$$
\nabla \times \nabla \times \underline{\underline{H}} = \dot{\mu} \sigma\_o \underline{\varepsilon\_r} \cdot (\nabla \times \underline{E})\_\prime \tag{51}
$$

$$\nabla \times \underline{\varepsilon\_r}^{-1} \cdot (\nabla \times \underline{H}) = k\_o^2 \underline{\mu\_r} \cdot \underline{H}.\tag{52}$$

Expanding the curl of (52)

$$\begin{bmatrix} \frac{1}{\varepsilon\_z} \frac{d\chi\_{zv}^{\prime\prime}}{dy} - \frac{1}{\varepsilon\_y} \frac{d\chi\_{yv}^{\prime\prime}}{dz} & 0 & 0\\ 0 & \frac{1}{\varepsilon\_x} \frac{d\chi\_{zv}^{\prime\prime}}{dz} - \frac{1}{\varepsilon\_z} \frac{d\chi\_{zv}^{\prime\prime}}{dx} & 0\\ 0 & 0 & \frac{1}{\varepsilon\_y} \frac{d\chi\_{yv}^{\prime\prime}}{dx} - \frac{1}{\varepsilon\_x} \frac{d\chi\_{xv}^{\prime\prime}}{dy} \end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1\\ 1 \end{bmatrix} = k\_o^2 \begin{bmatrix} \mu\_x H\_{xv}^{\prime\prime}\\ \mu\_y H\_{yv}^{\prime\prime}\\ \mu\_z H\_{zv}^{\prime\prime} \end{bmatrix},\tag{53}$$

$$
\varepsilon\_y \mid d\mathbf{x} \qquad \varepsilon\_x \mid dy \quad \square\_z \quad \square\_{zx} \qquad \square\_{yz} \quad \square\_{zv}
$$

$$
\chi\_{zv}'' = (d/dy)H\_{zv}'' - (d/dz)H\_{yv'}'' \tag{54}
$$

$$
\chi\_{yv}'' = (d/dz)H\_{xv}'' - (d/dx)H\_{zv'}'' \tag{55}
$$

$$
\chi\_{z\nu}^{\prime\prime} = (d/dx)H\_{y\nu}^{\prime} - (d/dy)H\_{x\nu}^{\prime}.\tag{56}
$$

Isolating the zo-component of (53) gives the following relationship for H<sup>00</sup> zυ

$$\left[\left(d^2/dxdz\right)H\_{xv}'' - \left(d^2/dx^2\right)H\_{zv}''\right]/\varepsilon\_y + \left(\left(d^2/dydz\right)H\_{yv}'' - \left(d^2/dy^2\right)H\_{zv}''\right)/\varepsilon\_x - k\_o^2\mu\_zH\_{zv}'' = 0,\tag{57}$$

and substituting (48) and (49) for H<sup>00</sup> <sup>x</sup><sup>υ</sup> and H<sup>00</sup> <sup>y</sup><sup>υ</sup> yields the following differential equation that can be solved for H<sup>00</sup> zυ

$$k\_o k\_x^2 \mu\_x (d^2/dx^2) H\_{z\nu}^{\prime\prime} / \left(k\_o^2 \mu\_x \varepsilon\_{\mathcal{Y}} - k\_{z\nu}^2\right) + k\_o^2 \mu\_y (d^2/dy^2) H\_{z\nu}^{\prime\prime} / \left(k\_o^2 \mu\_y \varepsilon\_x - k\_{z\nu}^2\right) + k\_o^2 \mu\_z H\_{z\nu}^{\prime\prime} = 0. \tag{58}$$

Assuming a solution of the form

where ET and HT are the transverse electric and magnetic fields, V(z) and I(z) are the voltage and current at point z, e and h are the waveguide mode equations, and υ є [m, n] is the mode

> <sup>z</sup>υ= k 2 <sup>o</sup>μyε<sup>x</sup> � k

<sup>z</sup>υ<sup>=</sup> <sup>k</sup><sup>2</sup>

<sup>z</sup>υ<sup>=</sup> <sup>k</sup><sup>2</sup>

<sup>z</sup>υ= k 2 <sup>o</sup>μyε<sup>x</sup> � k 2 zυ

2 zυ

2 zυ

> 2 zυ

∇ � ∇ � H ¼ jωεoε<sup>r</sup> � ð Þ ∇ � E , (51)

1

<sup>¼</sup> <sup>k</sup><sup>2</sup> o

1

1

, (46)

: (49)

, (50)

μxH<sup>00</sup> xυ

, (53)

<sup>z</sup><sup>υ</sup> ¼ 0, (57)

μyH<sup>00</sup> yυ

μzH<sup>00</sup> zυ

<sup>y</sup>υ, (54)

<sup>z</sup>υ, (55)

<sup>x</sup>υ: (56)

zυ

=ε<sup>x</sup> � k 2 oμzH<sup>00</sup>

<sup>o</sup>μ<sup>r</sup> � H: (52)

� �, (47)

� �, (48)

� �

<sup>o</sup>μxε<sup>y</sup> � k

<sup>o</sup>μxε<sup>y</sup> � k

� �

� �

<sup>z</sup><sup>υ</sup> we formulate the anisotropic wave equation from (1) and (2) where (52)

Assuming only a TE type mode in the waveguide sets Ez = 0. Then (13)–(16) become

<sup>z</sup>υð Þ d=dx H<sup>00</sup>

<sup>z</sup>υð Þ d=dy H<sup>00</sup>

∇ � ∇ � H ¼ jωεoε<sup>r</sup> � �jωμoμ<sup>r</sup> � H

�<sup>1</sup> � ð Þ¼ <sup>∇</sup> � <sup>H</sup> <sup>k</sup><sup>2</sup>

εy

=dydz � �H<sup>00</sup>

dχ<sup>00</sup> yυ dx � <sup>1</sup> εx dχ<sup>00</sup> xυ dy

<sup>z</sup><sup>υ</sup> � ð Þ d=dz H<sup>00</sup>

<sup>x</sup><sup>υ</sup> � ð Þ d=dx H<sup>00</sup>

<sup>y</sup><sup>υ</sup> � ð Þ d=dy H<sup>00</sup>

<sup>y</sup><sup>υ</sup> � <sup>d</sup><sup>2</sup>

� �

=dy<sup>2</sup> � �H<sup>00</sup>

zυ

<sup>x</sup><sup>υ</sup> ¼ �jωμoμyð Þ d=dy H<sup>00</sup>

<sup>y</sup><sup>υ</sup> ¼ jωμoμxð Þ d=dx H<sup>00</sup>

number defined by the two indices m and n.

E00

E00

H<sup>00</sup>

H<sup>00</sup>

<sup>y</sup><sup>υ</sup> ¼ �jk<sup>2</sup>

<sup>y</sup><sup>υ</sup> ¼ �jk<sup>2</sup>

∇ � ε<sup>r</sup>

dz 0 0

χ00

χ00

χ00

zυ

Isolating the zo-component of (53) gives the following relationship for H<sup>00</sup>

<sup>x</sup><sup>υ</sup> ¼ ð Þ d=dy H<sup>00</sup>

<sup>y</sup><sup>υ</sup> ¼ ð Þ d=dz H<sup>00</sup>

<sup>z</sup><sup>υ</sup> ¼ ð Þ d=dx H<sup>00</sup>

εx dχ<sup>00</sup> xυ dz � <sup>1</sup> εz dχ<sup>00</sup> zυ dx <sup>0</sup>

0 0 <sup>1</sup>

4.1.1. Incident TE mode

176 Antennas and Wave Propagation

To solve for H<sup>00</sup>

Expanding the curl of (52)

dχ<sup>00</sup> yυ

<sup>0</sup> <sup>1</sup>

<sup>x</sup><sup>υ</sup> � <sup>d</sup><sup>2</sup>

=dx<sup>2</sup> � �H<sup>00</sup>

� �=ε<sup>y</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup>

1 εz dχ<sup>00</sup> zυ dy � <sup>1</sup> εy

d2 =dxdz � �H<sup>00</sup>

resembles (19)

$$H\_{z\upsilon}^{\prime\prime} = H\_{\upsilon}\cos(k\_{x\upsilon}x)\cos(k\_{y\upsilon}y)e^{-\vec{\beta}\_{z\upsilon}z},\tag{59}$$

$$k\_{x\upsilon} = m\pi\eta\,\text{a},\tag{60}$$

$$k\_{yv} = n\pi/b,\tag{61}$$

which meets the boundary conditions at the PEC walls of the waveguide, then plugging (58) into (53) imposes the following restriction on the values of the tensors in (3) and (4)

$$
\mu\_x k\_{x\upsilon}^2 / \left( k\_o^2 \mu\_x \varepsilon\_y - k\_{z\upsilon}^2 \right) + \mu\_y k\_{y\upsilon}^2 / \left( k\_o^2 \mu\_y \varepsilon\_x - k\_{z\upsilon}^2 \right) = \mu\_z. \tag{62}
$$

Solving (62) for kz<sup>υ</sup> gives the following equation which yields four solutions to the propagation constant for the ordinary and extraordinary waves described in Section 3.4

$$\begin{split} k\_{z\nu}^2 &= \left\{ k\_o^2 \left( \mu\_x \varepsilon\_\mathcal{Y} + \mu\_y \varepsilon\_x \right) - k\_{x\nu}^2 \mu\_x / \mu\_z - k\_{y\nu}^2 \mu\_y / \mu\_z \\ &\pm \sqrt{4 k\_{x\nu}^2 k\_{y\nu}^2 \mu\_x \mu\_y / \mu\_z^2 + \left[ k\_o^2 \left( \mu\_x \varepsilon\_\mathcal{Y} - \mu\_y \varepsilon\_x \right) - k\_{x\nu}^2 \mu\_x / \mu\_z + k\_{y\nu}^2 \mu\_y / \mu\_z \right]^2} \right\} \end{split} \tag{63}$$

Equations (62) and (63) provide the criteria for determining the cutoff frequency for the propagation of modes inside the waveguide. Plugging (59) into (46)–(49) yields the following equations for the TE mode vectors in (42) and (43)

$$\begin{split} \underline{e}\_{\nu}''(\mathbf{x}, \mathbf{y}) &= j\omega\mu\_o H\_o \left[ \underline{\omega}\_o \mu\_y k\_{y\nu} \cos(k\_{x\nu}\mathbf{x}) \sin(k\_{y\nu}\mathbf{y}) / \left( k\_o^2 \mu\_y \varepsilon\_x - k\_{z\nu}^2 \right) \right. \\ &\left. - \underline{\underline{y}}\_o \mu\_x k\_{x\nu} \sin(k\_{x\nu}\mathbf{x}) \cos(k\_{y\nu}\mathbf{y}) / \left( k\_o^2 \mu\_x \varepsilon\_y - k\_{z\nu}^2 \right) \right], \end{split} \tag{64}$$

$$\begin{split} \underline{\mathbf{h}}\_{\nu}^{\prime\prime}(\mathbf{x}, \boldsymbol{y}) &= j \mathbb{k}\_{z\nu} H\_o \left[ \underline{\mathbf{x}}\_o k\_{x\nu} \text{sin}(k\_{x\nu} \mathbf{x}) \cos(k\_{y\nu} \boldsymbol{y}) / \left( k\_o^2 \mu\_x \epsilon\_y - k\_{z\nu}^2 \right) \\ &+ \underline{\mathbf{y}}\_o k\_{y\nu} \cos(k\_{x\nu} \mathbf{x}) \text{sin}(k\_{y\nu} \mathbf{y}) / \left( k\_o^2 \mu\_y \epsilon\_x - k\_{z\nu}^2 \right) \right]. \end{split} \tag{65}$$

#### 4.1.2. Incident TM mode

Assuming only a TM type mode in the waveguide sets Hz = 0. Then (13)–(16) become

$$\mathbf{E}'\_{z\nu} = -\mathbf{j}\mathbf{k}\_{z\nu}(\mathbf{d}/d\mathbf{x})\mathbf{E}'\_{z\nu} / \left(\mathbf{k}\_o^2 \boldsymbol{\mu}\_y \boldsymbol{\varepsilon}\_x - \mathbf{k}\_{z\nu}^2\right),\tag{66}$$

$$E\_{yv}^{'} = -jk\_{zv}(d/dy)E\_{zv}^{'} / \left(k\_o^2 \mu\_x \varepsilon\_y - k\_{zv}^2\right),\tag{67}$$

$$\boldsymbol{H}'\_{\rm x\nu} = j\omega\varepsilon\_o \varepsilon\_y (\boldsymbol{d}/d\boldsymbol{y}) \mathbf{E}'\_{\rm z\nu} / (k\_o^2 \mu\_x \varepsilon\_y - k\_{\rm z\nu}^2) \,\tag{68}$$

At any length z along the waveguide, the assumption that the horizontal distance between two resonant walls represented as a partially filled parallel plate waveguide is a valid presumption. We can calculate Lg(z) as the unknown distance between the edge of the anisotropic medium and the cavity wall based on a transverse resonance condition in the xo-direction. However, we first need to derive the characteristic impedance of the anisotropic region in the transmission

Calculating the fields in the free space region of the waveguide begins with Maxwell's source free Eqs. (1) and (2) and the equations for the individual vector components of the electromagnetic fields (13)–(16). Using the standard derivation of the wave equation for Hz in free space

<sup>¼</sup> <sup>∇</sup><sup>T</sup> <sup>∇</sup><sup>T</sup> � <sup>H</sup><sup>υ</sup> ð Þ� <sup>∇</sup><sup>2</sup>

2 o

Hz<sup>υ</sup> <sup>¼</sup> <sup>0</sup>: (78)

Hz <sup>¼</sup> <sup>0</sup>: (79)

<sup>T</sup>Hzυ, (76)

)Hz = 0 then (78)

<sup>T</sup>Hz<sup>υ</sup> ¼ 0, (77)

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123 179

/dy<sup>2</sup>

, (80)

<sup>o</sup> <sup>¼</sup> Zo Ae�jkox � Beþjkox : (81)

line model.

becomes

expression for Ey

Ez, respectively.

kyo, then Ex = 0, Hx = 0 and Hy = 0 as well.

4.2.2. Electromagnetic fields in anisotropic region

from (1) and (2) shows

4.2.1. Electromagnetic fields in free space regions

∇<sup>T</sup> � ∇<sup>T</sup> � H<sup>υ</sup> ¼ jωε<sup>o</sup> ∇<sup>T</sup> � E<sup>υ</sup>

d2

jωε<sup>o</sup> �jωμoHz<sup>υ</sup>

<sup>=</sup>dx<sup>2</sup> <sup>þ</sup> <sup>d</sup><sup>2</sup>

With kyo = 0 no variation of the fields in the y<sup>o</sup> direction and (d<sup>2</sup>

d2

Equation (80) is a standard differential equation with a known solution [17]

Ey <sup>¼</sup> koωμ<sup>o</sup> Ae�jkox � Beþjkox =<sup>k</sup>

<sup>=</sup>dx<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2 o

Hz<sup>υ</sup> <sup>¼</sup> Ae�jkox <sup>þ</sup> Beþjkox

where A and B are yet to be determined coefficients. Substituting (80) into (13)–(16) yields the

Accounting for the restrictions imposed by the transverse resonance conditions on Ez, kzo and

Starting with (1) and (2) for the source free Maxwell's equations in an anisotropic medium, the vector components (13)–(16) led to the derivation of the dispersion Eqs. (26) and (38) for Hz and

2

<sup>þ</sup> <sup>∇</sup><sup>2</sup>

<sup>=</sup>dy<sup>2</sup> <sup>þ</sup> <sup>k</sup>

Utilizing (52) and setting kzo = 0 due to the assumption of the transverse resonance condition in the free space region of the waveguide will lead to the solution to Hz. Assuming the dominate mode to be TE10 because a ≥ 2b, then kyo = 0 for the first resonance at cutoff [1].

$$\boldsymbol{H}'\_{y\boldsymbol{v}} = -j\omega\varepsilon\_o \varepsilon\_x (\boldsymbol{d}/d\mathbf{x}) \boldsymbol{E}'\_{z\boldsymbol{v}} / \left(k\_o^2 \mu\_y \varepsilon\_x - k\_{z\boldsymbol{v}}^2\right). \tag{69}$$

Solving (66)–(69) for Ez and substituting (1) for ∇ � H formulates the anisotropic wave equation for E where (72) resembles (20)

$$
\nabla \times \nabla \times \underline{\mathbf{E}} = -j\omega\mu\_o\underline{\mu\_r} \cdot (\nabla \times \underline{\mathbf{H}})\_\prime \tag{70}
$$

$$\nabla \times \nabla \times \underline{\mathbf{E}} = -j\omega\mu\_o \underline{\mu\_r} \cdot \left( j\omega\varepsilon\_o \underline{\varepsilon\_r} \cdot \underline{\mathbf{E}} \right) . \tag{71}$$

$$
\nabla \times \underline{\mu\_r}^{-1} \cdot (\nabla \times \underline{E}) = k\_o^2 \underline{\varepsilon\_r} \cdot \underline{E}.\tag{72}
$$

Equation (72) represents the anisotropic wave equation for the time harmonic electric field. Expanding the curl of (72) and isolating the zo component as we did for Hz in Section 4.1.1 yields the following solution for the Ez component

$$\boldsymbol{E}\_{z\boldsymbol{\nu}}^{'} = E\_{\boldsymbol{\nu}} \sin(k\_{\boldsymbol{x}\boldsymbol{\nu}} \boldsymbol{x}) \sin(k\_{\boldsymbol{y}\boldsymbol{\nu}} \boldsymbol{y}) e^{-\vec{\boldsymbol{\mu}}\_{z\boldsymbol{\nu}} z}. \tag{73}$$

Plugging (73) into (66)–(69) yields the following equations for the TM mode vectors in (42) and (43)

$$\boldsymbol{\Xi}\_{\boldsymbol{\nu}}^{'}(\mathbf{x},\boldsymbol{y}) = -j\mathbf{k}\_{z\boldsymbol{\nu}}E\_{o}\left[\underline{\mathbf{x}}\_{\boldsymbol{\nu}}k\_{x\boldsymbol{\nu}}\cos(k\_{x\boldsymbol{\nu}}\mathbf{x})\sin(k\_{y\boldsymbol{\nu}}\boldsymbol{y})/\left(k\_{o}^{2}\boldsymbol{\mu}\_{y}\boldsymbol{\varepsilon}\_{x} - k\_{z\boldsymbol{\nu}}^{2}\right) + \underline{\mathbf{y}}\_{\boldsymbol{\nu}}\sin(k\_{x\boldsymbol{\nu}}\mathbf{x})\cos(k\_{y\boldsymbol{\nu}}\boldsymbol{y})/\left(k\_{o}^{2}\boldsymbol{\mu}\_{y}\boldsymbol{\varepsilon}\_{x} - k\_{z\boldsymbol{\nu}}^{2}\right)\right],\tag{74}$$

$$\underline{\boldsymbol{h}}\_{\nu}^{\prime}(\mathbf{x},\boldsymbol{y}) = j\omega\boldsymbol{v}\_{o}\boldsymbol{E}\_{o}\left[\underline{\boldsymbol{\omega}}\_{o}\boldsymbol{\varepsilon}\_{y}k\_{y\nu}\mathrm{s}\mathrm{s}\mathrm{s}(k\_{x\nu}\boldsymbol{x})\cos\left(k\_{y\nu}\boldsymbol{y}\right)/\left(k\_{o}^{2}\boldsymbol{\mu}\_{x}\boldsymbol{\varepsilon}\_{y} - k\_{z\nu}^{2}\right) - \underline{\boldsymbol{y}}\_{o}\,\boldsymbol{\varepsilon}\_{x}k\_{x\nu}\cos\left(k\_{x\nu}\boldsymbol{x}\right)\mathrm{s}\mathrm{s}\left(k\_{y\nu}\boldsymbol{y}\right)/k\_{o}^{2}\boldsymbol{\mu}\_{y}\,\boldsymbol{\varepsilon}\_{x} - k\_{z\nu}^{2}\right].\tag{75}$$

#### 4.2. Anisotropic transverse resonance

This section describes the derivation of an anisotropic transverse resonance condition established between resonant walls of a rectangular waveguide. Assume an infinite rectangular waveguide partially loaded with an anisotropic medium, then w(z) represents the width of the anisotropic medium at any point z along the direction of propagation as shown in Figure 4.

Figure 4. Symmetrically loaded transmission line model with a short at either end.

At any length z along the waveguide, the assumption that the horizontal distance between two resonant walls represented as a partially filled parallel plate waveguide is a valid presumption. We can calculate Lg(z) as the unknown distance between the edge of the anisotropic medium and the cavity wall based on a transverse resonance condition in the xo-direction. However, we first need to derive the characteristic impedance of the anisotropic region in the transmission line model.

#### 4.2.1. Electromagnetic fields in free space regions

H0

H0

yields the following solution for the Ez component

<sup>υ</sup>ð Þ¼� <sup>x</sup>; <sup>y</sup> jkzυEo xokxυcosð Þ kxυ<sup>x</sup> sin kyυ<sup>y</sup> � �<sup>=</sup> <sup>k</sup><sup>2</sup>

<sup>υ</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>j</sup>ωεoEo xoεykyυsinð Þ kxυ<sup>x</sup> cos kyυ<sup>y</sup> � �<sup>=</sup> <sup>k</sup>

h

h

4.2. Anisotropic transverse resonance

E 0

Figure 4. Symmetrically loaded transmission line model with a short at either end.

tion for E where (72) resembles (20)

178 Antennas and Wave Propagation

(43)

e 0

h 0 <sup>x</sup><sup>υ</sup> ¼ jωεoεyð Þ d=dy E

<sup>y</sup><sup>υ</sup> ¼ �jωεoεxð Þ d=dx E

∇ � μ<sup>r</sup>

0 <sup>z</sup>υ<sup>=</sup> <sup>k</sup><sup>2</sup>

> 0 <sup>z</sup>υ= k 2 <sup>o</sup>μyε<sup>x</sup> � k

Solving (66)–(69) for Ez and substituting (1) for ∇ � H formulates the anisotropic wave equa-

∇ � ∇ � E ¼ �jωμoμ<sup>r</sup> � jωεoε<sup>r</sup> � E

�<sup>1</sup> � ð Þ¼ <sup>∇</sup> � <sup>E</sup> <sup>k</sup><sup>2</sup>

Equation (72) represents the anisotropic wave equation for the time harmonic electric field. Expanding the curl of (72) and isolating the zo component as we did for Hz in Section 4.1.1

<sup>z</sup><sup>υ</sup> <sup>¼</sup> Eosinð Þ kxυ<sup>x</sup> sin kyυ<sup>y</sup> � �<sup>e</sup>

Plugging (73) into (66)–(69) yields the following equations for the TM mode vectors in (42) and

<sup>o</sup>μyε<sup>x</sup> � k 2 zυ

2 <sup>o</sup>μxε<sup>y</sup> � k 2 zυ � �� <sup>y</sup>

� �

This section describes the derivation of an anisotropic transverse resonance condition established between resonant walls of a rectangular waveguide. Assume an infinite rectangular waveguide partially loaded with an anisotropic medium, then w(z) represents the width of the anisotropic medium at any point z along the direction of propagation as shown in Figure 4.

<sup>o</sup>μxε<sup>y</sup> � k

� �

2 zυ

> 2 zυ

∇ � ∇ � E ¼ �jωμoμ<sup>r</sup> � ð Þ ∇ � H , (70)

� �

�jkzυz

sinð Þ kxυ<sup>x</sup> cos kyυ<sup>y</sup> � �<sup>=</sup> <sup>k</sup>

<sup>ε</sup>xkxυcosð Þ kxυ<sup>x</sup> sin kyυ<sup>y</sup> � �=k<sup>2</sup>

<sup>þ</sup>yo

o

� �, (68)

: (69)

, (71)

: (73)

2 <sup>o</sup>μyε<sup>x</sup> � k 2 zυ � �i

> <sup>o</sup>μyε<sup>x</sup> � k 2 zυ i :

,

(74)

(75)

<sup>o</sup>ε<sup>r</sup> � E: (72)

Calculating the fields in the free space region of the waveguide begins with Maxwell's source free Eqs. (1) and (2) and the equations for the individual vector components of the electromagnetic fields (13)–(16). Using the standard derivation of the wave equation for Hz in free space from (1) and (2) shows

$$
\nabla\_T \times \nabla\_T \times \underline{H}\_v = j\omega \varepsilon\_o \left(\nabla\_T \times \underline{E}\_v\right) = \nabla\_T (\nabla\_T \cdot \underline{H}\_v) - \nabla\_T^2 H\_{zv} \tag{76}
$$

$$
\left(j\omega\varepsilon\_o\left(-j\omega\mu\_oH\_{zv}\right)+\nabla\_T^2H\_{zv}=0\right)\tag{77}
$$

$$\left[\left(d^2/d\mathbf{x}^2\right) + \left(d^2/dy^2\right) + k\_o^2\right]H\_{z\nu} = 0.\tag{78}$$

Utilizing (52) and setting kzo = 0 due to the assumption of the transverse resonance condition in the free space region of the waveguide will lead to the solution to Hz. Assuming the dominate mode to be TE10 because a ≥ 2b, then kyo = 0 for the first resonance at cutoff [1]. With kyo = 0 no variation of the fields in the y<sup>o</sup> direction and (d<sup>2</sup> /dy<sup>2</sup> )Hz = 0 then (78) becomes

$$(d^2/d\mathbf{x}^2 + k\_o^2)H\_z = 0.\tag{79}$$

Equation (80) is a standard differential equation with a known solution [17]

$$H\_{z\upsilon} = Ae^{-jk\_{\upsilon}x} + Be^{+jk\_{\upsilon}x},\tag{80}$$

where A and B are yet to be determined coefficients. Substituting (80) into (13)–(16) yields the expression for Ey

$$E\_y = k\_o \omega \mu\_o \left( A e^{-\vec{\mu}\_o \cdot \mathbf{x}} - B e^{+\vec{\mu}\_o \cdot \mathbf{x}} \right) / k\_o^2 = Z\_o \left( A e^{-\vec{\mu}\_o \cdot \mathbf{x}} - B e^{+\vec{\mu}\_o \cdot \mathbf{x}} \right). \tag{81}$$

Accounting for the restrictions imposed by the transverse resonance conditions on Ez, kzo and kyo, then Ex = 0, Hx = 0 and Hy = 0 as well.

#### 4.2.2. Electromagnetic fields in anisotropic region

Starting with (1) and (2) for the source free Maxwell's equations in an anisotropic medium, the vector components (13)–(16) led to the derivation of the dispersion Eqs. (26) and (38) for Hz and Ez, respectively.

The cutoff frequency or resonance of a rectangular waveguide is determined when the propagation constant in the direction of resonance, in the case the xo-direction, is 0 [1]. By definition, when the waveguide's dominant mode υ = 1 propagates, then kx1 > 0 and the guide is resonant whereas when the mode attenuates then kx1 < 0 and there is no resonance. Therefore, the resonance first manifests itself when kx1 = 0. For the dominate mode to be TE10 then a ≥ 2b and d2 Hz/dy2 = 0. Simplifying (28) with these substitutions produces a simpler form to solve for Hz

$$\left[\left(d^2/d\mathbf{x}^2\right) + k\_o^2 \mu\_z \varepsilon\_y\right] H\_z = 0,\tag{82}$$

$$
\beta = k\_o \sqrt{\mu\_z \epsilon\_y}.\tag{83}
$$

E� y � � � x¼�w=2

H� z � �

> Zoj �

�2jBe�jkoð Þ <sup>a</sup>=<sup>2</sup> sin½ �¼ koð Þ <sup>a</sup> � <sup>w</sup> <sup>=</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μz=ε<sup>y</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μz=ε<sup>y</sup>

for Z = �Ey/Hz from (89) and (90)

expression for r

and D.

Z<sup>1</sup> ¼ Zo

¼ Zo

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μz=ε<sup>y</sup> q� � <sup>1</sup> � <sup>r</sup><sup>e</sup>

<sup>j</sup>β1<sup>w</sup> � �=<sup>1</sup> <sup>þ</sup> <sup>r</sup><sup>e</sup>

resonance condition simplifies to Zin = 0 from either direction.

Zo

1 � re

4.2.4. Anisotropic transverse resonance condition

Plugging Eqs. (80) and (81) into (92) and (93) yields the following set of equations

¼ E<sup>þ</sup> y � � � x¼�w=2

> z � �

> > þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μz=ε<sup>y</sup>

q� � Deþjβ1<sup>x</sup> � Ce�jβ1<sup>x</sup> � �<sup>=</sup> Deþjβ1<sup>x</sup> <sup>þ</sup> Ce�jβ1<sup>x</sup> � � � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μz=ε<sup>y</sup>

Equations (95) and (96) give two equations to solve for three unknowns. Match equation (91) to the impedance in the anisotropic region at x = �w/2 to solve for the third unknown. Now solve

q� � <sup>1</sup> � <sup>r</sup>e�j2β1<sup>x</sup> � �<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>r</sup>e�j2β1<sup>x</sup> � � � � ,

where r = C/D. Now apply boundary condition (94) to (91) and (97) in order to yield an

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi εy=μ<sup>z</sup>

Substituting (101) into (97) yields the last equation along with (95) and (96) to solve for B, C

To simplify the calculation, consider Figure 4 as slice of Figure 3 in only one direction that is partially filled with an anisotropic medium. Figure 4 represents a transmission line representation that allows for a solution to Lg in terms of w for a given wavelength. Now use standard transmission line theory to calculate the input impedance Zin at x = 0 from both directions. Transmission line theory says that as we approach the same point in a transmission line from either direction the input impedances should be equal. Then by symmetry the transverse

j j Z<sup>1</sup> ¼ Zo

<sup>j</sup>β1<sup>w</sup> � �=<sup>1</sup> <sup>þ</sup> <sup>r</sup><sup>e</sup>

<sup>j</sup>β1<sup>w</sup> <sup>¼</sup> <sup>j</sup>

r ¼ e

<sup>2</sup>Be�jkoð Þ <sup>a</sup>=<sup>2</sup> cos½ �¼ koð Þ <sup>a</sup> � <sup>w</sup> <sup>=</sup><sup>2</sup> Ce�jβ1ð Þ <sup>w</sup>=<sup>2</sup> � Deþjβ1ð Þ <sup>w</sup>=<sup>2</sup> : (96)

<sup>x</sup>¼�w=<sup>2</sup> <sup>¼</sup> <sup>H</sup><sup>þ</sup>

<sup>x</sup>¼�w=<sup>2</sup> <sup>¼</sup> <sup>Z</sup>1<sup>j</sup>

, (92)

(97)

181

<sup>x</sup>¼�w=<sup>2</sup>, (93)

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123

<sup>x</sup>¼�w=<sup>2</sup>: (94)

q� � Ce�jβ1ð Þ <sup>w</sup>=<sup>2</sup> � Deþjβ1ð Þ <sup>w</sup>=<sup>2</sup> � �, (95)

q� �, (98)

q� �tan½ �¼ koð Þ <sup>a</sup> � <sup>w</sup> <sup>=</sup><sup>2</sup> <sup>j</sup>Ψ, (100)

�jβ1<sup>w</sup>ð Þ <sup>1</sup> � <sup>j</sup><sup>Ψ</sup> <sup>=</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>j</sup><sup>Ψ</sup> : (101)

<sup>j</sup>β1<sup>w</sup> � � <sup>¼</sup> jZotan½ � koð Þ <sup>a</sup> � <sup>w</sup> <sup>=</sup><sup>2</sup> , (99)

Solving (82) for Hz and plugging the result into (13)–(16) yields

$$H\_z = \mathbb{C}e^{-j\mathbb{f}\mathbf{x}} + De^{+j\mathbb{f}\mathbf{x}},\tag{84}$$

$$E\_y = Z\_v \beta \left(\mathbb{C}e^{-j\mathbb{f}\mathbf{x}} - \mathrm{De}^{+j\mathbb{f}\mathbf{x}}\right) / k\_o \varepsilon\_y = Z\_o \sqrt{\mu\_z / \varepsilon\_y} \left(\mathbb{C}e^{-j\mathbb{f}\mathbf{x}} - \mathrm{De}^{+j\mathbb{f}\mathbf{x}}\right),\tag{85}$$

We can see from (13)–(16) that based on our resonance conditions on Ez, kz1 and ky1 that Ex = 0, Hx = 0 and Hy = 0.

#### 4.2.3. Characteristic impedances of the two regions

The first boundary condition exists at the perfect electric conductor (PEC) boundary when x = �a/2 and E(x, y, z)=0

$$\left.E\_y\right|\_{\mathbf{x}=-a/2} = \mathbf{0} \to A e^{\mathbf{j}k\_o a/2} = B e^{-\mathbf{j}k\_o a/2},\tag{86}$$

$$A = B e^{-j\mathbf{k}\_s a}.\tag{87}$$

Plugging (87) into (80) and (81) yields

$$E\_y = Z\_o \text{Re}^{-\text{jk}\_o a/2} \left[ e^{-\text{jk}\_o(x+a/2)} - e^{+\text{jk}\_o(x+a/2)} \right]\_{'} \tag{88}$$

$$E\_y = -2Z\_oBe^{-jk\_o a/2} \sin[k\_o(\chi + a/2)].\tag{89}$$

Similarly,

$$H\_z = 2B e^{-jk\_o a/2} \cos[k\_o(\chi + a/2)].\tag{90}$$

Equations (89) and (90) solve for the impedance of the free space region as Z = �Ey/Hz

$$\mathbf{Z}\_o = -\mathbf{E}\_y/\mathbf{H}\_z = \mathbf{j}\mathbf{Z}\_o \tan[\mathbf{k}\_o(\mathbf{x} + a/2)],\tag{91}$$

within the region 0 ≤ (x + a/2) ≤ (a�w)/2. The second boundary condition exists at x = �w/2 where the tangential fields at the boundary are equal. In this case, there are two tangential fields in Ey and Hz. At the boundary, we have the following three conditions

$$E\_y^-\Big|\_{x=-w/2} = E\_y^+\Big|\_{x=-w/2} \tag{92}$$

$$\left.H\_z^{-}\right|\_{\mathbf{x}=-w/2} = \left.H\_z^{+}\right|\_{\mathbf{x}=-w/2'}\tag{93}$$

$$\left. \mathbf{Z}\_{o} \right|\_{\mathbf{x} = -w/2}^{-} = \mathbf{Z}\_{1}|\_{\mathbf{x} = -w/2}^{+}. \tag{94}$$

Plugging Eqs. (80) and (81) into (92) and (93) yields the following set of equations

$$-2j\mathcal{B}e^{-j\xi\_{v}(a/2)}\sin[k\_{v}(a-w)/2] = \sqrt{\left(\mu\_{z}/\varepsilon\_{\mathcal{Y}}\right)}\left(\mathcal{C}e^{-j\xi\_{1}(w/2)} - \mathcal{D}e^{+j\xi\_{1}(w/2)}\right),\tag{95}$$

$$\text{'} \mathbf{2} \mathbf{B} e^{-\not k\_{\boldsymbol{v}}(\boldsymbol{a}/2)} \cos[k\_{\boldsymbol{v}}(\boldsymbol{a} - \boldsymbol{w})/2] = \mathbf{C} e^{-\not \mathbb{B}\_{1}(\boldsymbol{w}/2)} - \mathbf{D} e^{+\not \mathbb{B}\_{1}(\boldsymbol{w}/2)}.\tag{96}$$

Equations (95) and (96) give two equations to solve for three unknowns. Match equation (91) to the impedance in the anisotropic region at x = �w/2 to solve for the third unknown. Now solve for Z = �Ey/Hz from (89) and (90)

$$\begin{split} Z\_1 &= Z\_o \sqrt{\left(\mu\_z/\varepsilon\_y\right)} \left[ \left( D e^{+j\theta\_1 \mathbf{x}} - C e^{-j\theta\_1 \mathbf{x}} \right) / \left( D e^{+j\theta\_1 \mathbf{x}} + C e^{-j\theta\_1 \mathbf{x}} \right) \right] \\ &= Z\_o \sqrt{\left(\mu\_z/\varepsilon\_y\right)} \left[ \left( 1 - \rho e^{-j2\theta\_1 \mathbf{x}} \right) / \left( 1 + \rho e^{-j2\theta\_1 \mathbf{x}} \right) \right] \end{split} \tag{97}$$
 
$$|Z\_1| = Z\_o \sqrt{\left( \mu\_z/\varepsilon\_y \right)} \tag{98}$$

where r = C/D. Now apply boundary condition (94) to (91) and (97) in order to yield an expression for r

$$\mathcal{Z}\_{\boldsymbol{\theta}}\sqrt{\left(\mu\_{\boldsymbol{z}}/\varepsilon\_{\boldsymbol{y}}\right)}\left[\left(1-\rho e^{j\mathfrak{E}\_{1}\boldsymbol{w}}\right)/1+\rho e^{j\mathfrak{E}\_{1}\boldsymbol{w}}\right]=j\mathcal{Z}\_{\boldsymbol{\theta}}\tan[k\_{\boldsymbol{\theta}}(\boldsymbol{a}-\boldsymbol{w})/2].\tag{99}$$

$$(1 - \rho e^{j\beta\_1 w})/1 + \rho e^{j\beta\_1 w} = j\sqrt{\left(\varepsilon\_y/\mu\_z\right)}\tan[k\_o(a - w)/2] = j\Psi,\tag{100}$$

$$
\rho = e^{-j\beta\_1 w} (1 - j\Psi) / (1 + j\Psi). \tag{101}
$$

Substituting (101) into (97) yields the last equation along with (95) and (96) to solve for B, C and D.

#### 4.2.4. Anisotropic transverse resonance condition

The cutoff frequency or resonance of a rectangular waveguide is determined when the propagation constant in the direction of resonance, in the case the xo-direction, is 0 [1]. By definition, when the waveguide's dominant mode υ = 1 propagates, then kx1 > 0 and the guide is resonant whereas when the mode attenuates then kx1 < 0 and there is no resonance. Therefore, the resonance first manifests itself when kx1 = 0. For the dominate mode to be TE10 then a ≥ 2b and

Hz/dy2 = 0. Simplifying (28) with these substitutions produces a simpler form to solve for Hz

ffiffiffiffiffiffiffiffiffi μzε<sup>y</sup>

> ffiffiffiffiffiffiffiffiffiffiffi μz=ε<sup>y</sup> q

� �Hz <sup>¼</sup> <sup>0</sup>, (82)

p : (83)

, (84)

Ce�jβ<sup>x</sup> � Deþjβ<sup>x</sup> � �, (85)

, (86)

, (88)

: (87)

sin½ � koð Þ x þ a=2 : (89)

cos½ � koð Þ x þ a=2 : (90)

Zo ¼ �Ey=Hz ¼ jZotan½ � koð Þ x þ a=2 , (91)

2 <sup>o</sup>μzε<sup>y</sup>

Hz <sup>¼</sup> Ce�jβ<sup>x</sup> <sup>þ</sup> Deþjβ<sup>x</sup>

We can see from (13)–(16) that based on our resonance conditions on Ez, kz1 and ky1 that Ex = 0,

The first boundary condition exists at the perfect electric conductor (PEC) boundary when

<sup>x</sup>¼�a=<sup>2</sup> <sup>¼</sup> <sup>0</sup> ! Aejkoa=<sup>2</sup> <sup>¼</sup> Be�jko <sup>a</sup>=<sup>2</sup>

�jko ð Þ <sup>x</sup>þa=<sup>2</sup> � <sup>e</sup> <sup>þ</sup>jkoð Þ <sup>x</sup>þa=<sup>2</sup> h i

<sup>A</sup> <sup>¼</sup> Be�jko <sup>a</sup>

β ¼ ko

d2 <sup>=</sup>dx<sup>2</sup> � � <sup>þ</sup> <sup>k</sup>

Solving (82) for Hz and plugging the result into (13)–(16) yields

Ey � �

Ey <sup>¼</sup> ZoBe�jko <sup>a</sup>=<sup>2</sup> <sup>e</sup>

Ey ¼ �2ZoBe�jkoa=<sup>2</sup>

Hz <sup>¼</sup> <sup>2</sup>Be�jkoa=<sup>2</sup>

fields in Ey and Hz. At the boundary, we have the following three conditions

Equations (89) and (90) solve for the impedance of the free space region as Z = �Ey/Hz

within the region 0 ≤ (x + a/2) ≤ (a�w)/2. The second boundary condition exists at x = �w/2 where the tangential fields at the boundary are equal. In this case, there are two tangential

4.2.3. Characteristic impedances of the two regions

Plugging (87) into (80) and (81) yields

Ey <sup>¼</sup> Zo<sup>β</sup> Ce�jβ<sup>x</sup> � Deþjβ<sup>x</sup> � �=koε<sup>y</sup> <sup>¼</sup> Zo

d2

180 Antennas and Wave Propagation

Hx = 0 and Hy = 0.

Similarly,

x = �a/2 and E(x, y, z)=0

To simplify the calculation, consider Figure 4 as slice of Figure 3 in only one direction that is partially filled with an anisotropic medium. Figure 4 represents a transmission line representation that allows for a solution to Lg in terms of w for a given wavelength. Now use standard transmission line theory to calculate the input impedance Zin at x = 0 from both directions. Transmission line theory says that as we approach the same point in a transmission line from either direction the input impedances should be equal. Then by symmetry the transverse resonance condition simplifies to Zin = 0 from either direction.

Starting with the short located at x = a/2, calculate Zin2 at x = w/2 as

$$Z\_{\rm in2} = jZ\_o \tan\left(k\_o L\_\wp\right). \tag{102}$$

5. Conclusions

Author details

Gregory Mitchell

References

106-117

Recently engineered materials have come to play an important role in state of the art designs electromagnetic devices and especially antennas. Many of these engineered materials have inherent anisotropic properties. Anisotropic media yield characteristics such as conformal surfaces, focusing and refraction of electromagnetic waves as they propagate through a material, high impedance surfaces for artificial magnetic conductors as well as high index, low loss, and lightweight ferrite materials. This chapter analyzes the properties of electromagnetic wave propagation in anisotropic media, and presents research including plane wave solutions to propagation in anisotropic media, a mathematical derivation of birefringence in anisotropic media, modal decomposition of rectangular waveguides filled with anisotropic media, and the

Anisotropic Propagation of Electromagnetic Waves http://dx.doi.org/10.5772/intechopen.75123 183

full derivation of anisotropic transverse resonance in a partially loaded waveguide.

[1] Pozar DM. Microwave Engineering. 3rd ed. New York: John Wiley and Sons; 2005. pp.

[2] Ince WJ, Stern E. Mint: Non-reciprocal remanence phase shifters in rectangular waveguide. IEEE Transactions on Microwave Theory and Techniques. 1967;MTT-15(2):87-95

[3] Collin R. Mint: A simple artificial anisotropic medium. IRE Transactions on Microwave

[4] Pozar D. Mint: Radiation and scattering from a microstrip patch on a uniaxial substrate.

[5] Graham J. Arbitrarily Oriented Biaxlly Anisotropic Media: Wave Behvaior and Microstrip

[6] Torrent D, Sanchez-Dehesa J. Radial wave crystals: Radially periodic structures from anisotropic metamaterials for engineering acoustic or electromagnetic waves. Physics

[7] Sanchez-Dehesa J, Torrent D, Carbonell J. Anisotropic metamaterials as sensing devices in acoustics and electromagnetism. In: The Proceedings of the International Society for

Optics and Photonics (SPIE). 2012; San Diego, California. Washington: SPIE

IEEE Transactions on Antennas and Propagation. 1987;AP-35(6):613-621

Antennas [thesis]. Syracuse: University of Syracuse; 2012

Address all correspondence to: gregory.a.mitchell1.civ@mail.mil

U.S. Army Research Laboratory, Adelphi, USA

Theory and Techniques. 1958;6:206-209

Review Letters. 2009;103

Now calculate Zin1 at x = 0 as

$$\mathbf{Z}\_{\dot{m}1} = \mathbf{Z}\_1 \left[ \mathbf{Z}\_{\dot{m}2} + \mathbf{j} \mathbf{Z}\_1 \tan(\beta\_1 \mathbf{w}/2) \right] / \left[ \mathbf{Z}\_1 + \mathbf{j} \mathbf{Z}\_{\dot{m}2} \tan(\beta\_1 \mathbf{w}/2) \right]. \tag{103}$$

The symmetric transverse resonance condition simplifies (103) to

$$\mathbf{Z}\_1 + \mathbf{j}\mathbf{Z}\_{\rm in2}\tan\left(\beta\_1 \mathbf{w}/2\right) = \mathbf{0}.\tag{104}$$

Plugging (98) and (101) into (104) yields the following equation for Lg [14]

$$L\_{\mathcal{Y}} = \lambda \tan^{-1} \left[ \sqrt{\left(\mu\_z/\varepsilon\_{\mathcal{Y}}\right)} / \tan \left(\pi w \sqrt{\mu\_z \varepsilon\_{\mathcal{Y}}} / \lambda\right) \right] / (2\pi). \tag{105}$$

where λ is wavelength. Importantly, the solution of (105) shows that the transverse resonance only depends on two of the six ε<sup>r</sup> and μ<sup>r</sup> components. This means that maintaining a constant resonance in a waveguide or cavity relies on the clever engineering of ε<sup>y</sup> and μ<sup>z</sup> and leaves designers free to adjust the other components as they see fit to enhance performance in other ways. Furthermore, if ε<sup>y</sup> = μ<sup>z</sup> = 1 then the resonance in the xo direction will see the anisotropic substrate as air, while other tensor elements can be utilized to achieve performance attributed to materials with an arbitrarily high refractive index.

#### 4.2.5. Suppression of birefringence in a rectangular waveguide

Section 3.4 discusses the phenomenon of birefringence in an unbounded anisotropic half-space by deriving the existence of a fourth order polynomial for the wavenumber in the propagation direction. However, for low order resonances, a rectangular waveguide suppresses the birefringence inherent to anisotropic media by suppressing propagation in the vertical direction of the waveguide. In other words, ky = 0 and d2 /dy2 = 0 assuming the horizontal dimension of the waveguide is at least twice the size of the vertical dimension or a ≥ 2b in Figure 1 [3]. The bound on the waveguide geometry simplifies (39), the dispersion equation for Ez in an anisotropic waveguide, to

$$k\_o^2 k\_x^2 \varepsilon\_x E\_z / \left(k\_o^2 \mu\_y \varepsilon\_x - k\_z^2\right) - k\_o^2 \varepsilon\_z E\_z = 0,\tag{106}$$

and results in the following second order polynomial for kz

$$k\_z^2 = \varepsilon\_x \left( k\_o^2 \mu\_y - k\_x^2 / \varepsilon\_z \right). \tag{107}$$

The suppression of the ky term in (39) yields a second order differential equation for the wave number in the propagation direction, thereby eliminating the property of birefringence for this case.

## 5. Conclusions

Starting with the short located at x = a/2, calculate Zin2 at x = w/2 as

The symmetric transverse resonance condition simplifies (103) to

Lg <sup>¼</sup> <sup>λ</sup>tan�<sup>1</sup>

to materials with an arbitrarily high refractive index.

the waveguide. In other words, ky = 0 and d2

tropic waveguide, to

case.

4.2.5. Suppression of birefringence in a rectangular waveguide

k2 o k2

and results in the following second order polynomial for kz

<sup>x</sup>εxEz= k

k2 <sup>z</sup> <sup>¼</sup> <sup>ε</sup><sup>x</sup> <sup>k</sup><sup>2</sup>

2 <sup>o</sup>μyε<sup>x</sup> � k

� �

Plugging (98) and (101) into (104) yields the following equation for Lg [14]

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μz=ε<sup>y</sup>

q� �=tan <sup>π</sup><sup>w</sup> ffiffiffiffiffiffiffiffiffi

where λ is wavelength. Importantly, the solution of (105) shows that the transverse resonance only depends on two of the six ε<sup>r</sup> and μ<sup>r</sup> components. This means that maintaining a constant resonance in a waveguide or cavity relies on the clever engineering of ε<sup>y</sup> and μ<sup>z</sup> and leaves designers free to adjust the other components as they see fit to enhance performance in other ways. Furthermore, if ε<sup>y</sup> = μ<sup>z</sup> = 1 then the resonance in the xo direction will see the anisotropic substrate as air, while other tensor elements can be utilized to achieve performance attributed

Section 3.4 discusses the phenomenon of birefringence in an unbounded anisotropic half-space by deriving the existence of a fourth order polynomial for the wavenumber in the propagation direction. However, for low order resonances, a rectangular waveguide suppresses the birefringence inherent to anisotropic media by suppressing propagation in the vertical direction of

waveguide is at least twice the size of the vertical dimension or a ≥ 2b in Figure 1 [3]. The bound on the waveguide geometry simplifies (39), the dispersion equation for Ez in an aniso-

> 2 z

<sup>o</sup>μ<sup>y</sup> � k 2 <sup>x</sup>=ε<sup>z</sup> � �

The suppression of the ky term in (39) yields a second order differential equation for the wave number in the propagation direction, thereby eliminating the property of birefringence for this

� <sup>k</sup><sup>2</sup>

<sup>p</sup> <sup>=</sup><sup>λ</sup> � � � �

Now calculate Zin1 at x = 0 as

182 Antennas and Wave Propagation

Zin<sup>2</sup> ¼ jZotan koLg

Zin<sup>1</sup> <sup>¼</sup> <sup>Z</sup><sup>1</sup> Zin<sup>2</sup> <sup>þ</sup> jZ1tan <sup>β</sup>1w=<sup>2</sup> � � � � <sup>=</sup> <sup>Z</sup><sup>1</sup> <sup>þ</sup> jZin2tan <sup>β</sup>1w=<sup>2</sup> � � � � : (103)

μzε<sup>y</sup>

<sup>Z</sup><sup>1</sup> <sup>þ</sup> jZin2tan <sup>β</sup>1w=<sup>2</sup> � � <sup>¼</sup> <sup>0</sup>: (104)

/dy2 = 0 assuming the horizontal dimension of the

<sup>o</sup>εzEz ¼ 0, (106)

: (107)

� �: (102)

=ð Þ 2π : (105)

Recently engineered materials have come to play an important role in state of the art designs electromagnetic devices and especially antennas. Many of these engineered materials have inherent anisotropic properties. Anisotropic media yield characteristics such as conformal surfaces, focusing and refraction of electromagnetic waves as they propagate through a material, high impedance surfaces for artificial magnetic conductors as well as high index, low loss, and lightweight ferrite materials. This chapter analyzes the properties of electromagnetic wave propagation in anisotropic media, and presents research including plane wave solutions to propagation in anisotropic media, a mathematical derivation of birefringence in anisotropic media, modal decomposition of rectangular waveguides filled with anisotropic media, and the full derivation of anisotropic transverse resonance in a partially loaded waveguide.

## Author details

Gregory Mitchell

Address all correspondence to: gregory.a.mitchell1.civ@mail.mil

U.S. Army Research Laboratory, Adelphi, USA

## References


[8] Ma YG, Wang P, Chen X, Ong CK. Mint: Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial. Applied Physics Letters. 2009;94

**Chapter 9**

Provisional chapter

**Magnetic Line Source Diffraction by a PEMC Step in**

DOI: 10.5772/intechopen.74938

In this chapter, we investigate a magnetic line source diffraction problem concerned with a step. To study the diffraction problem in lossy medium, we follow the Wiener-Hopf technique and steepest decent method to solve it for impedance step. By equating the impedances of the step to zero, the solution reduces for magnetic line source diffraction by PEC step. Then we transform the obtained solution for PEMC step by using duality transformation. Perfect electromagnetic conductor (PEMC) theory is novel idea developed by Lindell and Sihvola. This media is interlinked with two conductors namely perfect electric conductor (PEC) and perfect magnetic conductor (PMC). Both PEC and PMC are the limiting cases of perfect electromagnetic conductor (PEMC). We study the magnetic line source diffraction by PEMC step placed in different soils (i) gravel sand (ii) sand and (iii) clay. By using the permittivity, permeability and conductivity of these lossy mediums we predict the loss effect on the diffracted field. Such kind of study is very useful in antenna and wave propagation for subsurface targets and to investigate antenna

Keywords: Wiener-Hopf technique, Fourier transform, Green function, impedance, diffraction, line source, step, PEMC, PMC, PEC, Lossy medium, permeability,

In this chapter, we have studied the diffraction problem precisely and investigated the magnetic line source diffraction by a perfect electromagnetic conductor (PEMC) step [1–3] for the lossy medium. PEMC step is assumed to be placed in lossy medium. Discontinuity in diffraction theory is relevant to many engineering applications. The physical significance of the step

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Magnetic Line Source Diffraction by a PEMC Step in

**Lossy Medium**

Lossy Medium

Abstract

radiation patterns.

1. Introduction

conductivity, permittivity

Saeed Ahmed and Mona Lisa

Saeed Ahmed and Mona Lisa

http://dx.doi.org/10.5772/intechopen.74938

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter


#### **Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium** Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

DOI: 10.5772/intechopen.74938

Saeed Ahmed and Mona Lisa Saeed Ahmed and Mona Lisa

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74938

#### Abstract

[8] Ma YG, Wang P, Chen X, Ong CK. Mint: Near-field plane-wave-like beam emitting antenna fabricated by anisotropic metamaterial. Applied Physics Letters. 2009;94

[9] Cheng Q. Directive radiation of electromagnetic waves based on anisotropic metamaterials. In: The Proceedings of IEEE Asia-Pacific Conference on Antennas and Propagation. Singa-

[10] Schurig D, Mock JJ, Justice BJ, Cummer SA, Pendry JB, Starr AF, Smith DR. Metamaterial

[11] Wong J, Balmain K. A beam-steerable antenna based on the spatial filtering property of hyperbolically anisotropic metamaterials. In: The Proceedings of IEEE International Symposium of the Antennas and Propagation Society. Honolulu. New York: IEEE; 9–15 June

[12] Cai T, Wang GM. Polarization-controlled bifunctional antenna based on 2-D anisotropic gradient metasurface. In: The Proceedings of IEEE Conference on Microwave and Milli-

[13] Mitchell G, Weiss S. An overview of ARL's low profile antenna work utilizing anisotropic metaferrites. In: The Proceedings of the IEEE International Symposium on Phased Array

[14] Mitchell G, Wasylkiwskyj W. Mint: Theoretical anisotropic resonance technique for the design of low-profile wideband antennas. IET Microwaves, Antennas & Propagation.

[15] Meng FY, Wu Q, Li LW. Mint: Transmission characteristics of wave modes in a rectangular waveguide filled with anisotropic Metamaterial. Applied Physics A: Materials Science

[16] Meng FY, Wu Q, Fu JH. Mint: Miniaturized rectangular cavity resonator based on anisotropic metamaterials bilayer. Microwave and Optical Technology Letters. 2008;50:2016-

[17] Boyce WE, DiPrima RC. Elementary Differential Equations. 7th ed. New York: John Wiley

electromagnetic cloak at microwave frequencies. Science. 2006;314:977-980

meter Wave Technology. Beijing. New York: IEEE; 5–8 June 2016

Systems and Technology. Waltham. New York: IEEE; 18–21 October 2016

pore. New York: IEEE; 27–29 August 2012

2007

184 Antennas and Wave Propagation

2016;10:487-493

and Sons; 2001

2020

and Processing. 2009;94:747-753

In this chapter, we investigate a magnetic line source diffraction problem concerned with a step. To study the diffraction problem in lossy medium, we follow the Wiener-Hopf technique and steepest decent method to solve it for impedance step. By equating the impedances of the step to zero, the solution reduces for magnetic line source diffraction by PEC step. Then we transform the obtained solution for PEMC step by using duality transformation. Perfect electromagnetic conductor (PEMC) theory is novel idea developed by Lindell and Sihvola. This media is interlinked with two conductors namely perfect electric conductor (PEC) and perfect magnetic conductor (PMC). Both PEC and PMC are the limiting cases of perfect electromagnetic conductor (PEMC). We study the magnetic line source diffraction by PEMC step placed in different soils (i) gravel sand (ii) sand and (iii) clay. By using the permittivity, permeability and conductivity of these lossy mediums we predict the loss effect on the diffracted field. Such kind of study is very useful in antenna and wave propagation for subsurface targets and to investigate antenna radiation patterns.

Keywords: Wiener-Hopf technique, Fourier transform, Green function, impedance, diffraction, line source, step, PEMC, PMC, PEC, Lossy medium, permeability, conductivity, permittivity

### 1. Introduction

In this chapter, we have studied the diffraction problem precisely and investigated the magnetic line source diffraction by a perfect electromagnetic conductor (PEMC) step [1–3] for the lossy medium. PEMC step is assumed to be placed in lossy medium. Discontinuity in diffraction theory is relevant to many engineering applications. The physical significance of the step

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

problem regarding engineering application is due to the fact that it is used in many electronic devices such as solder pad which have many applications in them which are interconnected through a step like circuit, microwave oven etc. This configuration is significant for predicting the scattering caused by an abrupt change in the material as well as in the geometrical properties of a surface. This problem is concerned with the diffraction of plane, cylindrical and surface waves by different impedance step discontinuities, such as step discontinuities made of plasmonic materials. Specially diffraction by a step in a perfectly conducting plane makes a canonical problem for the geometrical theory of diffraction (GTD) analysis of scattering by metallic tapes on paneled compact range reflectors [4]. The scattering of surface waves by the junction of two semi-infinite planes joined together by a step was first introduced by Johansen [5] in the case where both the half planes and the step are characterized by the same surface impedances. This problem is solved by using Wiener-Hopf technique, Green function and steepest descent method. The diffracted far field is investigated by the method of steepest descent. Some of the other researchers like Büyükaksoy and Birbir, Büyükaksoy and Tayyar, Büyükaksoy and Tayyar, Aksoy and Alkumru [6–16] have been investigated the scattering problems which can also be considered for the diffraction of plane, cylindrical and surface waves by different impedance step discontinuities, such as step discontinuities made of plasmonic materials.

step height a ¼ λ=4, from which the effects of these parameters on the diffraction phenomenon are studied and compared with the PEMC analytical theory [3]. Next we study the solution magnetic line source diffraction by PEMC step. PEMC is novel metamaterial developed by Lindell and Sihvola [21, 22]. Its constitutive relations and salient features are

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

http://dx.doi.org/10.5772/intechopen.74938

187

ð Þi D ¼ MB

ð Þii H ¼ �ME

PEMC behaves as an example of an ideal boundary. As a check, we obtain the PMC and PEC

M ! 0 ð Þ PMC : n � H ¼ 0, n:D ¼ 0

M ! �∞ð Þ PEC : n � E ¼ 0, n � B ¼ 0:

This medium is characterized by a scalar parameter M known as admittance of the surface. PEMC is a generalization of both perfect electric conductor (PEC) and perfect magnetic conductor (PMC) media. Therefore, the medium is known as PEMC. Defining a certain class of duality transformations, this medium corresponds to PEC or PMC media. PEMC medium allows some nonzero fields, it rejects electromagnetic field propagation and acts as a boundary to electromagnetic waves just like the PEC and PMC media. Denoting the unit normal between air and PEMC by, from the continuity of tangential component, the electric field E and the

H þ ME ¼ 0

It is also continuous through the PEMC air interface, because this vanishes in the PEMC

n � ð Þ¼ H þ ME 0

D � MB ¼ 0

n � ð Þ¼ D � MB 0:

Because the normal component of the Poynting vector at the PEMC boundary vanishes and is nonreciprocal, except in the PMC and PEC limiting cases M ¼ 0, � ∞ respectively. PEMC

boundary conditions as the two limiting case of PEMC:

magnetic field H satisfy the equation

medium, and the boundary condition becomes

and is continuous across the boundary for

Similarly, the normal component of the field satisfies

given below:

and

and

The importance of present work stems from the facts that: (a) the scattering properties of a surface are functions of both its geometrical and material properties. (b) The edge scattering by dihedral structures whose surfaces can be modeled by the impedance boundary condition has been the focus of attention of many scientists for both acoustic and electromagnetic waves [17]. (c) The step geometry constitutes a canonical problem for scattering because a step geometry is used as an interconnection circuit of many electronic devices such as solder pad, microwave oven and frequency selective surface etc. [18]. A diffraction problem due to a magnetic line source is considered as better substitute than the plane waves. It is pertinent to mention here that the problem of diffraction of plane or line source diffraction of electromagnetic waves from a step is both mathematically difficult and physically important because the solution of the problem involves determination of n unknowns which in turn requires the solution of system of n linear equations.

It is clear that in the case of the line source incidence, the results of plane wave diffraction

by impedance step are modified by a multiplicative factor of the form <sup>2</sup><sup>π</sup> kr<sup>0</sup> <sup>1</sup> 2 exp ikr<sup>0</sup> <sup>þ</sup> <sup>i</sup> <sup>π</sup> 4 , which agrees with the results already known [19, 20]. In far zone, the solution of magnetic line source diffraction reduces to plane wave as kr ! ∞. Using the method described in [16], the concerned problem "magnetic line source diffraction by an impedance step" is first reduced to a modified Wiener-Hopf equation of the second kind whose solution contains infinitely many constants satisfying an infinite system of linear algebraic equations. From this far field solution we obtain analytical solution for magnetic line source diffraction by perfect electric conductor (PEC) step, by taking the surface impedances η<sup>1</sup> and η<sup>2</sup> equal to zero. Next, we apply the duality transformation introduced by Lindell and Sihvola. Transformations have been made from the diffracted fields by a PEC step plane to PEMC step. Numerical solution of this system is obtained for various values of parameter M with step height a ¼ λ=4, from which the effects of these parameters on the diffraction phenomenon are studied and compared with the PEMC analytical theory [3]. Next we study the solution magnetic line source diffraction by PEMC step. PEMC is novel metamaterial developed by Lindell and Sihvola [21, 22]. Its constitutive relations and salient features are given below:

$$(i) \quad \mathbf{D} = M\mathbf{B}$$

and

problem regarding engineering application is due to the fact that it is used in many electronic devices such as solder pad which have many applications in them which are interconnected through a step like circuit, microwave oven etc. This configuration is significant for predicting the scattering caused by an abrupt change in the material as well as in the geometrical properties of a surface. This problem is concerned with the diffraction of plane, cylindrical and surface waves by different impedance step discontinuities, such as step discontinuities made of plasmonic materials. Specially diffraction by a step in a perfectly conducting plane makes a canonical problem for the geometrical theory of diffraction (GTD) analysis of scattering by metallic tapes on paneled compact range reflectors [4]. The scattering of surface waves by the junction of two semi-infinite planes joined together by a step was first introduced by Johansen [5] in the case where both the half planes and the step are characterized by the same surface impedances. This problem is solved by using Wiener-Hopf technique, Green function and steepest descent method. The diffracted far field is investigated by the method of steepest descent. Some of the other researchers like Büyükaksoy and Birbir, Büyükaksoy and Tayyar, Büyükaksoy and Tayyar, Aksoy and Alkumru [6–16] have been investigated the scattering problems which can also be considered for the diffraction of plane, cylindrical and surface waves by different impedance step discontinuities, such as step discontinuities made of

The importance of present work stems from the facts that: (a) the scattering properties of a surface are functions of both its geometrical and material properties. (b) The edge scattering by dihedral structures whose surfaces can be modeled by the impedance boundary condition has been the focus of attention of many scientists for both acoustic and electromagnetic waves [17]. (c) The step geometry constitutes a canonical problem for scattering because a step geometry is used as an interconnection circuit of many electronic devices such as solder pad, microwave oven and frequency selective surface etc. [18]. A diffraction problem due to a magnetic line source is considered as better substitute than the plane waves. It is pertinent to mention here that the problem of diffraction of plane or line source diffraction of electromagnetic waves from a step is both mathematically difficult and physically important because the solution of the problem involves determination of n unknowns which in turn requires the solution of

It is clear that in the case of the line source incidence, the results of plane wave diffraction

which agrees with the results already known [19, 20]. In far zone, the solution of magnetic line source diffraction reduces to plane wave as kr ! ∞. Using the method described in [16], the concerned problem "magnetic line source diffraction by an impedance step" is first reduced to a modified Wiener-Hopf equation of the second kind whose solution contains infinitely many constants satisfying an infinite system of linear algebraic equations. From this far field solution we obtain analytical solution for magnetic line source diffraction by perfect electric conductor (PEC) step, by taking the surface impedances η<sup>1</sup> and η<sup>2</sup> equal to zero. Next, we apply the duality transformation introduced by Lindell and Sihvola. Transformations have been made from the diffracted fields by a PEC step plane to PEMC step. Numerical solution of this system is obtained for various values of parameter M with

kr<sup>0</sup> <sup>1</sup> 2

exp ikr<sup>0</sup> <sup>þ</sup> <sup>i</sup> <sup>π</sup>

4 ,

by impedance step are modified by a multiplicative factor of the form <sup>2</sup><sup>π</sup>

plasmonic materials.

186 Antennas and Wave Propagation

system of n linear equations.

ð Þii H ¼ �ME

PEMC behaves as an example of an ideal boundary. As a check, we obtain the PMC and PEC boundary conditions as the two limiting case of PEMC:

$$M \rightarrow 0 \ (PMC): \mathbf{n} \times \mathbf{H} = 0, \ \mathbf{n}. \mathbf{D} = 0$$

and

$$M \rightarrow \pm \approx (PEC) : \mathbf{n} \times \mathbf{E} = \mathbf{0}, \ \mathbf{n} \cdot B = \mathbf{0}.$$

This medium is characterized by a scalar parameter M known as admittance of the surface. PEMC is a generalization of both perfect electric conductor (PEC) and perfect magnetic conductor (PMC) media. Therefore, the medium is known as PEMC. Defining a certain class of duality transformations, this medium corresponds to PEC or PMC media. PEMC medium allows some nonzero fields, it rejects electromagnetic field propagation and acts as a boundary to electromagnetic waves just like the PEC and PMC media. Denoting the unit normal between air and PEMC by, from the continuity of tangential component, the electric field E and the magnetic field H satisfy the equation

$$\mathbf{H} + \mathbf{M}\mathbf{E} = 0$$

It is also continuous through the PEMC air interface, because this vanishes in the PEMC medium, and the boundary condition becomes

$$\mathbf{n} \times (\mathbf{H} + \mathbf{M}\mathbf{E}) = 0$$

Similarly, the normal component of the field satisfies

$$\mathbf{D} - \mathbf{M}\mathbf{B} = 0$$

and is continuous across the boundary for

$$\mathbf{n} \cdot (\mathbf{D} - M\mathbf{B}) = 0.$$

Because the normal component of the Poynting vector at the PEMC boundary vanishes and is nonreciprocal, except in the PMC and PEC limiting cases M ¼ 0, � ∞ respectively. PEMC is truly isotropic, but due to the cross-components in addition to the co-components in the scattered field, it is bi-isotropic. Due to its particular property of short-circuiting a linear combination of the tangential electric and magnetic fields, the PEMC media can be exploited in microwave engineering applications. Some examples of such are, e.g., ground planes for low-profile antennas, field pattern purifiers for aperture antennas, polarization transformers, radar reflectors, and generalized high-impedance surfaces. PEMC medium can be artificially constructed by building a structure which forces the boundary conditions on the surface of a sample to be the same as those of PEMC. Many authors have worked on PEMC and metamaterial [21–47]. Next we will study the magnetic line diffraction by PEMC step in lossy medium and the fields are obtained analytically for more general solution.

2. Mathematical model

The time dependence e�iω<sup>t</sup>

with the horizontal.

3. Boundary conditions

Consider the diffraction due to a magnetic line source located at x0; y<sup>0</sup>

ance η2: The geometry of the line source diffraction is shown in Figure 1.

half planes S<sup>1</sup> ¼ fx < 0, y ¼ a, z∈ ð Þg �∞; ∞ and S<sup>2</sup> ¼ fx > 0, y ¼ 0, z∈ ð Þg �∞; ∞ with relative surface impedance η<sup>1</sup> joined together by a step of height }a} with relative surface imped-

, is suppressed throughout the solution.

Figure 1. Geometry of problem: a line source located at (x0, y0) making an angle θ<sup>0</sup> with the horizontal, is incident upon impedance step of surface impedances η<sup>1</sup> and η2, respectively, as shown. Here, (x, y) is the observation point at an angle θ

<sup>u</sup><sup>T</sup>ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>δ</sup>ð Þ <sup>x</sup> � <sup>x</sup><sup>0</sup> <sup>δ</sup> <sup>y</sup> � <sup>y</sup><sup>0</sup>

The Helmholtz equation concerned with the diffraction problem is given below

subject to the boundary conditions at two half planes and a step given by:

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup> ik ∂ ∂y

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> ik ∂ ∂x

∂2 ∂x<sup>2</sup> þ

∂2 <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2 , illuminated by two

189

http://dx.doi.org/10.5772/intechopen.74938

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

, (1)

uTð Þ¼ <sup>x</sup>; <sup>a</sup> <sup>0</sup>, x <sup>&</sup>lt; <sup>0</sup> (2)

u<sup>T</sup>ð Þ¼ <sup>0</sup>; <sup>y</sup> <sup>0</sup>, <sup>0</sup> <sup>&</sup>lt; <sup>y</sup> <sup>&</sup>lt; <sup>a</sup> (3)

We extend the problem reported by [3] for the lossy background medium. Several canonical objects in lossy media has been investigated over the years by many authors [44–56] by applying approximate values of electric conductivity and dielectric constants of various materials. The concept of subsurface scattering of EM waves for detecting the cavities and targets buried in soil has important applications in the areas of nonproliferation of weapons, environmental monitoring, hazardous-waste site location and assessment, and even archeology. To have information about this potential, it is first essential to understand the behavior of the soil by applying EM wave, and how the targets within the soil give response. We analyze the response of the soil to an EM wave by using complex dielectric permittivity of the soil in finding radar range resolution. This leads to a concept of an optimum frequency and bandwidth for imaging in a particular soil. The radar cross section of several canonical objects in lossy media is derived, and examples are given for several objects like scattering of buried PEC sphere, PEC cylinder, and PEC plate [44] and similarly scattering by PEMC plate [54], PEMC strip [55] and PEMC cylinder [56]. Furthermore, we can study the diffraction by PEC and PEMC half plane [39] and step is also made by semiinfinite half planes with a step height a, so they can also be investigated for diffraction by using the electric parameters of soils. Also characteristics of radar cross section can be further studied with different objects for PEC, PMC and PEMC cases in lossy medium. In addition to RCS of various PEC, PMC and PEMC objects [59] in lossy medium can also be investigated in future.

The objective of this chapter is to determine the diffracted field by PEMC step excited by a line source in lossy medium and to investigate surface and borehole techniques for detecting and mapping subsurface cavities, targets and to evaluate the results of surface and borehole radar probings performed at the test sites. Detection of subsurface cavities is concerned with ground-probing radar. A number of factors that control the velocity, absorption and attenuation characteristics of a radar wave and plane EM wave propagating through a dielectric as well as lossy medium like the earth. The imaging of objects buried in soil has potentially valuable applications in many diverse areas, such as nonproliferation of weapons, environmental monitoring, hazardous-waste site location and assessment, and even archeology. We study the magnetic line source diffraction by PEMC step placed in different soils (i) gravel sand (ii) sand and (iii) clay. By using the approximate value of permittivity, permeability and conductivity of these lossy mediums, we predict and analyze the loss effect on the diffracted field.

## 2. Mathematical model

is truly isotropic, but due to the cross-components in addition to the co-components in the scattered field, it is bi-isotropic. Due to its particular property of short-circuiting a linear combination of the tangential electric and magnetic fields, the PEMC media can be exploited in microwave engineering applications. Some examples of such are, e.g., ground planes for low-profile antennas, field pattern purifiers for aperture antennas, polarization transformers, radar reflectors, and generalized high-impedance surfaces. PEMC medium can be artificially constructed by building a structure which forces the boundary conditions on the surface of a sample to be the same as those of PEMC. Many authors have worked on PEMC and metamaterial [21–47]. Next we will study the magnetic line diffraction by PEMC step in lossy medium and the fields are obtained analytically for more general

We extend the problem reported by [3] for the lossy background medium. Several canonical objects in lossy media has been investigated over the years by many authors [44–56] by applying approximate values of electric conductivity and dielectric constants of various materials. The concept of subsurface scattering of EM waves for detecting the cavities and targets buried in soil has important applications in the areas of nonproliferation of weapons, environmental monitoring, hazardous-waste site location and assessment, and even archeology. To have information about this potential, it is first essential to understand the behavior of the soil by applying EM wave, and how the targets within the soil give response. We analyze the response of the soil to an EM wave by using complex dielectric permittivity of the soil in finding radar range resolution. This leads to a concept of an optimum frequency and bandwidth for imaging in a particular soil. The radar cross section of several canonical objects in lossy media is derived, and examples are given for several objects like scattering of buried PEC sphere, PEC cylinder, and PEC plate [44] and similarly scattering by PEMC plate [54], PEMC strip [55] and PEMC cylinder [56]. Furthermore, we can study the diffraction by PEC and PEMC half plane [39] and step is also made by semiinfinite half planes with a step height a, so they can also be investigated for diffraction by using the electric parameters of soils. Also characteristics of radar cross section can be further studied with different objects for PEC, PMC and PEMC cases in lossy medium. In addition to RCS of various PEC, PMC and PEMC objects [59] in lossy medium can also be

The objective of this chapter is to determine the diffracted field by PEMC step excited by a line source in lossy medium and to investigate surface and borehole techniques for detecting and mapping subsurface cavities, targets and to evaluate the results of surface and borehole radar probings performed at the test sites. Detection of subsurface cavities is concerned with ground-probing radar. A number of factors that control the velocity, absorption and attenuation characteristics of a radar wave and plane EM wave propagating through a dielectric as well as lossy medium like the earth. The imaging of objects buried in soil has potentially valuable applications in many diverse areas, such as nonproliferation of weapons, environmental monitoring, hazardous-waste site location and assessment, and even archeology. We study the magnetic line source diffraction by PEMC step placed in different soils (i) gravel sand (ii) sand and (iii) clay. By using the approximate value of permittivity, permeability and conductivity of these lossy mediums, we predict and analyze the loss effect on the diffracted

solution.

188 Antennas and Wave Propagation

investigated in future.

field.

Consider the diffraction due to a magnetic line source located at x0; y<sup>0</sup> , illuminated by two half planes S<sup>1</sup> ¼ fx < 0, y ¼ a, z∈ ð Þg �∞; ∞ and S<sup>2</sup> ¼ fx > 0, y ¼ 0, z∈ ð Þg �∞; ∞ with relative surface impedance η<sup>1</sup> joined together by a step of height }a} with relative surface impedance η2: The geometry of the line source diffraction is shown in Figure 1.

Figure 1. Geometry of problem: a line source located at (x0, y0) making an angle θ<sup>0</sup> with the horizontal, is incident upon impedance step of surface impedances η<sup>1</sup> and η2, respectively, as shown. Here, (x, y) is the observation point at an angle θ with the horizontal.

The time dependence e�iω<sup>t</sup> , is suppressed throughout the solution.

The Helmholtz equation concerned with the diffraction problem is given below

$$
\left(\frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial y^2} + k^2\right) u^T(\mathbf{x}, y) = \delta(\mathbf{x} - \mathbf{x}\_0) \delta(y - y\_0), \tag{1}
$$

subject to the boundary conditions at two half planes and a step given by:

## 3. Boundary conditions

$$\left[1+\frac{\eta\_1}{i\hbar}\frac{\partial}{\partial y}\right]u^T(\mathbf{x},a)=0,\ \mathbf{x}<0\tag{2}$$

$$
\mu \left[ 1 + \frac{\eta\_2}{ik} \frac{\partial}{\partial x} \right] \mu^T(0, y) = 0, \ 0 < y < a \tag{3}
$$

and

$$\left[1+\frac{\eta\_1}{ik}\frac{\partial}{\partial y}\right]u^T(\mathbf{x},0)=0,\ \mathbf{x}>0\tag{4}$$

with continuity equations:

$$
\mu^T(\mathbf{x}, \mathbf{a}^-) = \mu^T(\mathbf{x}, \mathbf{a}^+) \tag{5}
$$

∂2 ∂x<sup>2</sup> þ

> ∂2 ∂x<sup>2</sup> þ

and

and

where

and

where

4. Fourier transform

∂2 <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2

> ∂2 <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2

> > <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup> ik ∂ ∂y

> > <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup> ik ∂ ∂y

η<sup>1</sup> sin ϕ<sup>0</sup> þ 1

η<sup>1</sup> sin ϕ<sup>0</sup> þ 1

<sup>b</sup> ¼ � <sup>1</sup> 4i

Φð Þ¼ α; y

u1ð Þ¼ x; y

e

e

r

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> ð∞ �∞

ð∞ �∞ u1ð Þ x; y e

Φð Þ α; y e

iαx dx,

�iαx dα,

ffiffiffiffiffiffiffiffiffi 2 πkr<sup>0</sup>

e i kr ð Þ <sup>0</sup>�π=4

<sup>u</sup><sup>1</sup> <sup>x</sup>; <sup>a</sup><sup>þ</sup> ð Þþ <sup>2</sup>bη<sup>1</sup> sin <sup>ϕ</sup><sup>0</sup>

<sup>∂</sup><sup>y</sup> � <sup>2</sup>bikη<sup>1</sup> sin <sup>ϕ</sup><sup>0</sup>

∂u<sup>1</sup> x; a<sup>þ</sup> ð Þ

Taking Fourier transform of the Eq. (10) such that:

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> ik ∂ ∂x

� �u1ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>0</sup>, x∈ð Þ �∞; <sup>∞</sup> : (10)

� �u2ð Þ¼ <sup>x</sup>; <sup>y</sup> <sup>0</sup>, x<sup>∈</sup> ð Þ <sup>0</sup>; <sup>∞</sup> : (11)

� �u1ð Þ¼ <sup>x</sup>; <sup>a</sup> <sup>0</sup>, x <sup>&</sup>lt; <sup>0</sup> (12)

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

http://dx.doi.org/10.5772/intechopen.74938

191

� �u2ð Þ¼ <sup>0</sup>; <sup>y</sup> <sup>0</sup>, <sup>0</sup> <sup>&</sup>lt; <sup>y</sup> <sup>&</sup>lt; <sup>a</sup> (13)

� �u2ð Þ¼ <sup>x</sup>; <sup>0</sup> <sup>0</sup>, x <sup>&</sup>gt; <sup>0</sup> (14)

�ik x cos <sup>ϕ</sup>0þ<sup>a</sup> sin <sup>ϕ</sup><sup>0</sup> ½ � <sup>¼</sup> <sup>∂</sup>u<sup>2</sup> <sup>x</sup>; <sup>a</sup>� ð Þ

�ik x cos <sup>ϕ</sup>0þ<sup>a</sup> sin <sup>ϕ</sup><sup>0</sup> ½ � <sup>¼</sup> <sup>u</sup><sup>2</sup> <sup>x</sup>; <sup>a</sup>� ð Þ (15)

<sup>∂</sup><sup>y</sup> (16)

and

$$\frac{\partial u^T(\mathbf{x}, a^-)}{\partial y} = \frac{\partial u^T(\mathbf{x}, a^+)}{\partial y} \tag{6}$$

where uT is the total field. For the mathematical analysis purpose, it is easy to express the total field uTð Þ <sup>x</sup>; <sup>y</sup> as follows:

$$u^T(\mathbf{x}, y) = \begin{cases} u\_i(\mathbf{x}, y) + u\_1^r(\mathbf{x}, y) + u\_1(\mathbf{x}, y), & y > a, \\ u\_2(\mathbf{x}, y), 0 < y < a, & \end{cases} \tag{7}$$

Here, k ¼ ω=c is the wave number and supposed to have positive imaginary part. The lossless case can be obtained by making Imk ! 0 in the final expressions. By substituting Eq. (7) in Eqs. (1)–(6), we arrive at

$$
\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + k^2\right) u^i(x, y) = \delta(x - x\_0) \delta(y - y\_0). \tag{8}
$$

and

$$
\left(\frac{\partial^2}{\partial \mathbf{x}^2} + \frac{\partial^2}{\partial y^2} + k^2\right) u\_1'(\mathbf{x}, y) = \delta(\mathbf{x} - \mathbf{x}\_0) \delta(y + y\_0). \tag{9}
$$

The solution of the incident field and reflected field from [11] can be written as

$$u^i(\mathbf{x}, \mathbf{y}) = b e^{-ik\left[\mathbf{x}\cos\phi\_0 + \mathbf{y}\sin\theta\_0\right]}$$

$$u\_1^r(\mathbf{x}, \mathbf{y}) = b \frac{\eta\_1 \sin\phi\_0 - 1}{\eta\_1 \sin\theta\_0 + 1} e^{-ik\left[\mathbf{x}\cos\phi\_0 + (y - 2a)\sin\phi\_0\right]}$$

where

$$b = -\frac{1}{4i} \sqrt{\frac{2}{\pi k r\_0}} e^{i(kr\_0 - \pi/4)}$$

The diffracted field u1ð Þ x; y and u2ð Þ x; y satisfy the Helmholtz equations

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium http://dx.doi.org/10.5772/intechopen.74938 191

$$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + k^2\right) u\_1(x, y) = 0,\qquad \qquad \qquad \mathfrak{x} \in (-\infty, \infty). \tag{10}$$

$$
\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + k^2\right) u\_2(x, y) = 0,\qquad \qquad \mathbf{x} \in (0, \infty). \tag{11}
$$

$$\left[1+\frac{\eta\_1}{ik}\frac{\partial}{\partial y}\right]\mu\_1(\mathbf{x},a)=0,\ \mathbf{x}<0\tag{12}$$

$$
\mu \left[ 1 + \frac{\eta\_2}{ik} \frac{\partial}{\partial x} \right] \mu\_2(0, y) = 0, \ 0 < y < a \tag{13}
$$

and

and

and

and

where

with continuity equations:

190 Antennas and Wave Propagation

field uTð Þ <sup>x</sup>; <sup>y</sup> as follows:

Eqs. (1)–(6), we arrive at

uTð Þ¼ <sup>x</sup>; <sup>y</sup>

∂2 ∂x<sup>2</sup> þ

∂2 ∂x<sup>2</sup> þ

ur

�

∂2 <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>k</sup><sup>2</sup> � �

> ∂2 <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2

> > ui

<sup>1</sup>ð Þ¼ x; y b

� �

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup> ik ∂ ∂y � �

<sup>∂</sup>uT <sup>x</sup>; <sup>a</sup>� ð Þ

uið Þþ <sup>x</sup>; <sup>y</sup> ur

u2ð Þ x; y , 0 < y < a,

ui

ur

η<sup>1</sup> sin ϕ<sup>0</sup> � 1 <sup>η</sup><sup>1</sup> sin <sup>θ</sup><sup>0</sup> <sup>þ</sup> <sup>1</sup> <sup>e</sup>

ð Þ¼ <sup>x</sup>; <sup>y</sup> be�ik x½ � cos <sup>ϕ</sup>0þ<sup>y</sup> sin <sup>θ</sup><sup>0</sup>

ffiffiffiffiffiffiffiffiffi 2 πkr<sup>0</sup>

e i kr ð Þ <sup>0</sup>�π=4

r

The solution of the incident field and reflected field from [11] can be written as

<sup>b</sup> ¼ � <sup>1</sup> 4i

The diffracted field u1ð Þ x; y and u2ð Þ x; y satisfy the Helmholtz equations

<sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>∂</sup>u<sup>T</sup> <sup>x</sup>; <sup>a</sup><sup>þ</sup> ð Þ

<sup>1</sup>ð Þþ x; y u1ð Þ x; y , y > a,

ð Þ¼ x; y δð Þ x � x<sup>0</sup> δ y � y<sup>0</sup>

<sup>1</sup>ð Þ¼ x; y δð Þ x � x<sup>0</sup> δ y þ y<sup>0</sup>

�ik x cos <sup>ϕ</sup>0þð Þ <sup>y</sup>�2<sup>a</sup> sin <sup>ϕ</sup><sup>0</sup> ½ �

where uT is the total field. For the mathematical analysis purpose, it is easy to express the total

Here, k ¼ ω=c is the wave number and supposed to have positive imaginary part. The lossless case can be obtained by making Imk ! 0 in the final expressions. By substituting Eq. (7) in

uTð Þ¼ <sup>x</sup>; <sup>0</sup> <sup>0</sup>, x <sup>&</sup>gt; <sup>0</sup> (4)

<sup>∂</sup><sup>y</sup> (6)

� �: (8)

� �: (9)

(7)

uT <sup>x</sup>; <sup>a</sup>� ð Þ¼ uT <sup>x</sup>; <sup>a</sup><sup>þ</sup> ð Þ (5)

$$
\mu \left[ 1 + \frac{\eta\_1}{ik} \frac{\partial}{\partial y} \right] \mu\_2(\mathbf{x}, 0) = 0, \quad \mathbf{x} > 0 \tag{14}
$$

$$\mu\_1(\mathbf{x}, \mathbf{a}^+) + \frac{2b\eta\_1 \sin \phi\_0}{\eta\_1 \sin \phi\_0 + 1} e^{-ik[\mathbf{x} \cos \phi\_0 + a \sin \phi\_0]} = \mu\_2(\mathbf{x}, \mathbf{a}^-) \tag{15}$$

and

$$\frac{\partial u\_1(\mathbf{x}, a^+)}{\partial y} - \frac{2bi k \eta\_1 \sin \phi\_0}{\eta\_1 \sin \phi\_0 + 1} e^{-ik[x \cos \phi\_0 + a \sin \phi\_0]} = \frac{\partial u\_2(\mathbf{x}, a^-)}{\partial y} \tag{16}$$

where

$$b = -\frac{1}{4i} \sqrt{\frac{2}{\pi k r\_0}} e^{i(kr\_0 - \pi/4)}$$

#### 4. Fourier transform

Taking Fourier transform of the Eq. (10) such that:

$$\Phi(\alpha, y) = \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{\infty} u\_1(\mathbf{x}, y) e^{i\alpha \mathbf{x}} d\mathbf{x} \dots$$

and

$$\mu\_1(\alpha, y) = \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{\infty} \Phi(\alpha, y) e^{-i\alpha x} d\alpha \dots$$

$$\Phi(\alpha, y) = \Phi\_-(\alpha, y) + \Phi\_+(\alpha, y)$$

$$\Phi\_+(\alpha, y) = \frac{1}{\sqrt{2\pi}} \int\_0^\infty u\_1(\mathbf{x}, y) e^{i\alpha \mathbf{x}} d\mathbf{x},$$

$$\Phi\_-(\alpha, y) = \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^0 u\_1(\mathbf{x}, y) e^{i\alpha \mathbf{x}} d\mathbf{x},$$

and Eq. (10) reduce to

$$\frac{d^2\phi}{dy^2} + \gamma^2 \phi(\alpha, y) = 0,\tag{17}$$

<sup>H</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> 8π<sup>2</sup>

expression <sup>Φ</sup>þð Þþ <sup>α</sup>; <sup>y</sup> <sup>Φ</sup>�ð Þ <sup>α</sup>; <sup>y</sup> and <sup>∂</sup>Φþð Þ <sup>α</sup>;<sup>y</sup>

where

where

where prime denotes differentiation with respect to y:

∂Ψþð Þ x; a

¼ F<sup>0</sup> iη1 k

> þ k η1

þ X∞ n¼1

k η2 � T

k η2 þ T � � sin ka

k η2 � α<sup>n</sup>

k η2 þ α<sup>n</sup>

2η<sup>1</sup> sin ϕ<sup>0</sup> η<sup>1</sup> sin ϕ<sup>0</sup> þ 1

The solution of Eq. (17) satisfying the radiation condition for y > a can be written as:

Φð Þ¼ α; y Bð Þ α e

e

where Bð Þ α is the unknown coefficient to be determined by substituting y ¼ a in the following

ik <sup>Φ</sup><sup>0</sup>

<sup>∂</sup><sup>y</sup> <sup>þ</sup> <sup>∂</sup>Φ�ð Þ <sup>α</sup>;<sup>y</sup>

<sup>R</sup>þð Þ¼ <sup>α</sup> <sup>B</sup>ð Þ <sup>α</sup> <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup>

<sup>R</sup>þð Þ¼ <sup>α</sup> <sup>Φ</sup>þð Þþ <sup>α</sup>; <sup>a</sup> <sup>η</sup><sup>1</sup>

<sup>∂</sup><sup>y</sup> <sup>¼</sup> <sup>R</sup>þð Þ <sup>α</sup> <sup>i</sup>γ αð Þ <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup>

From the Eqs (22), (23), (26) and (27), we obtain the following Wiener-Hopf functional equations

<sup>k</sup> γ αð Þþ ik η1

The corrected solution of Wiener-Hopf equation [2] in case of line source can be expressed as

Rþð Þ α ð Þ α þ T Gþð Þ α

� �Gþð Þ <sup>T</sup> <sup>R</sup>þð Þ <sup>T</sup> <sup>e</sup>

� �Gþð Þ <sup>α</sup><sup>n</sup> <sup>R</sup>þð Þ <sup>α</sup><sup>n</sup> <sup>n</sup><sup>π</sup>

� �aαnð Þ <sup>T</sup> � <sup>α</sup><sup>n</sup> ð Þ <sup>α</sup> <sup>þ</sup> <sup>α</sup><sup>n</sup>

ð Þ k cos θ<sup>0</sup> � T G�ð Þ k cos θ<sup>0</sup> α � k cos θ<sup>0</sup>

> η1 � �ð Þ <sup>α</sup> <sup>þ</sup> <sup>T</sup>

ik <sup>Φ</sup><sup>0</sup>

<sup>Φ</sup>�ð Þþ <sup>α</sup>; <sup>a</sup> ik

η1

�ika η1

a � �<sup>2</sup>

,

F0 α�k cos ϕ<sup>0</sup> � (28)

<sup>Φ</sup>�ð Þþ <sup>α</sup>; <sup>a</sup> <sup>η</sup><sup>1</sup>

�ika sin <sup>ϕ</sup><sup>0</sup> <sup>2</sup><sup>π</sup>

ffiffiffiffiffiffi kr<sup>0</sup> p e

i kr ð Þ <sup>0</sup>�π=<sup>4</sup> :

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

<sup>i</sup>γ αð Þj j <sup>y</sup>�<sup>a</sup> , (24)

http://dx.doi.org/10.5772/intechopen.74938

193

<sup>∂</sup><sup>y</sup> , and with the help of boundary condition;

�ð Þ¼ <sup>α</sup>; <sup>a</sup> <sup>0</sup>, (25)

<sup>k</sup> γ αð Þ � �, (26)

þð Þ <sup>α</sup>; <sup>a</sup> : (27)

where γ αð Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k <sup>2</sup> � <sup>α</sup><sup>2</sup> p and α is a complex transform variable. Apply half range Fourier transforms to the Eq. (11)

$$
\left[\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + k^2\right] \Psi\_+(a, y) = 0, \ 0 < y < a. \tag{18}
$$

where

$$
\Psi\_+(\alpha, y) = \frac{1}{\sqrt{2\pi}} \int\_0^\infty \mu\_2(\alpha, y) e^{i\alpha x} dx
$$

Fourier transforms of the Eqs. (12)–(16) can be written as

$$\left[1+\frac{\eta\_1}{ik}\frac{\partial}{\partial y}\right]\Phi\_-(a,a) = 0, \ x<0\tag{19}$$

$$\left[1+\frac{\eta\_2}{ik}\frac{\partial}{\partial x}\right]\Psi\_+(\alpha, a) = 0, \ 0 < y < a \tag{20}$$

and

$$\left[1+\frac{\eta\_1}{ik}\frac{\partial}{\partial y}\right]\Psi\_+(x,0) = 0, \ \text{x}>0\tag{21}$$

$$
\Phi\_{-}(\alpha, a^{+}) - \frac{H\_{0}}{\alpha - k \cos \phi\_{0}} = \Psi\_{+}(\mathbf{x}, a^{-}) \tag{22}
$$

and

$$\frac{\partial \Phi\_{-}(\alpha, a^{+})}{\partial y} + \frac{ik}{\eta\_{1}} \frac{H\_{0}}{\alpha - k \cos \phi\_{0}} = \frac{\partial \Psi\_{+}(\mathbf{x}, a^{-})}{\partial y} \tag{23}$$

$$H\_0 = \frac{1}{8\pi^2} \frac{2\eta\_1 \sin\phi\_0}{\eta\_1 \sin\phi\_0 + 1} e^{-ikr\sin\phi\_0} \frac{2\pi}{\sqrt{kr\_0}} e^{i(kr\_0 - \pi/4)} \dots$$

The solution of Eq. (17) satisfying the radiation condition for y > a can be written as:

$$\Phi(\alpha, y) = B(\alpha)e^{j\gamma(\alpha)|y-a|},\tag{24}$$

where Bð Þ α is the unknown coefficient to be determined by substituting y ¼ a in the following expression <sup>Φ</sup>þð Þþ <sup>α</sup>; <sup>y</sup> <sup>Φ</sup>�ð Þ <sup>α</sup>; <sup>y</sup> and <sup>∂</sup>Φþð Þ <sup>α</sup>;<sup>y</sup> <sup>∂</sup><sup>y</sup> <sup>þ</sup> <sup>∂</sup>Φ�ð Þ <sup>α</sup>;<sup>y</sup> <sup>∂</sup><sup>y</sup> , and with the help of boundary condition;

$$
\Phi\_{-}(\alpha, a) + \frac{\eta\_{1}}{i\hbar} \Phi\_{-}^{'}(\alpha, a) = 0,\tag{25}
$$

where prime denotes differentiation with respect to y:

$$R\_+(\alpha) = B(\alpha) \left( 1 + \frac{\eta\_1}{k} \gamma(\alpha) \right),\tag{26}$$

where

Φð Þ¼ α; y Φ�ð Þþ α; y Φþð Þ α; y

ð∞ 0

ð0 �∞

u1ð Þ x; y e

u1ð Þ x; y e

iαx dx,

> iαx dx,

ϕ αð Þ¼ ; y 0, (17)

Ψþð Þ¼ α; y 0, 0 < y < a: (18)

Φ�ð Þ¼ α; a 0, x < 0 (19)

Ψþð Þ¼ α; a 0, 0 < y < a (20)

Ψþð Þ¼ α; 0 0, x > 0 (21)

<sup>¼</sup> <sup>∂</sup>Ψ<sup>þ</sup> <sup>x</sup>; <sup>a</sup>� ð Þ

¼ Ψ<sup>þ</sup> x; a� ð Þ (22)

<sup>∂</sup><sup>y</sup> (23)

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

and α is a complex transform variable.

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> ð∞ 0

u2ð Þ x; y e

iαx dx

Φþð Þ¼ α; y

Φ�ð Þ¼ α; y

d2 ϕ dy<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

and Eq. (10) reduce to

192 Antennas and Wave Propagation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k <sup>2</sup> � <sup>α</sup><sup>2</sup>

Apply half range Fourier transforms to the Eq. (11)

∂2 ∂x<sup>2</sup> þ

Fourier transforms of the Eqs. (12)–(16) can be written as

∂2 <sup>∂</sup>y<sup>2</sup> <sup>þ</sup> <sup>k</sup> 2

Ψþð Þ¼ α; y

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup> ik ∂ ∂y � �

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup> ik ∂ ∂y � �

∂Φ� α; a<sup>þ</sup> ð Þ ∂y

<sup>Φ</sup>� <sup>α</sup>; <sup>a</sup><sup>þ</sup> ð Þ� <sup>H</sup><sup>0</sup>

þ ik η1

α � k cos ϕ<sup>0</sup>

H<sup>0</sup> α � k cos ϕ<sup>0</sup>

<sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>2</sup> ik ∂ ∂x � �

� �

p

where γ αð Þ¼

where

and

and

where

$$R\_+(a) = \Phi\_+(a, a) + \frac{\eta\_1}{i\hbar} \Phi\_+^{'}(a, a). \tag{27}$$

From the Eqs (22), (23), (26) and (27), we obtain the following Wiener-Hopf functional equations

$$\frac{\partial \Psi\_{+}(\mathbf{x},a)}{\partial y} = \frac{R\_{+}(a)\dot{\mathbf{y}}\gamma(a)}{\left(1 + \frac{\eta\_{1}}{k}\gamma(a) + \frac{\mathrm{i\bar{k}}}{\eta\_{1}}\Phi\_{-}(\alpha,a) + \frac{\mathrm{i\bar{k}}}{\eta\_{1}}\frac{F\_{0}}{a - k\cos\phi\_{0}}\right)}\tag{28}$$

The corrected solution of Wiener-Hopf equation [2] in case of line source can be expressed as

$$\frac{R\_{+}(\alpha)}{(\alpha+T)G\_{+}(\alpha)}$$

$$=F\_{0}\frac{i\eta\_{1}}{k}\frac{(k\cos\theta\_{0}-T)G\_{-}(k\cos\theta\_{0})}{\alpha-k\cos\theta\_{0}}$$

$$+\frac{k}{\eta\_{1}}\frac{\left(\frac{k}{\eta\_{2}}-T\right)G\_{+}(T)R\_{+}(T)e^{-\frac{ka}{\eta\_{1}}}}{\left(\frac{k}{\eta\_{2}}+T\right)\sin\left(\frac{ka}{\eta\_{1}}\right)(\alpha+T)}$$

$$+\sum\_{n=1}^{\infty}\frac{\left(\frac{k}{\eta\_{2}}-\alpha\_{n}\right)G\_{+}(\alpha\_{n})R\_{+}(\alpha\_{n})\left(\frac{n\pi}{a}\right)^{2}}{\left(\frac{k}{\eta\_{2}}+\alpha\_{n}\right)a\alpha\_{n}(T-\alpha\_{n})(\alpha+\alpha\_{n})},$$

$$F\_0 = \frac{i}{8\pi^2} \frac{2\eta\_1 \sin\theta\_0}{1 + \eta\_1 \sin\theta\_0} e^{-ikr\sin\theta\_0} \frac{2\pi}{\sqrt{kr\_0}} e^{-i\left(kr\_0 - \frac{\pi}{4}\right)}.$$

and αn, Gð Þ α and Gþð Þ α are defined in [11]. The function Rþð Þ α depends upon the unknown series of constants Rþð Þ T , Rþð Þ α<sup>1</sup> , Rþð Þ α<sup>2</sup> , Rþð Þ α<sup>3</sup> :… To find an approximate value for Rþð Þ α , substitute α ¼ T, α1, α2,:…,α<sup>m</sup> in Eq. (29) to get m þ 1 equations in m þ 1 unknowns. The simultaneous solution of these equations yields approximate solutions for Rþð Þ T , Rþð Þ α<sup>1</sup> , Rþð Þ α<sup>2</sup> ,…Rþð Þ α<sup>m</sup> :

#### 5. Far zone solution

The unknown constant Bð Þ α can be obtained by taking inverse Fourier transform of Eq. (24), the final expression for the diffracted field is written as

$$u\_1(x,y) = \frac{1}{\sqrt{2\pi}} \int\_L \frac{R\_+(a)}{\left(1 + \frac{\eta\_1}{k}\gamma(a)\right)} e^{j\gamma(a)(y-a)} e^{-iax} d\alpha,\tag{29}$$

Φþð Þ¼ α; a Gþð Þ α i sin ϕ<sup>0</sup>

and

such that

step can be transformed.

e i kr0�<sup>π</sup> ð Þ<sup>4</sup> 4π ffiffiffiffiffiffi kr<sup>0</sup> p

<sup>a</sup> <sup>p</sup> <sup>e</sup>

� Y∞ n¼1

7. Magnetic line source diffraction by PEMC step

Es d Hs d � �

¼

Es

Hs <sup>d</sup> ¼ � <sup>1</sup> η0

<sup>¼</sup> <sup>1</sup> Mη<sup>0</sup> � �<sup>2</sup> <sup>þ</sup> <sup>1</sup>

<sup>η</sup>oH<sup>s</sup>

where E<sup>s</sup> and H<sup>s</sup> are the diffracted fields and Es

E H � �

from the PEC step by satisfy the condition,

Moreover, the transformation

gives

0 @

½ � <sup>γ</sup>a=πln ð Þ <sup>α</sup>þi<sup>γ</sup> <sup>=</sup><sup>k</sup> e

<sup>1</sup> � ka nπ � �<sup>2</sup>

<sup>G</sup>þð Þ¼ <sup>α</sup> ffiffi

G� k cos ϕ<sup>0</sup> � � α � k cos ϕ<sup>0</sup>

" #

G�ð Þ¼ α Gþð Þ �α

Next we transform magnetic line source diffracted field from PEC to PEMC step under the duality transformations in the [21]. The field diffracted by perfectly electric conducting (PEC)

We obtain a solution for magnetic line source diffraction by PEMC step by applying a transformation introduced by Lindell and Sihvola, that is known as duality transformation [21]:

> Mη<sup>0</sup> η<sup>0</sup> �1 η0

Mη<sup>0</sup>

<sup>d</sup> and H<sup>s</sup>

Mη<sup>0</sup> �η<sup>0</sup>

Mη<sup>0</sup>

1 A

Es d Hs d

<sup>d</sup> ¼ �uz � <sup>E</sup><sup>s</sup>

1 η0

0 @ 1 A

Es Hs

<sup>d</sup> <sup>¼</sup> <sup>M</sup>η0E<sup>s</sup> <sup>þ</sup> <sup>η</sup>0H<sup>s</sup> (35)

Es <sup>þ</sup> <sup>M</sup>η0H<sup>s</sup> (36)

þX<sup>∞</sup> n¼1

½ � iαa=πð Þ 1�Cþln 2ð Þ π =kaþiπ=2

e iαa

" #

� <sup>i</sup>α<sup>a</sup> nπ <sup>G</sup>þð Þ <sup>α</sup><sup>n</sup> <sup>n</sup><sup>π</sup> a � �<sup>2</sup>

aαnð Þ α<sup>n</sup> þ α

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

Φþð Þ αn; a

http://dx.doi.org/10.5772/intechopen.74938

<sup>n</sup><sup>π</sup> (33)

� �, (34)

<sup>d</sup> are the intermediate fields obtained

<sup>d</sup>: (37)

� � (38)

(32)

195

where L is a straight line parallel to the real axis, lying in the strip Im k½ � cos θ<sup>0</sup> < Im½ � α < Im k½ �: To determine the far field behavior of the scattered field, introducing the following substitutions x ¼ r cos θ, y � a ¼ r sin θ and α ¼ �k cos ð Þ θ þ it , where t is real. The contour of integration over <sup>α</sup> in Eq. (30) goes into the branch of hyperbola around �ik if <sup>π</sup> <sup>2</sup> < θ < π. We further observe that in deforming the contour into a hyperbola the pole α ¼ ξ may be crossed. If we make another transformation ξ ¼ k cos ð Þ θ<sup>0</sup> þ it<sup>1</sup> the contour over ξ also goes into a hyperbola. The two hyperbolae will not cross each other if θ < θ0: However, if the inequality is reversed there will be a contribution from pole which, in fact, cancels the incident wave in the shadow region in [11]. Simply the asymptotic evaluation of the integral in Eq. (30) using the method of steepest descent, we find the following solution for far field diffracted by an impedance step due to a line source at a large distance from the edge:

$$
\mu\_1(r,\phi) \, e^{-i\pi/4} k \sin\phi \mathcal{R} + (\alpha) \frac{e^{ikr}}{\sqrt{kr}} \tag{30}
$$

#### 6. Magnetic line source diffraction by PEC step

The asymptotic solution for the field diffracted by perfect electric conductor (PEC) step is obtained by equating <sup>η</sup><sup>1</sup> <sup>¼</sup> <sup>η</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> � � as

$$u\_1(r, \phi) = e^{-i\pi/4} k \sin\phi \Phi\_+(-k\cos\phi, a) \frac{e^{ikr}}{\sqrt{kr}}\tag{31}$$

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium http://dx.doi.org/10.5772/intechopen.74938 195

$$\Phi\_{+}(a,a) = G\_{+}(a)\left[i\sin\phi\_{0}\frac{e^{i\left(k\tau\_{0}-\frac{\pi}{4}\right)}G\_{-}\left(k\cos\phi\_{0}\right)}{4\pi\sqrt{kr\_{0}}}+\sum\_{n=1}^{\infty}\frac{G\_{+}(a\_{n})\left(\frac{n\pi}{a}\right)^{2}}{a a\_{n}(a\_{n}+a)}\Phi\_{+}(a\_{n},a)\right] \tag{32}$$

and

<sup>F</sup><sup>0</sup> <sup>¼</sup> <sup>i</sup> 8π<sup>2</sup>

the final expression for the diffracted field is written as

u1ð Þ¼ x; y

due to a line source at a large distance from the edge:

6. Magnetic line source diffraction by PEC step

<sup>u</sup><sup>1</sup> <sup>r</sup>; <sup>ϕ</sup> � � <sup>¼</sup> <sup>e</sup>

�iπ=4

obtained by equating <sup>η</sup><sup>1</sup> <sup>¼</sup> <sup>η</sup><sup>2</sup> <sup>¼</sup> <sup>0</sup> � � as

where

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> ð L

tion over <sup>α</sup> in Eq. (30) goes into the branch of hyperbola around �ik if <sup>π</sup>

u<sup>1</sup> r; ϕ � � e

�iπ=4

k sin ϕR þ ð Þ α

<sup>k</sup> sin <sup>ϕ</sup>Φ<sup>þ</sup> �<sup>k</sup> cos <sup>ϕ</sup>; <sup>a</sup> � � <sup>e</sup>ikr

ffiffiffiffi kr

The asymptotic solution for the field diffracted by perfect electric conductor (PEC) step is

eikr ffiffiffiffi kr

Rþð Þ α<sup>2</sup> ,…Rþð Þ α<sup>m</sup> :

194 Antennas and Wave Propagation

5. Far zone solution

2η<sup>1</sup> sin θ<sup>0</sup> 1 þ η<sup>1</sup> sin θ<sup>0</sup>

e

and αn, Gð Þ α and Gþð Þ α are defined in [11]. The function Rþð Þ α depends upon the unknown series of constants Rþð Þ T , Rþð Þ α<sup>1</sup> , Rþð Þ α<sup>2</sup> , Rþð Þ α<sup>3</sup> :… To find an approximate value for Rþð Þ α , substitute α ¼ T, α1, α2,:…,α<sup>m</sup> in Eq. (29) to get m þ 1 equations in m þ 1 unknowns. The simultaneous solution of these equations yields approximate solutions for Rþð Þ T , Rþð Þ α<sup>1</sup> ,

The unknown constant Bð Þ α can be obtained by taking inverse Fourier transform of Eq. (24),

Rþð Þ α <sup>1</sup> <sup>þ</sup> <sup>η</sup><sup>1</sup> <sup>k</sup> γ αð Þ � � <sup>e</sup>

where L is a straight line parallel to the real axis, lying in the strip Im k½ � cos θ<sup>0</sup> < Im½ � α < Im k½ �: To determine the far field behavior of the scattered field, introducing the following substitutions x ¼ r cos θ, y � a ¼ r sin θ and α ¼ �k cos ð Þ θ þ it , where t is real. The contour of integra-

observe that in deforming the contour into a hyperbola the pole α ¼ ξ may be crossed. If we make another transformation ξ ¼ k cos ð Þ θ<sup>0</sup> þ it<sup>1</sup> the contour over ξ also goes into a hyperbola. The two hyperbolae will not cross each other if θ < θ0: However, if the inequality is reversed there will be a contribution from pole which, in fact, cancels the incident wave in the shadow region in [11]. Simply the asymptotic evaluation of the integral in Eq. (30) using the method of steepest descent, we find the following solution for far field diffracted by an impedance step

�ika sin <sup>θ</sup><sup>0</sup> 2π

ffiffiffiffiffiffi kr<sup>0</sup> p e

<sup>i</sup>γ αð Þð Þ <sup>y</sup>�<sup>a</sup> e

�iαx

dα, (29)

<sup>2</sup> < θ < π. We further

p (30)

p (31)

�i kr0�<sup>π</sup> ð Þ<sup>4</sup> :

$$G\_{+}(\alpha) = \sqrt{a}e^{[\gamma a/\pi \ln(\alpha + i\gamma)/k]}e^{[\sin/\pi(1-\mathcal{C}+\ln(2\pi)/ka + i\pi/2)]}$$

$$\times \prod\_{n=1}^{\infty} \left[1 - \left(\frac{ka}{n\pi}\right)^{2} - \frac{i\alpha a}{n\pi}\right]e^{\frac{i\omega}{n\pi}}\tag{33}$$

such that

$$G\_{-}(a) = G\_{+}(-a)$$

Next we transform magnetic line source diffracted field from PEC to PEMC step under the duality transformations in the [21]. The field diffracted by perfectly electric conducting (PEC) step can be transformed.

#### 7. Magnetic line source diffraction by PEMC step

We obtain a solution for magnetic line source diffraction by PEMC step by applying a transformation introduced by Lindell and Sihvola, that is known as duality transformation [21]:

$$
\begin{pmatrix} E\_d^s \\ H\_d^s \end{pmatrix} = \begin{pmatrix} M\eta\_0 & \eta\_0 \\ -1 & M\eta\_0 \end{pmatrix} \begin{pmatrix} E^s \\ H^s \end{pmatrix} \tag{34}
$$

$$E\_d^s = M\eta\_0 E^s + \eta\_0 H^s \tag{35}$$

$$H\_d^s = -\frac{1}{\eta\_0} E^s + M \eta\_0 H^s \tag{36}$$

where E<sup>s</sup> and H<sup>s</sup> are the diffracted fields and Es <sup>d</sup> and H<sup>s</sup> <sup>d</sup> are the intermediate fields obtained from the PEC step by satisfy the condition,

$$
\eta\_o H\_d^s = -\mu\_z \times E\_d^s. \tag{37}
$$

Moreover, the transformation

$$
\begin{pmatrix} E \\ H \end{pmatrix} = \frac{1}{\left(M\eta\_0\right)^2 + 1} \begin{pmatrix} M\eta\_0 & -\eta\_0 \\ 1 & M\eta\_0 \end{pmatrix} \begin{pmatrix} E\_d^s \\ H\_d^s \end{pmatrix} \tag{38}
$$

gives

$$E = \frac{1}{\left(M\eta\_0\right)^2 + 1} \left[ \left(\left(M\eta\_0\right)^2 - 1\right)E^\circ - 2M\eta\_0 E^\circ \right] \tag{39}$$

α ¼ ω ffiffiffiffiffiffiffiffiffi ε0μ<sup>0</sup> <sup>p</sup> ffiffiffiffi εr 2

β ¼ ω ffiffiffiffiffiffiffiffiffi ε0μ<sup>0</sup> <sup>p</sup> ffiffiffiffi εr 2

(iii) clay: σ ¼ 0:01 mho/m and ε<sup>r</sup> ¼ 7ε0, respectively.

9. Results and discussion

an angle θ with the horizontal.

<sup>r</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ

<sup>r</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ

where ε<sup>0</sup> and μ<sup>0</sup> are the permittivity and the permeability of free space. For the hosted medium, we use three type of soil models [58] namely: (i) gravel sand having its conductivity σ ¼ 0:001 mho/m and its relative permittivity ε<sup>r</sup> ¼ 10:5ε0; (ii) sand: σ ¼ 0:0001 mho/m and ε<sup>r</sup> ¼ 8ε<sup>0</sup> and

In this section we discuss some graphical results which have been presented in [3] to predict the effects of the admittance parameters M and step height a and line source parameter r<sup>0</sup> on the diffraction phenomenon. It can be observed from [3] that the amplitude of the diffracted field increases with increase in step height. The graphs show that the amplitude of the diffracted field decreases as the source is taken away from the origin, which is a natural phenomenon and verifying the results. Through Mathematica software we have reproduced the results given by Lindell and Sihvola [21] and the results have retrieved. Here, an attempt is made to develop the theoretical results for lossy medium using the analytical solution for magnetic line source diffraction by PEMC step. As the step is assumed to be surrounded by different soils (i) gravel sand

Figure 2. Geometry of the diffraction problem: a line source located at (x0, y0) making an angle θ<sup>0</sup> with the horizontal, is incident upon PEMC step surrounded by lossy medium, as illustrated in this figure. Here, (x, y) is the observation point at

s

s

2 4

2 4

18σ ωεrε<sup>0</sup> � �<sup>2</sup>

18σ ωεrε<sup>0</sup> � �<sup>2</sup> � 1

þ 1

3 5

3 5

1=2

1=2

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

http://dx.doi.org/10.5772/intechopen.74938

197

$$H = \frac{1}{\left(M\eta\_0\right)^2 + 1} \left[ \left( \left(M\eta\_0\right)^2 - 1 \right) H^s - 2M\eta\_0 H^s \right] \tag{40}$$

where E and H are the fields diffracted (scattered) by the PEMC step which is written as

$$E = \frac{1}{\left(M\eta\_0\right)^2 + 1} \left[ \left(\left(M\eta\_0\right)^2 - 1\right)E^\circ - 2M\eta\_0 E^\circ \right] \tag{41}$$

and

$$E^{\circ} = e^{-i\pi/4} k \sin \phi \Phi\_{+} \left( -k \cos \phi, a \right) \frac{e^{ikr}}{\sqrt{kr}} \tag{42}$$

where

$$\Phi\_{+}(a,a) = \mathcal{G}\_{+}(a) \left[ ik \sin \phi\_{0} \frac{e^{i\left(k\tau\_{0} - \frac{\pi}{4}\right)}}{4\pi\sqrt{kr\_{0}}} \frac{G\_{-}(k\cos\phi\_{0})}{a - k\cos\phi\_{0}} + \sum\_{n=1}^{\infty} \frac{G\_{+}(a\_{n})\left(\frac{n\pi}{a}\right)^{2}}{a\alpha\_{n}(a\_{n} + a)} \Phi\_{+}(a\_{n},a) \right] \tag{43}$$

#### 8. Magnetic line source diffraction by PEMC step in lossy medium

When we study magnetic line source diffraction by PEMC step in lossy medium, we just replace free-space wave number k by γ, then the solution obtained from the PEMC step for the lossy medium can be expressed such that (Figure 2)

$$E = \frac{1}{\left(M\eta\_0\right)^2 + 1} \left[ \left(\left(M\eta\_0\right)^2 - 1\right)E^\circ - 2M\eta\_0 E^\circ \right] \tag{44}$$

and

$$E^s = e^{-i\pi/4} \gamma \sin\phi \Phi\_+ \left( -\gamma \cos\phi, a \right) \frac{e^{i\gamma r}}{\sqrt{\gamma r}} \tag{45}$$

where

$$\Phi\_{+}(a,a) = G\_{+}(a)\left[i\gamma\sin\phi\_{0}\frac{e^{i\left(\gamma r\_{0} - \overline{\frac{\pi}{4}}\right)}}{4\pi\sqrt{\gamma r\_{0}}}\frac{G\_{-}\left(\gamma\cos\phi\_{0}\right)}{a-\gamma\cos\phi\_{0}} + \sum\_{n=1}^{\infty}\frac{G\_{+}(a\_{n})\left(\frac{n\pi}{a}\right)^{2}}{a\alpha\_{n}(a\_{n}+a)}\Phi\_{+}(a\_{n},a)\right] \tag{46}$$

where γ ¼ β � α, here α is attenuation factor and β is propagation constant defined as in [57].

$$\alpha = \omega \sqrt{\varepsilon\_0 \mu\_0} \sqrt{\frac{\varepsilon\_r}{2}} \left[ \sqrt{1 + \left( \frac{18\sigma}{\omega \varepsilon\_r \varepsilon\_0} \right)^2} - 1 \right]^{1/2}$$

$$\beta = \omega \sqrt{\varepsilon\_0 \mu\_0} \sqrt{\frac{\varepsilon\_r}{2}} \left[ \sqrt{1 + \left( \frac{18\sigma}{\omega \varepsilon\_r \varepsilon\_0} \right)^2} + 1 \right]^{1/2}$$

where ε<sup>0</sup> and μ<sup>0</sup> are the permittivity and the permeability of free space. For the hosted medium, we use three type of soil models [58] namely: (i) gravel sand having its conductivity σ ¼ 0:001 mho/m and its relative permittivity ε<sup>r</sup> ¼ 10:5ε0; (ii) sand: σ ¼ 0:0001 mho/m and ε<sup>r</sup> ¼ 8ε<sup>0</sup> and (iii) clay: σ ¼ 0:01 mho/m and ε<sup>r</sup> ¼ 7ε0, respectively.

Figure 2. Geometry of the diffraction problem: a line source located at (x0, y0) making an angle θ<sup>0</sup> with the horizontal, is incident upon PEMC step surrounded by lossy medium, as illustrated in this figure. Here, (x, y) is the observation point at an angle θ with the horizontal.

#### 9. Results and discussion

<sup>E</sup> <sup>¼</sup> <sup>1</sup> Mη<sup>0</sup> � �<sup>2</sup> <sup>þ</sup> <sup>1</sup>

<sup>H</sup> <sup>¼</sup> <sup>1</sup> Mη<sup>0</sup> � �<sup>2</sup> <sup>þ</sup> <sup>1</sup>

<sup>E</sup> <sup>¼</sup> <sup>1</sup> Mη<sup>0</sup> � �<sup>2</sup> <sup>þ</sup> <sup>1</sup>

Es <sup>¼</sup> <sup>e</sup>

Φþð Þ¼ α; a Gþð Þ α ik sin ϕ<sup>0</sup>

the lossy medium can be expressed such that (Figure 2)

<sup>E</sup> <sup>¼</sup> <sup>1</sup> Mη<sup>0</sup> � �<sup>2</sup> <sup>þ</sup> <sup>1</sup>

<sup>E</sup><sup>s</sup> <sup>¼</sup> <sup>e</sup>

Φþð Þ¼ α; a Gþð Þ α iγ sin ϕ<sup>0</sup>

�iπ=4

e <sup>i</sup> <sup>γ</sup>r0�<sup>π</sup> ð Þ<sup>4</sup> 4π ffiffiffiffiffiffiffi γr<sup>0</sup> p

�iπ=4

e i kr0�<sup>π</sup> ð Þ<sup>4</sup> 4π ffiffiffiffiffiffi kr<sup>0</sup> p

8. Magnetic line source diffraction by PEMC step in lossy medium

When we study magnetic line source diffraction by PEMC step in lossy medium, we just replace free-space wave number k by γ, then the solution obtained from the PEMC step for

> Mη<sup>0</sup> � �<sup>2</sup> � <sup>1</sup> � �

and

196 Antennas and Wave Propagation

where

and

where

Mη<sup>0</sup> � �<sup>2</sup> � <sup>1</sup> � �

Mη<sup>0</sup> � �<sup>2</sup> � <sup>1</sup> � �

Mη<sup>0</sup> � �<sup>2</sup> � <sup>1</sup> � �

where E and H are the fields diffracted (scattered) by the PEMC step which is written as

<sup>E</sup><sup>s</sup> � <sup>2</sup>Mη0E<sup>s</sup> h i

<sup>H</sup><sup>s</sup> � <sup>2</sup>Mη0H<sup>s</sup> h i

<sup>E</sup><sup>s</sup> � <sup>2</sup>Mη0E<sup>s</sup> h i

<sup>k</sup> sin <sup>ϕ</sup>Φ<sup>þ</sup> �<sup>k</sup> cos <sup>ϕ</sup>; <sup>a</sup> � � <sup>e</sup>ikr

G� k cos ϕ<sup>0</sup> � � α � k cos ϕ<sup>0</sup>

ffiffiffiffi kr

<sup>G</sup>þð Þ <sup>α</sup><sup>n</sup> <sup>n</sup><sup>π</sup> a � �<sup>2</sup>

aαnð Þ α<sup>n</sup> þ α

þX<sup>∞</sup> n¼1

" #

<sup>E</sup><sup>s</sup> � <sup>2</sup>Mη0E<sup>s</sup> h i

ffiffiffiffiffi

<sup>G</sup>þð Þ <sup>α</sup><sup>n</sup> <sup>n</sup><sup>π</sup> a � �<sup>2</sup>

aαnð Þ α<sup>n</sup> þ α

þX<sup>∞</sup> n¼1

" #

<sup>γ</sup> sin <sup>ϕ</sup>Φ<sup>þ</sup> �<sup>γ</sup> cos <sup>ϕ</sup>; <sup>a</sup> � � <sup>e</sup><sup>i</sup>γ<sup>r</sup>

G� γ cos ϕ<sup>0</sup> � � α � γ cos ϕ<sup>0</sup>

where γ ¼ β � α, here α is attenuation factor and β is propagation constant defined as in [57].

p (42)

Φþð Þ αn; a

<sup>γ</sup><sup>r</sup> <sup>p</sup> (45)

Φþð Þ αn; a

(39)

(40)

(41)

(43)

(44)

(46)

In this section we discuss some graphical results which have been presented in [3] to predict the effects of the admittance parameters M and step height a and line source parameter r<sup>0</sup> on the diffraction phenomenon. It can be observed from [3] that the amplitude of the diffracted field increases with increase in step height. The graphs show that the amplitude of the diffracted field decreases as the source is taken away from the origin, which is a natural phenomenon and verifying the results. Through Mathematica software we have reproduced the results given by Lindell and Sihvola [21] and the results have retrieved. Here, an attempt is made to develop the theoretical results for lossy medium using the analytical solution for magnetic line source diffraction by PEMC step. As the step is assumed to be surrounded by different soils (i) gravel sand (ii) sand and (iii) clay. By using the permittivity, permeability and conductivity of these lossy mediums we predict the loss effect on the diffracted field. The computed fields are obtained analytically for more general solution. The said problem is first reduced to a modified Wiener-Hopf equation of second kind whose solution contains an infinite set of constants satisfying an infinite system of linear algebraic equations. A numerical solution of this system is obtained for various values of admittance parameter M and the height of the step a versus observation angle. Further, the effect of these parameters on the diffraction phenomenon is studied. It is observed that if the source is shifted to a large distance these results differ from those of by a multiplicative factor to the part of the scattered field containing the effects of incident and reflected waves. The diffraction analysis of magnetic line source by a PEMC step provides explicit formulas for electric and magnetic field amplitude and the polarization. Here, it is interesting to note that the copolarized and the cross-polarized fields depend on the parameter M.

Author details

References

2015;PP(99):1

139(3):291-305

gation. 1967;15(3):442-448

1987;MTF35(11):956-963

TX, USA. 1983; 21. pp.13-17

Saeed Ahmed\* and Mona Lisa

\*Address all correspondence to: saeedqau@gmail.com

Department of Earth Sciences, Quaid-i-Azam University, Islamabad, Pakistan

IEEE Transactions on Antennas and Propagation. 2009;57(4):1289-1293

conducting (PEMC) step. Journal of Modern Optics. 2014;62(3):175-178

[1] Ayub M, Ramzan M, Mann AB. Magnetic line source diffraction by an impedance step.

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

http://dx.doi.org/10.5772/intechopen.74938

199

[2] Ahmed S. Comments on magnetic line source diffraction by an impedance step. IEEE.

[3] Ahmed S. Magnetic line source diffraction of a plane wave by a perfectly electromagnetic

[4] Sommers GA, Pathak PH. GTD solution for the diffraction by metallic tapes on paneled compact range reflectors. Proceedings of the Institution of Electrical Engineers. 1992;

[5] Johansen E. Surface wave scattering by a step. IEEE Transactions on Antennas and Propa-

[6] Jamid HA, Al-Bader SJ. Reflection and transmission of surface Plasmon mode at a step

[7] Valagiannopoulos CA, Uzunoglu NK. Rigorous analysis of a metallic circular post in a rectangular waveguide with step discontinuity of sidewalls. IEEE Transactions on Micro-

[8] Yang H-Y, Alexopoulos NG. Characterization of the finline step discontinuity on anisotropic substrates. IEEE Transactions on Microwave Theory and Techniques. November

[9] Valagiannopoulos CA. High selectivity and controllability of a parallel-plate component with a filled rectangular ridge. Progress in Electromagnetics Research. 2011;119:497-511

[10] Pannon W, Uslenghi PLE. Exact and asymptotic scattering by a step discontinuity in an impedance plane, antennas and Propagation Society International Symposium, Houston,

[11] Büyükaksoy A, Birbir F. Plane wave diffraction by an impedance step. IEEE Transactions

[12] Büyükaksoy A, Birbir F. Correction to plane wave diffraction by an impedance step. IEEE

Transactions on Antennas and Propagation. Mar. 1996;44(3):422-422

discontinuity. IEEE Photonics Technology Letters. February 1997;9(2):220-222

wave Theory and Techniques. August 2007;55(8):1673-1684

on Antennas and Propagation. 1993;41(8):1160-1164

#### 10. Conclusion

It is concluded that the both coupled electric and magnetic fields excitation can be observed analytically for PEMC theory that leads to a most general case for the magnetic line source diffraction by step embedded in lossy medium. The lossy medium is assumed to be made of three different soils (i) gravel sand, (ii) sand and (iii) clay. We see from their respective electric parameters namely permittivity, permeability and conductivity, as the loss increases the amplitude of the diffracted field decreases. By applying this technique to detect the subsurface targets, we can use various soil models. Further, in this chapter at a time we studied diffraction by step using PEMC theory and loss effect on the field patterns. Here, we can predict the behavior of the fields diffracted by magnetic line source. This is the most general solution and is more useful rather a plane wave solution. In far zone, we can obtain a solution for the diffraction of a plane wave by PEMC step placed in lossy medium under the condition kr ! ∞. It is also concluded that the parameter M plays a significant role in PEMC theory to interlink the PEC and PMC media. The cross-polarized scattered fields vanish in the presence of PEC and PMC cases and they are maximal for Mη<sup>0</sup> ¼ �1. If M ¼ �∞, correspond to PEC case and M ¼ 0, correspond to a PMC case. The impulse response of the soil is important in investigating the best operating frequency and bandwidth for a subsurface-imaging SAR. Due to dispersion and loss in the soil, the impulse response deviated from the free-space impulse response. The following conclusions can be drawn from examination of the soil's impulse response: (i) an optimum bandwidth exists; (ii) loss increases as bandwidth increases; (iii) very large bandwidths are not useful for imaging objects at large depths; (iv) vertical polarization is best for large angles of incidence and (v) lower frequencies seem best.

#### Acknowledgements

The author Dr. Saeed Ahmed acknowledges the financial support from the Department of Earth Sciences, Quaid-i-Azam University, Islamabad, Pakistan, during the Post Doctoral studies for the year (20 January, 2017–19 January, 2018).

## Author details

(ii) sand and (iii) clay. By using the permittivity, permeability and conductivity of these lossy mediums we predict the loss effect on the diffracted field. The computed fields are obtained analytically for more general solution. The said problem is first reduced to a modified Wiener-Hopf equation of second kind whose solution contains an infinite set of constants satisfying an infinite system of linear algebraic equations. A numerical solution of this system is obtained for various values of admittance parameter M and the height of the step a versus observation angle. Further, the effect of these parameters on the diffraction phenomenon is studied. It is observed that if the source is shifted to a large distance these results differ from those of by a multiplicative factor to the part of the scattered field containing the effects of incident and reflected waves. The diffraction analysis of magnetic line source by a PEMC step provides explicit formulas for electric and magnetic field amplitude and the polarization. Here, it is interesting to note that the co-

It is concluded that the both coupled electric and magnetic fields excitation can be observed analytically for PEMC theory that leads to a most general case for the magnetic line source diffraction by step embedded in lossy medium. The lossy medium is assumed to be made of three different soils (i) gravel sand, (ii) sand and (iii) clay. We see from their respective electric parameters namely permittivity, permeability and conductivity, as the loss increases the amplitude of the diffracted field decreases. By applying this technique to detect the subsurface targets, we can use various soil models. Further, in this chapter at a time we studied diffraction by step using PEMC theory and loss effect on the field patterns. Here, we can predict the behavior of the fields diffracted by magnetic line source. This is the most general solution and is more useful rather a plane wave solution. In far zone, we can obtain a solution for the diffraction of a plane wave by PEMC step placed in lossy medium under the condition kr ! ∞. It is also concluded that the parameter M plays a significant role in PEMC theory to interlink the PEC and PMC media. The cross-polarized scattered fields vanish in the presence of PEC and PMC cases and they are maximal for Mη<sup>0</sup> ¼ �1. If M ¼ �∞, correspond to PEC case and M ¼ 0, correspond to a PMC case. The impulse response of the soil is important in investigating the best operating frequency and bandwidth for a subsurface-imaging SAR. Due to dispersion and loss in the soil, the impulse response deviated from the free-space impulse response. The following conclusions can be drawn from examination of the soil's impulse response: (i) an optimum bandwidth exists; (ii) loss increases as bandwidth increases; (iii) very large bandwidths are not useful for imaging objects at large depths; (iv) vertical polarization is

polarized and the cross-polarized fields depend on the parameter M.

best for large angles of incidence and (v) lower frequencies seem best.

ies for the year (20 January, 2017–19 January, 2018).

The author Dr. Saeed Ahmed acknowledges the financial support from the Department of Earth Sciences, Quaid-i-Azam University, Islamabad, Pakistan, during the Post Doctoral stud-

10. Conclusion

198 Antennas and Wave Propagation

Acknowledgements

Saeed Ahmed\* and Mona Lisa

\*Address all correspondence to: saeedqau@gmail.com

Department of Earth Sciences, Quaid-i-Azam University, Islamabad, Pakistan

## References


[13] Büyükaksoy A, Birbir F. Plane wave diffraction by a reactive step. International Journal of Engineering Science. 1997;35:311-319

[30] Lindell IV, Ruotanen LH. Duality transformations and Green dyadics for bi-anisotropic media. Journal of Electromagnetic Waves and Applications. 1998;12:1131-1152

Magnetic Line Source Diffraction by a PEMC Step in Lossy Medium

http://dx.doi.org/10.5772/intechopen.74938

201

[31] Lindell IV, Olyslager F. Duality in electromagnetics. Journal of Communications Technol-

[32] Ahmed S, Akbar M, Shafiq M. Diffraction by a perfectly electromagnetic conducting

[33] Ahmed S, Mehmood I. Diffraction of a plane wave by a perfectly electromagnetic conducting

[34] Ahmed S, Manan F. Scattering by randomly placed line source in the presence of perfectly electromagnetic conducting plane. American International Journal of Contemporary Research.

[35] Ahmed S, Manan F. Scattering by randomly placed perfectly electromagnetic conducting half plane. International Journal of Applied Science and Technology. November 2011;1(6):

[36] Ahmed S, Manan F. Scattering by perfectly electromagnetic conducting random grating. American International Journal of Contemporary Research. November 2011;1(3):66-71 [37] Ahmed S, Manan F. Scattering by perfectly electromagnetic conducting random width strip. American International Journal of Contemporary Research. November 2011;1(6):

[38] Ahmed S, Manan F. Scattering by randomly placed perfectly electromagnetic conducting random width strip. International Journal of Applied Science and Technology. November

[39] Ahmed S. Diffraction by perfect electromagnetic conductor (PEMC) half plane. Interna-

[40] Ahmed S, Mann AB, Nawaz R, Tiwana MH. Diffraction of electromagnetic plane wave by a slit in a homogeneous bi-isotropic medium. Waves in Random and Complex Media.

[41] Ahmed S. Comments on electromagnetic scattering from chiral coated nihility cylinder.

[42] Ahmed S. Comments on electromagnetic scattering from a chiral coated PEMC cylinder.

[43] Ahmed S. Comments on electromagnetic response of a circular DB cylinder in the presence of chiral and chiral nihility metamaterials. Progress in Electromagnetics Research

[44] Brock BC, Sorensen KW. Electromagnetic Scattering from Buried Objects. Sandia National

[45] Kuloglu M, Chen H-C. Ground penetrating radar for tunnel detection. Geoscience and

Remote Sensing Symposium (IGARSS), IEEE International, 2010;4314-4317

tional Journal of Electronics Letters. January 2017;5(3):255-260

Progress In Electromagnetics Research Letters. 2015;53:123

Progress in Electromagnetics Research Letters. 2015;53:101

Laboratories, SAND94-2361; September 1994

ogy and Electronics. 2000;45(2):S260-S268

November 2011;1(3):168-172

311-317

305-310

2011;1(6):300-304

2017;27(2):325-338

Letters. 2015;54:1

(PEMC) step. Journal of Modern Optics. 2013;60:637-640

(PEMC) slot. Journal of Modern Optics. 2014;61:335-338


[30] Lindell IV, Ruotanen LH. Duality transformations and Green dyadics for bi-anisotropic media. Journal of Electromagnetic Waves and Applications. 1998;12:1131-1152

[13] Büyükaksoy A, Birbir F. Plane wave diffraction by a reactive step. International Journal of

[14] Büyükaksoy A, Tayyar IH. High frequency diffraction by a rectangular impedance cylinder on an impedance plane. IEE Proceedings–Science, Measurement and Technology.

[15] Tayyar IH, Aksoy S, Alkumru A. Surface wave scattering by a rectangular impedance cylinder located on a reactive plane. Mathematical Methods in the Applied Sciences. 2005;

[17] Rojas RG. Wiener-Hopf analysis of the EM diffraction by an impedance discontinuity in a planar surface and by an impedance half-plane. IEEE Transactions on Antennas and

[18] Clavel E, Schanen L, Roudet J, Arechal YM. Influence of an impedance step in interconnection inductance calculation. IEEE Transactions on Magnetics. May 1996;32(3)

[20] Hohmann GW. Electromagnetic scattering by conductors in the earth near a line source of

[21] Lindell IV, Sihvola AH. Transformation method for problems involving perfect electromagnetic conductor (PEMC) structures. IEEE Transactions on Antennas and Propagation.

[22] Lindell IV, Sihvola AH. Perfect electromagnetic conductor. Journal of Electromagnetic

[23] Lindell IV. Electromagnetic fields in self-dual media in differential-form representation.

[24] Ruppin R. Scattering of electromagnetic radiation by a perfect electromagnetic conductor cylinder. Journal of Electromagnetic Waves and Applications. 2006;20(13):1853-1860 [25] Lindell IV, Sihvola AH. Reflection and transmission of waves at the interface of perfect electromagnetic conductor (PEMC). Progress in Electromagnetics Research B. 2008;5:

[26] Lindell IV, Sihvola AH. Realization of the PEMC boundary. IEEE Transactions on Anten-

[27] Lindell IV. Differential Forms in Electromagnetics. New York: Wiley and IEEE Press; 2004 [28] Lindell IV, Sihvola AH. Losses in PEMC boundary. IEEE Transactions on Antennas and

[29] Jancewicz B. Plane electromagnetic wave in PEMC. Journal of Electromagnetic Waves

[16] Noble B. Methods Based on Wiener Hopf Techniques. New York: Pergamon; 1958

[19] Jones DS. The Theory of Electromagnetism. London: Pergamon Press; 1964

Progress in Electromagnetics Research, PIER. 2006;58:319-333

Engineering Science. 1997;35:311-319

Propagation. 1988;36(1):71-83

2005;53(9):3005-3011

169-183

current. Geophysics. 1971;36(1):101-131

Waves and Applications. 2005;19:861-869

nas and Propagation. 2005;53:3012-3018

Propagation. 2006;54(9):2553-2558

and Applications. 2006;20(5):647-659

2002;149:49-59

200 Antennas and Wave Propagation

28:525-549


[46] Brock BC, Patitz WE. Optimum Frequency for Subsurface-Imaging Synthetic Aperture Radar, SAND93-0815. Springfield, USA: National Technical Information Service, US

[47] Doerry AW. A Model for Forming Airborne Synthetic Aperture Radar Images of Underground Targets, SAND94-0139. Albuquerque, New Mexico: Sandia National Laborato-

[48] Von Hippel AR, editor. Dielectric Materials and Applications. New York: The Technology

[49] Radzevicius SJ, Daniels JJ. Ground penetrating radar polarization and scattering from

[50] Frezza F, Pajewski L, Ponti C, Schettini G, Tedeschi N. Cylindrical-wave approach for electromagnetic scattering by subsurface metallic targets in a lossy medium. Journal of

[51] Armin WD. A Model for Forming Airborne Synthetic Aperture Radar Images of Underground Targets, Synthetic Aperture Radar Department, 2345, Sandia National Laborato-

[52] Bradford JH. Frequency-dependent attenuation analysis of ground-penetrating radar

[53] Zhenhua M. Advanced Feature Based Techniques for Landmine Detection Using Ground

[54] Ahmed S, Khan MK, ur Rehman A. Scattering by a perfect electromagnetic conductor plate embedded in lossy medium. International Journal of Electronics. 2015;103(07):1228-1235 [55] Ahmed S. The study of the radar cross section of perfect electromagnetic conductor strip.

[56] Ahmed S, Rehaman AU, Zain Iftekhar M, Lisa M. Scattering by a PEMC cylinder embed-

[57] Ballard RB Jr. Electromagnetic (RADAR) Techniques Applied to Cavity Detection, Technical Report No. 5, Geotechnical Laboratory, P.O. BOX 631, Vicksburg, Miss. 39180; 1983

[58] Carcione JM. Ground-radar numerical modelling applied to engineering problems. Euro-

[59] Rajyalakshmi P, Raju GSN. Characteristics of radar cross section with different objects. International Journal of Electronics and Communication Engineering. 2011;4(2):205-216

pean Journal of Environmental and Engineering Geophysics. 1996;1:65-81

Department of Commerce; May 1993

Applied Geophysics. 2013;97:55-59

data. Geophysics. May–June 2007;72(3)

Optik. 2015;126(23):4191-4194

ded in lossy medium. Optik. 2016;127(19):8011-8018

Press of M.I.T., and John Wiley & Sons, Inc.; 1954

cylinders. Journal of Applied Geophysics. 2000;45:111-125

ries Albuquerque, NM 87185-0529, Technical Report; January 1994

Penetrating Radar, MS Thesis, University of Missouri-Columbia; 2007

ries; January 1994

202 Antennas and Wave Propagation

## *Edited by Pedro Pinho*

Antennas and radio propagation are continuously and rapidly evolving and new challenges arise every day. As a result of these rapid changes the need for up-to-date texts that address this growing field from an interdisciplinary perspective persists. This book, organized into nine chapters, presents new antenna designs and materials that will be used in the future, due to the trend for higher frequencies, as well as a bird's eye view of some aspects related to radio propagation channel modeling. The book covers the theory but also the practical aspects of technology implementation in a way that is suitable for undergraduate and graduate-level students, as well as researchers and professional engineers.

Published in London, UK © 2018 IntechOpen © Panuwat Sikham / iStock

Antennas and Wave Propagation

Antennas and Wave

Propagation