Applications of the Wavelets in Waveguide Analysis and Improvement of Medical Images

## **Chapter 8**

## Application to Medical Image Processing

*Anthony Y. Aidoo, Gloria A. Botchway and Matilda A.S.A. Wilson*

### **Abstract**

Medical images are often corrupted by white noise, blurring and contrast defects. Consequently, important medical information may be degraded or completely masked. Advanced medical diagnostics and pathological analysis utilize information obtained from medical images. Consequently, the best techniques must be applied to capture, compress, store, retrieve and share these images. Recently, the wavelet transform technique has been applied to enhance and compress medical images. This review focuses on the trends of wavelet-based medical image processing techniques. A summary of the application of wavelets to enhance and compress medical images such as magnetic resonance imaging (MRI), computerized tomography (CT), positron emission tomography (PET), single photon emission computed tomography (SPECT), and X-ray is provided. Morphological techniques such as closing, thinning and pruning are combined with wavelets methods to extract the features from the medical images.

#### **Keywords:**

#### **1. Introduction**

The goal of this chapter is to provide a review of the applications of wavelets to medical imaging. The focus will be on medical image denoising and compression. Advanced medical diagnostics utilize information obtained from technologies such as magnetic resonance imaging (MRI), computerized tomography (CT), positron emission tomography (PET), single photon emission computed tomography (SPECT), and X-ray [1, 2]. However, such images are corrupted by white noise, blurring and contrast defects. Consequently, important medical information may be degraded or completely masked. Recently, wavelet-based techniques have been applied to achieve superior image denoising and economical image compression.

#### **1.1 Wavelet properties in medical imaging: multiresolution analysis**

A multiresolution analysis is a decomposition of the Hilbert space *<sup>H</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ð Þ *R* into a chain of closed subspaces *V <sup>j</sup>* , *j*∈*Z* which form a sequence of successive approximation subspaces of *H* such that the following hold:

$$\begin{aligned} \text{1. } V\_j \subset V\_{j+1} \text{ for all } j \in Z \\\\ \text{2. } \bigcup\_{j=-\infty}^{\infty} V\_j \text{ is dense in } L^2(R) \text{ and } \bigcap\_{j=-\infty}^{\infty} V\_j = \{0\}. \\\\ \text{3.} f(\mathbf{x}) \in V\_j \Leftrightarrow f(2\mathbf{x}) \in V\_{j+1} \text{ for all } j \in Z \\\\ \text{4.} f(\mathbf{x}) \in V\_j \Leftrightarrow f(\mathbf{x} - k) \in V\_j \text{ for all } j, k \in Z \end{aligned}$$


#### **1.2 Wavelet properties in medical imaging: wavelet bases**

One of the special qualities of wavelets which is exploited in medical image analysis is the ability to construct *L*<sup>2</sup> bases which are simply dilations and translations of a single compactly supported function given by *<sup>ψ</sup> <sup>j</sup>*,*<sup>k</sup>* <sup>¼</sup> <sup>2</sup>�*j=*<sup>2</sup> *<sup>ψ</sup> <sup>x</sup>=*<sup>2</sup> *<sup>j</sup>* � *<sup>k</sup>* n o � � where *j*, *k*∈ *Z*. This enables any image function *f* to be represented by:

$$f = \sum\_{j \in Zk} \sum\_{j,k} c\_{j,k} \nu\_{j,k} \tag{1}$$

### **2. Undecimated wavelet transform**

Conventional methods for medical image enhancement have very limited versatility in application and their use could lead to the loss of important medical image features of interest. This could be highly fatal in medical imaging applications [3]. The undecimated discrete wavelet transform (UDWT) method is a wavelet transform algorithm without the downsampling operations, resulting in both the original signal and the approximation and detailed coefficients having same length at each level of decomposition. The basic algorithm of the conventional UDWT is that it applies the transform at each point of the image and saves the detailed coefficients and uses the approximation coefficients for the next level. The size of the coefficients array does not diminish from level to level. This decomposition operation is further iterated up to a higher level. Various denoising methods using the DWT provide robust computational methods in denoising digital medical images. The only issue with the DWT is that it is shift variant. This disadvantage can may be ameriorated by using the UDWT to achieve shift invariance.

#### **3. Image enhancement: wavelets and medical image denoising**

Medical images are usually corrupted by noise inherrent in the processes of acquisition, trasmission, and retrieval [4, 5]. In particular, medical images such as those

*Application to Medical Image Processing DOI: http://dx.doi.org/10.5772/intechopen.102819*

obtained from MRI or X-rays are often complicated by random noise that occurs during the image acquisition stage [6]. Wavelet-based techniques overcome most of these limitations. The objective of enhancement is to remove the effects of signal degradation caused by the signal processing. Noise may be removed by smoothing the signal by subjecting it to a low-pass filter. Sharpening to remove blur is used to identify more detailed features by applying a high-pass filter.

#### **3.1 Wavelets methods**

With its inherrent properties of multiresolution structure, application of wavelets to medical images converts the noisy image in the time domain into the wavelet transform domain. Subsequently, essential image detail information is compressed into large coefficients that are retained at different resolution scales. The small coefficients represent the noise in the image as well as any redundant information. In medical image analysis, the boundary line between "large" and "small" coefficients is crucial since it determines whether the noise is significantly removed in addition to crucial detail being preserved.

#### **3.2 Hybrid methods for medical image denoising**

Spatial filters have the tendency of blurring images since the technique smoothens data in order to remove niose [7]. Tackling this problem by relying on wavelet transform alone sometimes does not satisfactorily address the image enhancement problem since wavelet tranform methods are plagued by oscillations, shift variance, aliasing and lack of directionality. Three methods that are combined with wavelets significantly eliminate the problems listed above the medical image enhancement outcomes considered here.

#### *3.2.1 Total variation denoising*

Total variation (TV) regularization is a deterministic method that minimizes the effect of discontinuities in image processing [8]. The TV technique is endowed with the power of preserving and even enhancing the edges. The use of TV for image denoising assumes that the observed image is made up of the sum of a piecewise smooth image and guassian noise.

A real valued function *f x*ð Þ, representing a signal is sampled using the partition *P* ¼ �f g ∞ < *x*<sup>0</sup> <*x*<sup>1</sup> < … <*xn*, *n* ∈ of the interval ½ � *x*0, *x* . The TV *T <sup>f</sup>* of *f* over the interval is defined by:

$$T\_f(\mathbf{x}) = \sup \left\{ \sum\_{i}^{n} |f(\mathbf{x}\_i) - f(\mathbf{x}\_{i-1})| \colon -\infty < \mathbf{x}\_0 < \mathbf{x}\_1 < \dots < \mathbf{x}\_n, n \in \mathbb{N} \right\} \tag{2}$$

If lim *<sup>x</sup>*!<sup>∞</sup>*<sup>T</sup> <sup>f</sup>*ð Þ *<sup>x</sup>* is finite, then *<sup>f</sup>* is of bounded variation. The TV of an *<sup>L</sup>*<sup>1</sup> function *<sup>f</sup>* of several variables, in an open subset Ω of *<sup>n</sup>*, is defined as by:

$$T\_f(\mathbf{x}) = \sup \left\{ \int\_{\Omega} f(\mathbf{x}) \text{div}\phi(\mathbf{x}) \text{d}\mathbf{x} : \phi(\mathbf{x}) \in \mathbb{C}\_{\epsilon}^1(\Omega, \mathbb{R}^n), \||\phi\||\_{L^n(\Omega)} \le 1 \right\} \tag{3}$$

If *f* is a differentiable function defined on a bounded open domain Ω ⊂ *<sup>n</sup>* this reduces to:

$$T\_f(\mathbf{x}) = \int\_{\Omega} |\nabla f(\mathbf{x})| \mathbf{dx} \tag{4}$$

**Definition 3.1** *The total variation of an image is defined by the duality: for u*∈ *L*<sup>1</sup> *loc the total variation is given by T <sup>f</sup>* <sup>¼</sup> sup �div*ϕ*d*<sup>x</sup>* : *<sup>ϕ</sup>*∈*C*<sup>∞</sup> *<sup>c</sup>* <sup>Ω</sup>; *<sup>N</sup>* � �, <sup>j</sup>*ϕ*ð Þj *<sup>x</sup>* <sup>≤</sup> <sup>1</sup>∀*x*<sup>∈</sup> <sup>Ω</sup> � �*.*

This hybrid approach used here represents a noisy image in a simplified form by Eq. (1). The reconstruction of *u x*ð Þreduces to the optimization problem of minimizing the function

$$E(u) = \frac{\lambda}{2} \left\| u - z \right\|\_{L^2(\Omega)}^2 + R(u) \tag{5}$$

(see for example [9]). Here, the parameter *λ*> 0 and *R u*ð Þ is the regularization functional defined on the domain Ω. The disadvantage of this method is that despite removing noise adequately, it removes essential details from the image [8]. Since the efficiency of the method is controlled by the choice of the regularization functional, this is usually costly in medical imaging. The use of the TV of the image function below ameliorates this problem.

$$R(\mu) = T\_x(\mu) = \int\_{\Omega} |\Delta \mu| d\mathbf{x} \tag{6}$$

It leads to sharper reconstruction of the original image by both removing the imbedded noise and better preservation of its edges [10, 11]. TV minimization scheme takes the geometric information of the original images into account, and this helps to preserve and sharpen the edges significantly [11].

#### *3.2.2 The wavelet-total variation method*

**Proposition 1** *[12] Let K* <sup>¼</sup> *<sup>p</sup>*<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> ð Þ Ω : Ð <sup>Ω</sup>*p x*ð Þ*u x*ð Þd*x*≤*Tz*ð Þ *<sup>u</sup>* <sup>∀</sup>*<sup>u</sup>* <sup>∈</sup>*L*<sup>2</sup> ð Þ <sup>Ω</sup> � �*. If Tz is considered as a functional over the Hilbert space L*<sup>2</sup> ð Þ <sup>Ω</sup> *, we have <sup>∂</sup>Tz*ð Þ¼ *<sup>u</sup> p*∈*K* : Ð <sup>Ω</sup>*p x*ð Þ*u x*ð Þd*<sup>x</sup>* <sup>¼</sup> *Tz*ð Þ *<sup>u</sup>* � �*.*

**Proof 1** *If p*∈ *K and* Ð <sup>Ω</sup>*p x*ð Þ*u x*ð Þd*<sup>x</sup>* <sup>¼</sup> *Tz*ð Þ *<sup>u</sup> then p*∈*∂Tz*ð Þ *<sup>u</sup> . Clearly for any v* <sup>∈</sup>*L*<sup>2</sup> ð Þ Ω *we have Tz*ð Þ¼ *v* sup*<sup>p</sup>*∈*<sup>K</sup>* Ð <sup>Ω</sup>*p x*ð Þ*u x*ð Þd*x. Tz*ð Þ*<sup>v</sup>* <sup>≥</sup><sup>Ð</sup> <sup>Ω</sup>*p x*ð Þ*v x*ð Þd*x* ¼ *Tz*ð Þþ *u* Ð <sup>Ω</sup>ð Þ *v x*ð Þ� *u x*ð Þ *p x*ð Þd*x. Conversely, if p*∈*∂Tz*ð Þ *<sup>u</sup> , then for any t*<sup>&</sup>gt; <sup>0</sup> *and v* <sup>∈</sup> *N, with Tz*ð Þ¼ *tu tTz*ð Þ *u since Tz is positively one-homogeneous, we have: tT v*ð Þ¼ *Tz*ð Þ*v* ≥*Tz*ð Þþ *u* Ð <sup>Ω</sup>*p x*ð Þð Þ *tv x*ð Þ� *u x*ð Þ <sup>d</sup>*x. Dividing by t and letting t* ! <sup>∞</sup><sup>↦</sup> *leads to Tz*ð Þ*<sup>v</sup>* <sup>Ð</sup> <sup>Ω</sup>*p x*ð Þ*v x*ð Þd*x. Hence p*∈*K. On the other hand, letting t* ! <sup>0</sup> *gives Tz*ð Þ *<sup>u</sup>* <sup>≤</sup><sup>Ð</sup> <sup>Ω</sup>*p x*ð Þ*u x*ð Þd*x.*


**Table 1.** *TV vs TV and UDWT.* *Application to Medical Image Processing DOI: http://dx.doi.org/10.5772/intechopen.102819*

#### **Figure 1.**

*(Column 1) Original chest images with nodules, (column 2) wavelet decomposition of images in column 1, (column 3) reconstructed images from the decomposed images.*

The wavelet TV scheme represents the components of the function by orthogonal wavelet basis. The wavelet coefficients are then selected to achieve the goals of denoising and enhancement (**Table 1**). **Figure 1** shows the results on the chest radiograph images.

The Python 2.7 code is given below.

"""A code to implement a wavelet denoising and morphological enhancement """

```
#import math
#import numpy as np
```

```
import cv2
import mat ot as plt import pywt
kernel=cv2.getStructuringElement(cv2.MORPH_ELLIPSE,(11,11))
img1=cv2.imread('JPCLN001.jpg') coeff1=pywt.wavedec2(img1, 'bior1.3')
coeff11=pywt.waverec2(coeff1,'db2')
erosion1=cv2.erode(coeff11,kernel,iterations=2)
opening1=cv2.dil te(erosion1,kernel,iterations=3)
oinv1=1-opening1
fig 1=plt.figure()
fig 1.suptitle('Original, Decomposed and Reconstructed CR Images with Nodules')
```
plt.subplot(331),plt.imshow(img1),plt.title('a'),plt.xticks([]),plt.yt icks([]) plt.subplot(332),plt.imshow(coeff1[0]),plt.title('d'),plt.xticks([]),p lt.yticks([]) plt.subplot 333),plt.imshow(coeff11),plt.title('g'),plt.xticks([]),plt .yticks([]) fig 1.savefig('1CR i ages with nodules, decomposed and reconstructed.png') fig 3=plt.figure()

fig 3.suptitle('Decomposed nodule images eroded, opened and inversed') plt.subplot(331),plt.imshow(erosion1),plt.title('a'),plt.xticks([]),pl t.yticks([]) plt.subplot(332),plt.imshow(opening1),plt.title('d'),plt.xticks([]),pl t.yticks([]) plt.subpl t(333),plt.imshow(oinv1),plt.title('g'),plt.xticks([]),plt.y ticks([])

fig 3.savefig('1Decomposed nodule images eroded, opened and inversed.png')

#### *3.2.3 Mathematical morphology*

Mathematical morphology (MM) is a technique for extracting image components of interest Dilation and erosion are the two basic operations in mathematical mophology as well as thinning, opening, closing, and prunning. Wavelets combined with MM has recently been used to improve chest radiographs [13].

**Definition 3.2** *Erosion and Dilation: Let E be the Euclidean space, let A* : *<sup>E</sup>* <sup>⊆</sup> <sup>ℤ</sup><sup>2</sup> ! <sup>ℤ</sup> *be an image and B* : <sup>ℤ</sup><sup>2</sup> ! f g 0, 1 *be a structuring element. The translation of a set C by a point z* ¼ ð Þ *z*1, *z*<sup>2</sup> *, denoted by C*ð Þ*<sup>z</sup> is defined as C*ð Þ*<sup>z</sup>* ¼ *a*∣*a* ¼ *c* þ *z*,*c*∈ *A. The erosion of A by B, denoted by A*ð Þ ⊖ *B , is expressed as*

$$\pi(A \ominus B) = \underline{\pi}(B)\_x \subseteq A,\tag{7}$$

ie. The set of all pixel locations z in the image plane where ð Þ *B <sup>z</sup>* is contained in *A*. **Definition 3.3** *The dilation of A by B is denoted by A*ð Þ ⊕ *B and is expressed as*

$$\iota(A \oplus B) = \iota | (\hat{B})\_x \sqcap A \neq \mathcal{Q},\tag{8}$$

*where <sup>B</sup>*^ <sup>¼</sup> *<sup>w</sup>*∣*<sup>w</sup>* ¼ �*b, for b*∈*B is the reflection of B*.

This indicates the set of all pixel locations *z* in the image plane where the intersection of *B*^ with *A* is not empty [14].

Erosion shrinks an image or a region *A* by a template or a structuring element *B*. Dilation expands an image or a region *A* by a template or a structuring element *B*. The dilation process consists of obtaining the reflection of *B* about its origin and then shifting this reflection by some displacement *x*.

Other effects can be obtained by applying erosion and dilation in a loop. Closing and opening are two examples of basic erosion and dilation combinations.

#### **3.3 Opening and closing**

**Definition 3.4** *The opening of A by B, denoted by A* ∘ *B, is simply erosion of A by B, followed by dilation of the result by B, that is,*

$$(A \circ B) = (A \ominus B) \oplus B. \tag{9}$$

Visually, opening smoothens contours, breaks narrow isthmuses and eliminates small islands.

**Definition 3.5** *The closing of A by B, denoted by A* • *B, is a dilation followed by an erosion and is given as*

$$(A \bullet B) = (A \oplus B) \ominus B.\tag{10}$$

*Closing smoothens the contours, fills narrow gulfs and eliminates small holes. It is based on these operations that other operations are derived*.

#### **3.4 Thinning and pruning**

The thinning operation is related to the hit-or-miss transform and it can be expressed in terms of it. The thinning operation is derived by translating the origin of the structuring element to each possible pixel position in the image and comparing it with the underlying image pixels at each such position. Pruning is a post-processing technique that follows thinning. It removes parasitic components known as spurs which are unwanted branches, from the thinned image. There are specific structuring elements used for pruning.

Combined with MM wavelets can be used to decompose a fingerprint image in order to extract the areas with details. The results of this approach is shown in **Figures 2** and **3**.

$$\mathbf{Algorthm 1}$$

Denoising the fingerprint image


#### Algorithm 2

Processing the image for feature extraction


#### *3.4.1 Wavelet K-SVD approach*

The wavelet tranform technique (for image denoising) has several advantages such as sparsity, multiresolution structure, and similarity with human vision. Recently, it has been combined with K-Singular Valued Decomposition (K-SVD) algorithm, and an adaptive learning over the wavelet decomposition of a noisy medical image has resulted [15].

**Figure 2.**

*Wavelet analysis and synthesis of image: (a) original image, (b) decomposed image, (c) reconstructed image.*

**Figure 3.**

*Morphological operations applied to fingerprint images: (a) original image, (b) binary closing of image, (c) thinned image, (d) prunned image.*

### **4. Wavelet based compression methods for storing medical images**

In medical image compression, a mathematical transform is applied to the digital data. This is intended to compress the data for efficient storage, transmission, and


#### **Table 2.**

*Compression ratios.*

retrieval. Compression involves coding and image approximation and helps to reduce the quantity of information, improve the transmission rate and reduce the size of the equipment and storage space required. Data compression requires the choice of a transform such that as many of the transformed data as possible vanish. The ability to localize basis functions in wavelet applications make them suitable for compression. In addition to this property of wavelets, a wavelet decomposition of an image capitalizes on its multiresolution structure, a recursive method to compute the wavelet transform of an image [16]. The three components of the wavelet tranform based image are image decomposition, quantization, and decompression.

#### **4.1 Hybrid methods for medical image compression**

The DWT combined with vector quantization methods have recently be shown to achieve superior results in medical image compression than wavelet alone technique. For example, Ammah and Owusu [17] proposed an efficient medical hybrid image decompression technique for ultrasound and MRI images. The method consists of first preprocessing the image by noise removal. This is follwed by filtering the image using the DWT with hard thresholding. The image is subsequently vector quantized and then Huffman encoded. The inverse operations are then applied to obtain the decompressed image.

The metrics used to evaluate the efficiency of the compression methods are the compression ratio (CR) and the peak signal to noise ratio (PSNR) given by:

$$\text{CR} = \frac{I(\mathbf{x}, \mathbf{y})}{I'(\mathbf{x}, \mathbf{y})} \tag{11}$$

and

$$\text{PSNR} = 20 \log\_{10} \left\{ \frac{255}{\text{RMSE}} \right\} \tag{12}$$

where the RMSE is the root mean squared error.

Using a new class of spline wavelet filters, more effective data compression techniques have been devised to compress massive quantities of medical image data leading to a more economical storage process and enhanced medical image quality when retrieved. Combined with other methods, the inherrent properties of wavelets such as sparsity and multiresolution structure produce superior medical image data compresssion results than the competition. CRs for three most used methods are is shown in **Table 2**.

#### **5. Conclusion**

Modern radiology techniques are essential in advanced medical diagnostics and pathological analysis [18]. Applications of the wavelet tranform for medical imaging techniques and current advances in research in this direction has been highlighted in this chapter. This includes the combination of the standard wavelet techniques with TV, MM and other methods.

## **Acknowledgements**

Author Anthony Y. Aidoo acknowledges the support from the CSU-AAUP Faculty Research Fund 2021-22.

## **Classifications**

**2020 AMS Subject Classification**: 68U10, 94A12, 28A99, 92C55, 65R10, 44A99.

## **Author details**

Anthony Y. Aidoo<sup>1</sup> \*, Gloria A. Botchway<sup>2</sup> and Matilda A.S.A. Wilson<sup>3</sup>

1 Department of Mathematical Sciences, Eastern Connecticut State University, Willimantic, USA

2 Department of Mathematics, University of Ghana, Accra, Ghana

3 Department of Computer Science, University of Ghana, Accra, Ghana

\*Address all correspondence to: aidooa@easternct.edu

© 2022 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.

*Application to Medical Image Processing DOI: http://dx.doi.org/10.5772/intechopen.102819*

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## **Chapter 9**

Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section and for Complementary Shape of the Cross Section

*Zion Menachem*

## **Abstract**

This chapter presents wave propagation along a straight rectangular waveguide for practical applications where there are two complementary shapes of the dielectric profile in the cross section. In the first case, the cross section consists of circular dielectric material in the center of the cross section. In the second case, the cross section consists of a circular hollow core in the center of the cross section. These examples show two discontinuous cross sections and complementary shapes that cannot be solved by analytical methods. We will explain in detail the special technique for calculating the dielectric profile for all cases. The method is based on Laplace and Fourier transforms and inverse Laplace and Fourier transform. In order to solve any inhomogeneous problem in the cross section, more than one technique can be proposed for the same mode-model method. We will explain in detail how and where the technique can be integrated into the proposed mode-model. The image method and periodic replication are needed for fulfilling the boundary condition of the metallic waveguide. The applications are useful for straight rectangular waveguides in millimeter regimes, where the circular dielectric material is located in the center of the cross section, and also for hollow waveguides, where the circular hollow core is located in the center of the cross section.

**Keywords:** wave propagation, dielectric profiles, rectangular waveguide, circular dielectric material, circular hollow core

## **1. Introduction**

We begin with a review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides with emphasis on recent developments as published in [1]. Six groups of methods were reviewed: the finite-element method, the finite-difference method, the integral-equation method, methods based on series expansion, approximate methods based on separation of variables, and methods that do not fit the above groups.

The use of wavelet-like basis functions for solving electromagnetics problems is demonstrated in [2]. The modes of an arbitrarily shaped hollow metallic waveguide use a surface integral equation and the method of moments. A class of wavelet-like basis functions produces a sparse method of moments. A technique for efficient computation of an integral wavelet transform of a finite-energy function on a dense set of the time-scale domain is proposed [3] by using compactly supported spline wavelets. Application of principal component analysis and wavelet transform to fatigue crack detection in waveguides is proposed in [4]. Ultrasonic guided waves are a useful tool in structural health monitoring applications that can benefit from built-in transduction, moderately large inspection ranges, and high sensitivity to small flaws. An accurate full-wave integral formulation was developed [5] for the study of integrated planar dielectric waveguide structures with printed metalized sections, which are of practical interest for millimeter-wave and submillimeter-wave applications. An advantageous finite element method for the rectangular waveguide problem was developed [6] by which complex propagation characteristics may be obtained for arbitrarily shaped waveguides. The finite-element method has been used to derive approximate values of the possible propagation constant for each frequency. The impedance characteristics of the fundamental mode in a rectangular waveguide were computed using this finite element method. The extension to higher-order elements is straightforward, and by modifications of the method it is possible to treat other types of waveguides as well, e.g., dielectric waveguides with impedance walls and open unbounded dielectric waveguides properties treating the region of infinity.

A comprehensive study of the design and performance of a multilayer dielectric rod waveguide with a rectangular cross section is proposed in [7]. The design is comprised of a high permittivity core encased by a low permittivity cladding. A mathematical model was proposed to predict the fundamental mode cutoff frequency in terms of the core dimensions and the core and cladding permittivity. The model is useful for design purposes and it offers an excellent match to full-wave electromagnetic simulation results.

The characteristics of the effective-medium-clad dielectric waveguides, including dispersion, cross-polarization, crosstalk between parallel waveguides, bending loss, and wave leakage at the crossing, have been comprehensively investigated and measured [8].

Mode matching has been done at all the air and dielectric interfaces and thus the characteristic equations have been derived [9]. Two ratios are introduced in the characteristic equations and the new set of characteristic equations thus obtained are then plotted and graphical solutions are obtained for the propagation parameters assuming certain numerical values for the introduced ratios.

A fundamental and accurate technique to compute the propagation constant of waves in a dielectric rectangular waveguide was proposed [10]. The formulation is based on matching the fields to the constitutive properties of the material at the boundary.

The method of lines for the analysis of dielectric waveguides was proposed [11]. These waveguides are uniform along the direction of propagation, are loss-free and passive. Hybrid-mode dispersion curves, field and intensity distributions for integrated optical waveguides were presented.

The problem of normal waves in a closed regular waveguide of arbitrary cross section has been considered [12]. It was reduced to a boundary value problem for the

#### *Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

longitudinal components of the electromagnetic field in Sobolev spaces. The solutions were defined using the variational formulation of the problem. The problem was reduced to the study of an operator function. The properties of the operators involved in the operator function were examined. Theorems were proved concerning the discrete character of the spectrum and the distribution of characteristic numbers of the operator function on the complex plane. The completeness of the system of Eigen- and associated vectors of the operator function was investigated.

TE-wave propagation in a hollow waveguide with a graded dielectric layer using a hyperbolic tangent function is proposed in Ref. [13]. General formulas for the electric field components of the TE-waves, applicable to hollow waveguides with arbitrary cross sectional shapes were presented. The exact analytical results for the electric field components were illustrated in the special case of a rectangular waveguide. The exact analytical results for the reflection and transmission coefficients are valid for waveguides of arbitrary cross sectional shapes. The obtained reflection and transmission coefficients are in exact asymptotic agreement with those obtained for a very thin homogeneous dielectric layer using mode-matching and cascading. The proposed method gives analytical results that are directly applicable without the need of modematching, and it has the ability to model realistic, smooth transitions.

Rectangular waveguides were the earliest mode of transmission lines used for compact systems like radars and inside equipment shelters [14]. An air-filled rectangular waveguide WR-90 is simulated using HFSS simulation software to obtain different parameters. The electric and magnetic field patterns are analyzed: intrinsic impedance and wavelength for the first four modes of the waveguide are also obtained.

The diffraction of electromagnetic waves by rectangular waveguides with a longitudinal slit has been simulated [15]. The results allow determining the patterns of change in frequency bands in which the structure can be used as a directional coupler and as a power divider when changing the number of slots, their sizes and provisions. Modeling the characteristics of such kinds of structures allows predicting the creation of directional couplers and power dividers with high integral characteristics.

Several methods of propagation along the straight waveguides were developed, based on Maxwell's equations. A transfer matrix function for the analysis of electromagnetic wave propagation along the straight dielectric waveguide with arbitrary profiles has been proposed in Ref. [16].

In this chapter, the main objective is to generalize the mode model method [16] in order to solve also complicated and practical problems of circular dielectric material and a circular hollow core in the center of the cross section of the rectangular waveguide. It is important to distinguish between the mode-model method and the proposed technique. The proposed technique deals only with calculating the dielectric profile in the cross section of the inhomogeneous case. In order to solve any inhomogeneous problem in the cross section, more than one technique can be proposed for the same mode-model method. The technique proposed in this chapter will refer to two interesting practical applications. In the first case, the cross section consists of circular dielectric material in the center of the cross section. In the second case, the cross section shows the complementary shape of the cross section of the first case, as an example of a hollow waveguide in which the circular hollow core is located in the center of the cross section. These examples show two discontinuous cross sections and complementary shapes that cannot be solved by analytical methods. We will explain in detail the special technique for calculating the dielectric profile in all cases. After receiving the expressions of the proposed technique for each inhomogeneous problem in the cross section, we will explain how and where the technique can be integrated into the proposed mode-model. The second objective is to find the relevant parameters in order to obtain the Gaussian behavior of the output field in the interesting cases of circular dielectic material and a circular hollow core in the rectangular cross section.

#### **2. Complementary shapes in the cross section for different applications**

The wavelet transform creates a representation of the signal in both the time and frequency domain in order to allow efficient access to localized information about the signal. A set of waveforms comprising a transform is called a basis function. Fourier transforms use only sine and cosine waves as their basis functions, namely a signal is decomposed into a series sine and cosine functions or wavelets by the FFT. Examples for the applications of wavelet transform are demonstrated in [2–5]. The proposed method in this chapter is based on the Fourier transform that creates a representation of the signal in the frequency domain. Two complicated and complementary shapes are given in this section.

**Figure 1(a)** and **(b)** shows two complementary shapes of profiles in the cross section of the straight rectangular waveguide and their relevant parameters. The circular dielectric material in the center of the cross section is shown in **Figure 1(a)** and the circular hollow core in the center of the cross section is shown in **Figure 1(b)**. The two examples are demonstrated as a response to a half-sine (TE10) input-wave profile. These two different complementary shapes of the cross section are demonstrated for two different applications. The first example (**Figure 1(a)**) is useful in millimeter regimes where the circular dielectric material is located in the center of the cross section. The second example (**Figure 1(b)**) is useful in the millimeter regimes where the circular hollow core is located in the center of the cross section.

The main objective is to generalize the mode model method [16] in order to also solve complicated problems of circular dielectric material and a circular hollow core in the rectangular cross section. All the mathematical development relates to the frequency domain. The main points are given in Appendix A.

It is important to separate between mode-model method and the proposed technique. The proposed technique refers only to calculating of the dielectric profile in the

#### **Figure 1.**

*Complementary shapes of profiles in the cross section of the straight rectangular waveguide and their relevant parameters. (a) Circular dielectric material in the center of the cross section. (b) A circular hollow core in the center of the cross section.*

*Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

cross section of the inhomogeneous problem. In order to solve any inhomogeneous problem in the cross section, more than one technique can be proposed for the same mode-model method. After receiving the expressions of the proposed technique for eace inhomogeneous problem in the cross section, we will explain how and where the technique can be integrated into the proposed mode-model. The second objective is to find the relevant parameters in order to obtain the Gaussian behavior of the output field in the interesting cases of circular dielectic profile and circular hollow profile in the rectangular cross section.

The method is based on Maxwell's equations for the computation of output fields at each point along the straight waveguide. This method relates the wave profile at the output to the input wave in the Laplace space. A Laplace transform is necessary to obtain convenient and simple input–output connections of the fields. The method consists of Fourier coefficients of the transverse dielectric profile and of the input– output profile. Thus, the accuracy of the method depends on the number of the modes in the system.

The output transverse field profiles are computed by the inverse Laplace and Fourier transforms. The output components of the electric field are given finally by

$$\mathbf{E\_x} = \left\{ \mathbf{D\_x} + a\_1 \mathbf{M\_1 M\_2} \right\}^{-1} \left( \hat{\mathbf{E\_{x\_0}}} - a\_2 \mathbf{M\_1} \hat{\mathbf{E\_{y\_0}}} \right), \tag{1}$$

$$\mathbf{E\_{y}} = \left\{\mathbf{D\_{y}} + a\_{1}\mathbf{M\_{3}}\mathbf{M\_{4}}\right\}^{-1} \left(\hat{\mathbf{E\_{y\_{0}}}} - a\_{3}\mathbf{M\_{3}}\hat{\mathbf{E\_{x\_{0}}}}\right),\tag{2}$$

$$\mathbf{E}\_{\mathbf{z}} = \mathbf{D}\_{\mathbf{z}}^{-1} \left\{ \hat{\mathbf{E}}\_{\mathbf{z}\_0} + \frac{\mathbf{1}}{2\mathbf{s}} \left( \mathbf{G}\_{\mathbf{x}} \mathbf{E}\_{\mathbf{x}\_0} + \mathbf{G}\_{\mathbf{y}} \mathbf{E}\_{\mathbf{y}\_0} \right) - \frac{1}{2} \left( \mathbf{G}\_{\mathbf{x}} \mathbf{E}\_{\mathbf{x}} + \mathbf{G}\_{\mathbf{y}} \mathbf{E}\_{\mathbf{y}} \right) \right\},\tag{3}$$

where **Ex0** , **Ey0** , **Ez0** are the initial values of the corresponding fields at z = 0, i.e., **Ex0** <sup>¼</sup> **Ex** (x, y, z = 0), and **<sup>E</sup>**^**x0** , **<sup>E</sup>**^**y0** , **<sup>E</sup>**^**z0** are the initial-value vectors.

The modified wave-number matrices are given by

$$\begin{aligned} \mathbf{D}\_{\mathbf{x}} & \equiv \mathbf{K}^{(0)} + \frac{k\_a^2 \chi\_0}{2\varepsilon} \mathbf{G} + \frac{jk\_{ox}}{2\varepsilon} \mathbf{N} \mathbf{G}\_{\mathbf{x}}, \qquad \mathbf{D}\_{\mathbf{y}} \equiv \mathbf{K}^{(0)} + \frac{k\_{o\mathcal{K}0}^2}{2\varepsilon} \mathbf{G} + \frac{jk\_{oy}}{2\varepsilon} \mathbf{M} \mathbf{G}\_{\mathbf{y}},\\ \mathbf{D}\_{\mathbf{z}} & \equiv \mathbf{K}^{(0)} + \frac{k\_{o\mathcal{K}0}^2}{2\varepsilon} \mathbf{G}, \end{aligned} \tag{4}$$

where the diagonal matrices **K**ð Þ **<sup>0</sup>** , **M**, and **N** are given by

$$\mathbf{K}\_{(\mathbf{n},\mathbf{m})(\mathbf{n}',\mathbf{m}')}^{(\mathbf{0})} = \left\{ \left[ \mathbf{k}\_{\mathbf{o}}^{2} - (\mathbf{n}\pi/\mathbf{a})^{2} - (\mathbf{m}\pi/\mathbf{b})^{2} + \mathbf{s}^{2} \right] / 2\mathbf{s} \right\} \delta\_{\mathbf{n}\mathbf{n}'} \delta\_{\mathbf{m}\mathbf{n}'},\tag{5}$$
 
$$\mathbf{M}\_{(\mathbf{n},\mathbf{m})(\mathbf{n}',\mathbf{m}')} = \mathbf{m} \delta\_{\mathbf{n}\mathbf{n}'} \delta\_{\mathbf{m}\mathbf{n}'}, \qquad \mathbf{N}\_{(\mathbf{n},\mathbf{m})(\mathbf{n}',\mathbf{m}')} = \mathbf{n} \delta\_{\mathbf{n}\mathbf{n}'} \delta\_{\mathbf{m}\mathbf{n}'},$$

and where

$$a\_1 = \frac{k\_{\alpha x} k\_{\alpha \mathbf{y}}}{4s^2}, \qquad a\_2 = \frac{jk\_{\alpha x}}{2s}, \qquad a\_3 = \frac{jk\_{\alpha \mathbf{y}}}{2s}, \tag{6}$$

$$\mathbf{M}\_1 = \mathbf{N} \mathbf{G}\_\mathbf{y} \mathbf{D}\_\mathbf{y}^{-1}, \qquad \mathbf{M}\_2 = \mathbf{M} \mathbf{G}\_\mathbf{x}, \qquad \mathbf{M}\_3 = \mathbf{M} \mathbf{G}\_\mathbf{x} \mathbf{D}\_\mathbf{x}^{-1}, \qquad \mathbf{M}\_4 = \mathbf{N} \mathbf{G}\_\mathbf{y}.$$

Similarly, the other components of the magnetic field are obtained. The output transverse field profiles are given by the inverse Laplace and Fourier transforms, as follows

$$\mathbf{E}\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \sum\_{\mathbf{n}} \sum\_{\mathbf{m}} \left[ \int\_{\sigma - \mathbf{j}\_{m}}^{\sigma + \mathbf{j}\_{m}} \mathbf{E}\_{\mathbf{y}}(\mathbf{n}, \mathbf{m}, \mathbf{s}) \exp\left[ \mathbf{j} \mathbf{n} \mathbf{k}\_{\alpha \mathbf{x}} \mathbf{x} + \mathbf{j} \mathbf{m} \mathbf{k}\_{\alpha \mathbf{y}} \mathbf{y} + \mathbf{s} \mathbf{z} \right] \mathbf{ds},\tag{7}$$

where the inverse Laplace transform is calculated according to the Salzer method [17, 18]. The inverse Laplace transform is performed in this study by a direct numerical integration on the Laplace transform domain by using the method of Gaussian Quadrature. The integration path in the right side of the Laplace transform domain includes all the singularities.

$$\int\_{\sigma - j\_{\text{on}}}^{\sigma + j\_{\text{on}}} e^{\varsigma \zeta} E\_{\text{\textquotedblleft}}(\varsigma) d\varsigma = \frac{1}{\zeta} \int\_{\sigma - j\_{\text{on}}}^{\sigma + j\_{\text{on}}} e^{p} E\_{\text{\textquotedblleft}}(p \,/ \zeta) dp = \frac{1}{\zeta} \sum\_{i=1}^{15} w\_{i} E\_{\text{\textquotedblright}}(\varsigma = p\_{i}/\zeta), \tag{8}$$

where *wi* and *pi* are the weights and zeros, respectively, of the orthogonal polynomials of order 15. The Laplace variable *s* is normalized by *pi =ζ* in the integration points, where Re *pi* � �>0 and all the poles should be localized on their left side on the Laplace transform domain. This approach of a direct integral transform does not require as in other methods, to deal with each singularity separately.

The relation between the functions f (t) and F (p) is given by

$$f(t) = \frac{1}{2\pi j} \int\_{\sigma - j\_{\text{os}}}^{\sigma + j\_{\text{os}}} e^{pt} F(p) dp. \tag{9}$$

The function F(p) may be either known only numerically or too complicated for evaluating f(t) by Cauchy's theorem. The function F(p) behaves like a Polynomial without a constant term, in the variable 1/p, along (*σ* � *j* <sup>∞</sup>, *σ* þ *j* <sup>∞</sup>). One may find f(t) numerically by using new quadrature formulas (analogous to those employing the zeros of the Laguerre polynomials in the direct Laplace transform). A suitable choice of *pi* yields an n-point quadrature formula that is exact when *p*2*n* is any arbitrary polynomial of the 2n(th) degree in *x* � 1*=p*, namely

$$\frac{1}{2\pi j}\int\_{\sigma - j\_{\text{os}}}^{\sigma + j\_{\text{os}}} e^p \rho(\mathbf{1}/p) dp = \sum\_{i=1}^{n} A\_i(n)\rho\_{2n}(\mathbf{1}/p\_i). \tag{10}$$

In Eq. (10), *xi* � 1*=pi* are the zeros of the orthogonal polynomials *pn*ð Þ� *x* Πð Þ *x* � *xi* where

$$\frac{1}{2\pi j}\int\_{\sigma - j\_{\rm o}}^{\sigma + j\_{\rm o}} \epsilon^p \left(\frac{1}{p}\right) p\_n \left(\frac{1}{p}\right) \left(\frac{1}{p}\right)^i dp = 0,\tag{11}$$

*i* = 0,1, … ,*n* � 1 and *Ai*ð Þ *n* correspond to the Christoffel numbers. The normalization *Pn*ð Þ� 1*=p* ð Þ 4*n* � 2 ð Þ 4*n* � 6 , … , 6*pn*ð Þ 1*=p* , for *n* ≥2, produces all integral coefficients. *Pn*ð Þ <sup>1</sup>*=<sup>p</sup>* is proven to be ð Þ �<sup>1</sup> *<sup>n</sup> <sup>e</sup>*�*ppndn <sup>e</sup><sup>p</sup>=pn* ð Þ*=dp<sup>n</sup>* . The numerical table gives us the values of the reciprocals of the zeros of *Pn*ð Þ *x* or *pi* ð Þ *n* , the zeros of *Pn*ð Þ *x* , or 1*=pi n*, and the corresponding Christoffel numbers *Ai*ð Þ *n* . By using these quantities in the quadrature formula that represents in Eq. (10), then the "Christoffel numbers" are given by

$$A\_i(n) \equiv \frac{1}{2\pi j} \int\_{\sigma - j\_{\text{os}}}^{\sigma + j\_{\text{os}}} e^p L\_i(n+1) \left(\frac{1}{p}\right) dp. \tag{12}$$

*Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

A sufficient condition for Eq. (12) to hold is obviously the "Orthogonality" of ð Þ 1*=p pn*ð Þ 1*=p* with respect to any "arbitrary" *ρ*ð Þ 1*=p* (see Eq. (11)). The points 1*=pi* are denoted by 1*= pi <sup>n</sup>* and they are the "zeros" of a certain set of "orthogonal polynomials" in the variable 1/*p*. By using these quantities in the "quadrature formula" we can obtain theoretically "exact accuracy" for "any polynomial" in 1/*p* up to the 16(th) degree.

A Fortran code is developed using NAG subroutines (The Numerical Algorithms Group (NAG)) [19].

The proposed technique will introduce details for all the interesting cases of a discontinuous cross section, as shown in **Figure 1(a)** and **(b)**.

#### **3. Calculation of the different inhomogeneous profiles**

This section explains the proposed technique for calculating the dielectric profile for the two different inhomogeneous and complicated shapes of the cross section, as shown in **Figure 1(a)** and **(b)**.

#### **3.1 Calculation for circular dielectric material in the center of the cross section**

The technique is based on Fourier transform and uses the image method and periodic replication for fulfilling the boundary conditions of the metallic waveguide. Periodicity and symmetry properties are chosen to force the boundary conditions at the location of the walls in a real problem, by extending the waveguide region (0 ≤*x*≤ *a*, and 0≤ *y*≤*b*) to regions that are four-fold larger (�*a*≤*x*≤*a*, and �*b*≤*y*≤*b*). The elements of the matrix g(n,m) are calculated for an arbitrary profile in the cross section of the straight waveguide according to **Figure 2 (a)** and **(b)**.

The dielectric profile *g x*ð Þ , *y* is calculated according to *ε*ð Þ¼ *x*, *y ε*0ð Þ 1 þ *g x*ð Þ , *y* and according to **Figure 2(a)** and **(b)** where *g x*ð Þ¼ , *y g*0. The specific case of circular dielectric material in the center of the cross section is shown in **Figure 2(b)** by using the image method. We obtain

#### **Figure 2.**

*The image method for (a) an arbitrary profile in the cross section, and (b) the specific case of circular dielectric material in the center of the cross section.*

$$\begin{aligned} \mathbf{g}(n,m) &= \frac{\mathbf{g}\_0}{abk\_\circ} \int\_{x\_{11}}^{x\_{12}} \left[ \sin\left(k\_\circ y\_{12}(\mathbf{x})\right) - \sin\left(k\_\circ y\_{11}(\mathbf{x})\right) \right] \cos\left(k\_\circ x\right) d\mathbf{x} \\\ = \frac{\mathbf{g}\_0}{am\pi} \int\_{x\_{11}}^{x\_{12}} \sin\left[\frac{m\pi}{2b}\left(y\_{12}(\mathbf{x}) - y\_{11}(\mathbf{x})\right)\right] \cos\left[\frac{m\pi}{2b}\left(y\_{12}(\mathbf{x}) + y\_{11}(\mathbf{x})\right)\right] \cos\left(\frac{n\pi}{a}x\right) d\mathbf{x}, \end{aligned} \tag{14}$$

$$y\_{11}(\mathbf{x}) = b/2 - \sqrt{r^2 - (\mathbf{x} - a/2)^2},\tag{15}$$

$$\mathcal{Y}\_{12}(\mathbf{x}) = b/2 + \sqrt{r^2 - \left(\mathbf{x} - a/2\right)^2} \tag{16}$$

*Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

The dielectric profile for the cross section (**Figure 1(a)**) is given by.

$$\log(n, m \neq 0) = \frac{2\mathbf{g}\_0}{am\pi} \int\_{x\_{11}}^{x\_{12}} \sin\left[\frac{m\pi}{2b} \left(y\_{12}(\mathbf{x}) - y\_{11}(\mathbf{x})\right)\right] \cos\left[\frac{m\pi}{2b} \left(y\_{12}(\mathbf{x}) + y\_{11}(\mathbf{x})\right)\right] \cos\left(\frac{n\pi}{a}\mathbf{x}\right) d\mathbf{x},\tag{17}$$

$$\mathbf{g}(n,m=\mathbf{0}) = \frac{\mathbf{g}\_0}{ab} \int\_{x\_{11}}^{x\_{12}} \left(\boldsymbol{\upchi}\_{12}(\mathbf{x}) - \boldsymbol{\upchi}\_{11}(\mathbf{x})\right) \cos\left(\frac{n\pi}{a}\mathbf{x}\right) d\mathbf{x},\tag{18}$$

where *y*12ð Þ� *x y*11ð Þ¼ *x* 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>r</sup>*<sup>2</sup> � ð Þ *<sup>x</sup>* � *<sup>a</sup>=*<sup>2</sup> <sup>2</sup> q and *y*12ð Þþ *x y*11ð Þ¼ *x b*.

The cyclic matrix **G** is given as follows. The Fourier transform is applied to the transverse dimension

$$\mathfrak{g}\left(k\_{\mathbf{x}},k\_{\mathbf{y}}\right) = F\{\mathbf{g}(\mathbf{x},\mathbf{y})\} = \int\_{\mathbf{x}}\int\_{\mathbf{y}} \mathbf{g}(\mathbf{x},\mathbf{y}) e^{-jk\_{\mathbf{x}}\mathbf{x} - jk\_{\mathbf{y}}\mathbf{y}} d\mathbf{x} d\mathbf{y}.\tag{19}$$

The components are organized in a vectorial notation as follows

$$\mathbf{E} = \begin{bmatrix} \overline{E}\_{-N,-M} \\ \vdots \\ \overline{E}\_{-N,+M} \\ \vdots \\ \overline{E}\_{+n,+m} \\ \vdots \\ \overline{E}\_{+N,+M} \end{bmatrix} . \tag{20}$$

The Fourier components of the dielectric profile are calculated in the Fourier space. The convolution operation

$$\overline{\mathbf{g}}^{\*}\overline{E} = \left\{ \sum\_{n'=-N}^{N} \sum\_{m'=-M}^{M} \mathbf{g}\_{n-n',m-m'} E\_{n',m'} \right\} \tag{21}$$

is written in a matrix form as **G***E* where

$$\overline{\mathbf{g}}(n,m)(n',m') = \mathbf{g}\_{n-n',m-m'} \tag{22}$$

and the matrix order is (2 N + 1)(2 M + 1), where *E* is the electric field.

The convolution operation is expressed by the cyclic matrix **G** which consists of Fourier components of the dielectric profile *gnm*. Thus, the cyclic matrix **G** is given by the form

$$\mathbf{G} = \begin{bmatrix} \mathcal{g}\_{00} & \mathcal{g}\_{-10} & \mathcal{g}\_{-20} & \cdots & \mathcal{g}\_{-nm} & \cdots & \mathcal{g}\_{-NM} \\\\ \mathcal{g}\_{10} & \mathcal{g}\_{00} & \mathcal{g}\_{-10} & \cdots & \mathcal{g}\_{-(n-1)m} & \cdots & \mathcal{g}\_{-(N-1)M} \\\\ \mathcal{g}\_{20} & \mathcal{g}\_{10} & \ddots & \ddots & \ddots & & \\ \vdots & \mathcal{g}\_{20} & \ddots & \ddots & \ddots & & \\ \mathcal{g}\_{nm} & \ddots & \ddots & \ddots & \mathcal{g}\_{00} & \vdots \\\\ \vdots & & & & & \\ \mathcal{g}\_{NM} & \cdots & \cdots & \cdots & \cdots & \mathcal{g}\_{00} \end{bmatrix}. \tag{23}$$

The derivative of the dielectric profile is given by

$$\mathbf{g}\_{\mathbf{x}}(n,m) = \frac{2\mathbf{g}\_{0}}{am\pi} \bigg|\_{\mathbf{x}\_{11}}^{\mathbf{x}\_{12}} \sin\left[\frac{m\pi}{2b}\left(y\_{12}(\mathbf{x}) - y\_{11}(\mathbf{x})\right)\right] \cos\left[\frac{m\pi}{2b}\left(y\_{12}(\mathbf{x}) + y\_{11}(\mathbf{x})\right)\right] \cos\left(\frac{m\pi}{a}x\right) d\mathbf{x},\tag{24}$$

where *y*<sup>11</sup> and *y*<sup>12</sup> are given according to Eqs (15) and (16). Similarly, we can calculate the value of *gy*ð Þ *n*, *m* , where *gy*ð Þ¼ *x*, *y* ð Þ 1*=ε*ð Þ *x*, *y* ð Þ *dε*ð Þ *x*, *y =dy* .

#### **3.2 Calculation for the circular hollow core in the center of the cross section**

**Figure 3(a)–(c)** shows the extending of the waveguide region in all cases to a four-fold larger region, according to the image method. The image method and periodic replication are needed for fulfilling the boundary condition of the metallic waveguide. **Figure 3(a)** shows the hollow waveguide where the circular hollow core is located in the center of the cross section. This figure represents an example of the complementary shape of **Figure 3(c)**. **Figure 3(b)** shows the cross section entirely filled with the dielectric material. **Figure 3(c)** shows the cross section where the circular dielectric material is located in the center.

Note that the problem shown in **Figure 3(a)** is more complicated than the problem shown in **Figure 3(c)**, and the technique for solving this inhomogeneous problem in the cross section based on the image method is not effective for the specific case shown in **Figure 3(a)**. Thus the proposed technique for calculating the dielectric profile of this problem is based on the fact that this figure represents an example of the complementary shape of **Figure 3(c)**.

In order to solve any inhomogeneous problem in the cross section (e.g., **Figure 3 (a)** and **(c)**), more than one technique can be proposed for the same mode-model method.

The proposed technique to calculate the dielectric profile for the cross section as shown in **Figure 3(a)** for hollow waveguide is based on subtracting the dielectric profile of the waveguide with the dielectric material in the core (**Figure 3(c)**) from the dielectric profile of the waveguide filled entirely with the dielectric material (**Figure 3(b)**).

**Figure 3.**

*Extending the waveguide region in all cases to a four-fold larger region, according to the image method. (a) The hollow waveguide where the circular hollow core is located in the center of the cross section. (b) The cross section entirely filled with dielectric material. (c) Circular dielectric material is located in the center of the cross section.*

*Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

#### **4. Numerical results**

This section presents several examples for the different geometries of two specific examples of the complementary shapes of dielectric profile in the cross section, as shown in **Figure 1(a) and (b)**. The solutions are demonstrated as a response to a halfsine (TE10) input-wave profile.

A comparison with the known transcendental Equation [20] according to **Figure 4(a)** is given in order to examine the validity of the theoretical model. The known solution for the dielectric slab modes based on the transcendental Equation [20] is given as follows:

$$E\_{\mathcal{V}^1} = j \frac{k\_{\pi}}{\varepsilon\_0} \sin \left( \omega \mathbf{x} \right) \qquad \quad \mathbf{0} < \mathbf{x} < t \tag{25}$$

$$E\_{p2} = j \frac{k\_x}{\varepsilon\_0} \frac{\sin\left(\nu t\right)}{\cos\left(\mu(t - a/2)\right)} \cos\left[\mu(\infty - a/2)\right] \qquad \qquad t < \infty < t + d \tag{26}$$

$$E\_{\mathcal{V}^3} = j \frac{k\_x}{\varepsilon\_0} \sin \left[ \nu (a - \varkappa) \right] \qquad \qquad t + d < \varkappa < a,\tag{27}$$

where *ν* � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *k*2 *<sup>o</sup>* � *<sup>k</sup>*<sup>2</sup> *z* q and *μ* � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *εrk*<sup>2</sup> *<sup>o</sup>* � *<sup>k</sup>*<sup>2</sup> *z* q result from the transcendental equation

$$\left(\frac{a-d}{d}\right)\frac{d\mu}{2}\tan\left(\frac{d\mu}{2}\right) - \left(t\nu\right)\cot\left(t\nu\right) = 0.\tag{28}$$

The solution obtained for the wave profile ((25)–(27)) describes a symmetrical mode of the dielectric slab. This mode is substituted as an input wave at z = 0 to the solution of the proposed theoretical model Eq. (2).

The comparison between the theoretical model (Eq. (2)) and the transcendental equation (Eqs (25)–(27)) is shown in **Figure 4(b)** for the dielectric slab in a rectangular metallic waveguide (**Figure 4(a)**) and the convergence of our theoretical results is shown in **Figure 4(c)**.

**Figure 4.**

*(a) A dielectric slab in a rectangular metallic waveguide. (b) A comparison between the theoretical model (Eq. (2)) and the transcendental equation (Eqs (25)–(27)) according to Ref. [20], where a = 2b = 2 cm, d = 3.3 mm, εr= 9, and λ= 6.9 cm. (c) The convergence of our theoretical results.*

The comparison is demonstrated for every order (*N* = 1, 3, 5, 7, and 9). The order *N* determines the accuracy of the solution. The convergence of the solution is verified by the criterion for the *Ey* component of the fields.

The convergence of the solution is verified by the criterion

$$C(N) \equiv \log \left\{ \frac{\max \left( |E\_{\mathcal{Y}}^{N+2} - E\_{\mathcal{Y}}^{N}| \right)}{|\max \left( E\_{\mathcal{Y}}^{N+2} \right) - \min \left( E\_{\mathcal{Y}}^{N}| \right)|} \right\}, \qquad \qquad N \ge 1. \tag{29}$$

where the number of the modes is equal to 2ð Þ *<sup>N</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> . The order *N* determines the accuracy of the solution.

#### **Figure 5.**

*The output field where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2.5 mm, r is the radius of the circular dielectric material, and for (a) ε<sup>r</sup> = 3, for (b) ε<sup>r</sup> = 5, for (c) ε<sup>r</sup> = 7, and for (d) ε<sup>r</sup> = 9. (e) The output field in the same cross section of the results (a)–(d) for* x*-axis and where y = b/2 = 10 mm, for the values of ε<sup>r</sup> = 3, 5, 7, and 9, respectively. The other parameters are a = b = 20 mm, k*<sup>0</sup> *= 167 1/m, λ = 3.75 cm, and* β *=58 1/m.*

#### *Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

If the value of the criterion (Eq. (29)) is less than �2, then the numerical solution is well converged. When *N* increases, then *Ey*ð Þ *N* approaches *Ey*. The value of the criterion between *N* = 7 and *N* = 9 is equal to �2.38 ≃ �2, namely a hundredth part. Comparison between the theoretical mode-model (Eq. (2)) and the known model [20] shows good agreement.

**Figure 5(a)–(e)** shows the output field where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, *ε<sup>r</sup>* = 3, 5, 7, 9, for r = 2.5 mm, where r is the radius of the circular dielectric material. The output field in the same cross section of the results **Figure 5 (a)–(d)** are shown in **Figure 5(e)** for the x-axis and where y = b/2 = 10 mm, for the

#### **Figure 6.**

*The output field where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2 mm, where r is the radius of the circular dielectric material, and for (a) ε<sup>r</sup> = 3, for (b) ε<sup>r</sup> = 5, for (c) ε<sup>r</sup> = 7, and for (d) ε<sup>r</sup> = 9. (e) The output field in the same cross section of the results (a)–(d) for* x*-axis and where y = b/2 = 10 mm, for the values of ε<sup>r</sup> = 3, 5, 7, and 9, respectively. The other parameters are a = b = 20 mm, k*<sup>0</sup> *= 167 1/m, λ = 3.75 cm, and* β *= 58 1/m.*

values of *ε<sup>r</sup>* = 3, 5, 7, and 9, respectively. **Figure 6(a)–(e)** demonstrates the output fields by changing only the parameter of the radius of the circular dielectric material from r = 2.5 to r = 2. The other parameters are a = b = 20 mm, *k*<sup>0</sup> = 167 1*=*m, *λ* = 3.75 cm, and *β* = 58 1*=*m.

**Figure 7(a)–(e)** shows the output field where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, *ε<sup>r</sup>* = 1.5, 1.6, 1.7, 1.8, for r = 2.5 mm, where r is the radius of the circular hollow core. The output field in the same cross section of the results **Figure 7 (a)–(d)** are shown in **Figure 7(e)** for *x*-axis and where y = b/2 = 10 mm, for the values of *ε<sup>r</sup>* = 1.5, 1.6, 1.7, and 1.8, respectively. **Figure 8(a)–(e)** demonstrates the output

#### **Figure 7.**

*The output field where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2.5 mm, where r is the radius of the circular hollow core, and for (a) ε<sup>r</sup> = 1.5, for (b) ε<sup>r</sup> = 1.6, for (c) ε<sup>r</sup> = 1.7, and for (d) ε<sup>r</sup> = 1.8. (e) The output field in the same cross section of the results (a)–(d) for* x*-axis and where y = b/2 = 10 mm, for the values of ε<sup>r</sup> = 1.5, 1.6, 1.7, and 1.8, respectively. The other parameters are a = b = 20 mm, k*<sup>0</sup> *= 167 1/m, λ = 3.75 cm, and* β *= 58 1/m.*

*Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

fields by changing only the parameter of the radius of the circular hollow core from r = 2.5 to r = 2. The other parameters are a = b = 20 mm, *k*<sup>0</sup> = 167 1/m, *λ* = 3.75 cm, and *β* = 58 1/m.

By increasing only the dielectric constant from *ε<sup>r</sup>* = 3 to *ε<sup>r</sup>* = 9, according to **Figures 5(a)–(e)** and **6(a)–(e)**, and from *ε<sup>r</sup>* = 1.5 to *ε<sup>r</sup>* = 1.8, according to **Figures 7(a)–(e)** and **8(a)–(e)**, the Gaussian shape of the output transverse profile of the field increased, the TE10 wave profile decreased, and the relative amplitude of the output field decreased.

We can predict the waveguide parameters (*ε<sup>r</sup>* and r) for obtaining the Gaussian behavior of the output field in all case. The cross section in the first interesting

#### **Figure 8.**

*The output field where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where a = b = 20 mm, and r = 2 mm, where r is the radius of the circular hollow core, and for (a) ε<sup>r</sup> = 1.5, for (b) ε<sup>r</sup> = 1.6, for (c) ε<sup>r</sup> = 1.7, and for (d) ε<sup>r</sup> = 1.8. (e) The output field in the same cross section of the results (a)–(d) for* x*-axis and where y = b/2 = 10 mm, for the values of ε<sup>r</sup> = 1.5, 1.6, 1.7, and 1.8, respectively. The other parameters are a = b = 20 mm, k*<sup>0</sup> *= 167 1/m, λ = 3.75 cm, and* β *=58 1/m.*

case consists of circular dielectric material in the center of the cross section (**Figure 1 (a)**). The cross section in the second interesting case consists of a circular hollow core in the center of the cross section (**Figure 1(b)**). The output results refer to the same parameters a = b = 20 mm, *k*<sup>0</sup> = 167 1/m, *λ* = 3.75 cm, and *β* = 58 1/m. According to the results of the first case, in order to obtain the Gaussian behavior, the values of *ε<sup>r</sup>* = 3, 5, 7, 9 and r = 2 or r = 2.5 are needed. In the second case, in order to obtain the Gaussian behavior, the values of *ε<sup>r</sup>* = 1.5, 1.6, 1.7, and 1.8 and r = 2 or r = 2.5 are needed.

These results are strongly affected by the different parameters *ε<sup>r</sup>* and r, and for the same other parameters of *k*<sup>0</sup> = 167 1/m, *λ* = 3.75 cm, *β* = 58 1/m, and the dimensions of the rectangular cross section.

### **5. Conclusions**

The wavelet transform creates a representation of the signal in both the time and frequency domain in order to allow efficient access of localized information about the signal. A set of waveforms comprising a transform is called a basis function. Fourier transforms use only sine and cosine waves as their basic functions, namely a signal is decomposed into a series of sine and cosine functions or wavelets by the FFT. Examples for the applications of wavelet transform are demonstrated in [2–5]. The proposed method in this chapter is based on the Fourier transform that creates a representation of the signal in the frequency domain.

Two specific examples of complementary shapes of dielectric profile in the cross section were introduced in this chapter. In the first case, the cross section consists of circular dielectric material in the center of the cross section. In the second case, the cross section shows the complementary shape of the cross section of the first case, as an example of a hollow waveguide in which the circular hollow core is located in the center of the cross section.

Note that the problem shown in **Figure 3(a)** is more complicated than the problem shown in **Figure 3(c)**, and the technique for solving this inhomogeneous problem in the cross section based on the image method is not effective for the specific case shown in **Figure 3(a)**. The proposed technique for calculating the dielectric profile of the problem shown in **Figure 3(a)** is based on the fact that this figure represents an example of the complementary shape of **Figure 3(c)**.

In order to solve any inhomogeneous problem in the cross section (e.g., **Figure 3(a)** and **(c)**), more than one technique can be proposed for the same mode-model method.

The proposed technique to calculate the dielectric profile for the cross section as shown in **Figure 3(a)** for hollow waveguide is based on subtracting the dielectric profile of the waveguide from the dielectric material in the core (**Figure 3(c)**) from the dielectric profile of the waveguide filled entirely with the dielectric material (**Figure 3(b)**).

**Figures 5(a)–(e)** and **6(a)–(e)** demonstrate the output fields, where the circular dielectric material is located in the center of the cross section of the straight rectangular waveguide, where the parameter r refers to the radius of the circular dielectric material. **Figures 7(a)–(e)** and **8(a)–(e)** demonstrate the output fields, where the circular hollow core is located in the center of the cross section of the straight rectangular waveguide, where the parameter r refers to the radius of the circular hollow core. The other parameters are a = b = 20 mm, *k*<sup>0</sup> = 167 1/m, *λ* = 3.75 cm, and *β* = 58 1/m.

*Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

By increasing only the dielectric constant from *ε<sup>r</sup>* = 3 to *ε<sup>r</sup>* = 9, according to **Figures 5(a)–(e)** and **6(a)–(e)**, and from *ε<sup>r</sup>* = 1.5 to *ε<sup>r</sup>* = 1.8, according to **Figures 7(a)–(e)** and **8(a)–(e)**, the Gaussian shape of the output transverse profile of the field increased, the TE10 wave profile decreased, and the relative amplitude of the output field decreased.

We can predict the waveguide parameters (*ε<sup>r</sup>* and r) for obtaining the Gaussian behavior of the output field in all cases. The output results refer to the same parameters a = b = 20 mm, *k*<sup>0</sup> = 167 1/m, *λ* = 3.75 cm, and *β* = 58 1/m. According to the results of the first case, in order to obtain the Gaussian behavior, the values of *ε<sup>r</sup>* = 3, 5, 7, 9 and r = 2 or r = 2.5 are needed. In the second case, in order to obtain the Gaussian behavior, the values of *ε<sup>r</sup>* = 1.5, 1.6, 1.7, and 1.8 and r = 2 or r = 2.5 are needed.

The results are strongly affected by the different parameters *ε<sup>r</sup>* and r, and for the same other parameters of *k*<sup>0</sup> = 167 1/m, *λ* = 3.75 cm, *β* = 58 1/m, and the dimensions of the rectangular cross section.

The applications are useful for straight rectangular waveguides in millimeter regimes, where the circular dielectric material is located in the center of the cross section, and also for hollow waveguides, where the circular hollow core is located in the center of the cross section.

#### **Appendix A**

The wavelet transform creates a representation of the signal in both the time and frequency domain in order to allow efficient access of localized information about the signal. A set of waveforms comprising a transform is called a basis function. Fourier transforms use only sine and cosine waves as its basic functions, namely a signal is decomposed into a series sine and cosine functions or wavelets by the FFT. Examples for the applications of wavelet transform are demonstrated in [2–5]. The proposed method in this chapter is based on the Fourier transform that creates a representation of the signal in the frequency domain. The main points of the proposed method and the proposed technique are:


$$
\tilde{a}(\varsigma) = L\{a(\zeta)\} = \int\_{\zeta=0}^{\infty} a(\zeta)e^{-\imath \zeta} d\zeta,\tag{30}
$$

is applied on the z-dimension, where *a z*ð Þ represents any z-dependent variables of the wave equations.

3.A Fourier transform is applied on the transverse dimension

$$\overline{\mathbf{g}}\left(\mathbf{k}\_{\mathbf{x}},\mathbf{k}\_{\mathbf{y}}\right) = F\{\mathbf{g}(\mathbf{x},\mathbf{y})\} = \int\_{\mathbf{x}}\int\_{\mathbf{y}} \mathbf{g}(\mathbf{x},\mathbf{y}) e^{-jk\_{\mathbf{x}}\mathbf{x} - jk\_{\mathbf{y}}\mathbf{y}} d\mathbf{x} d\mathbf{y},\tag{31}$$

and the differential equations are transformed to an algebraic form in the (*ω*, *s*, *kx*, *ky*) space.

4.The method of images is applied to satisfy the conditions *n*^ � *E* ¼ 0 and *n*^ � ð Þ¼ ∇ � *E* 0 on the surface of the ideal metallic waveguide walls, where *n*^ is a unit vector perpendicular to the surface. The dielectric profile, *g x*ð Þ , *y* , is defined inside the waveguide boundaries, 0 ≤*x*≤*a* and 0 ≤*y*≤ *b*. In order to maintain the boundary conditions without physical metallic walls, a substitute physical problem is constructed with infinite transverse extent. The periodicity and the symmetry properties are chosen to force the boundary conditions at the location of the walls in the real problem. This is done by extending the waveguide region 0≤ *x*≤*a*, 0≤*y*≤ *b* to a four-fold larger region. Hence, the following relations are yielded

$$\mathbf{g}(-\mathbf{x}, \mathbf{y}) = \mathbf{g}(\mathbf{x}, -\mathbf{y}) = \mathbf{g}(\mathbf{x}, \mathbf{y}) = \mathbf{g}(-\mathbf{x}, -\mathbf{y}),\tag{32}$$

$$E\_{\mathbf{x}}(\mathbf{x}, -\mathbf{y}) = -E\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}) \quad , \quad E\_{\mathbf{x}}(-\mathbf{x}, \mathbf{y}) = E\_{\mathbf{x}}(\mathbf{x}, \mathbf{y}). \tag{33}$$

The region �*a*≤*x*≤*a*, � *b*≤ *y*≤*b* is then further extended to infinity by periodic replication, *g x*ð þ 2ℓ*a*, *y* þ 2*kb*Þ ¼ *g x*ð Þ , *y* , where �∞ <ℓ, *k*< ∞. The field components are periodically, namely, *Ex*ð*x* þ 2ℓ*a*, *y* þ 2*kb*Þ ¼ *Ex*ð Þ *x*, *y* for �∞ <ℓ, *k*< ∞. The substitution of the physical problem is equivalent to the original problem in the region 0 ≤*x*≤*a*, 0≤*y* ≤*b*, and satisfies the same boundary conditions on the boundary of this region. The discrete Fourier transform series is given with *kx* ¼ *nπ=a* and *ky* ¼ *mπ=b*, and the transverse wavenumbers are given by *kox* ¼ *π=a*, and *koy* ¼ *π=b*, where *a* and *b* are the transverse dimensions of the rectangular boundaries. We substitute *kx* ¼ *nkox* and *ky* ¼ *mkoy*, where the integers n and m are truncated by �*N* ≤ *n*≤ *N* and �*M* ≤ *m* ≤ *M*, respectively. The orders *N* and *M* determine the accuracy of the solution.

5.The output transverse field profiles are given by the inverse Laplace and Fourier transforms, as follows

$$\mathbf{E}\_{\mathbf{y}}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \sum\_{\mathbf{n}} \sum\_{\mathbf{m}} \left[ \int\_{\sigma - \mathbf{j}\_{\text{v}}}^{\sigma + \mathbf{j}\_{\text{v}}} \mathbf{E}\_{\mathbf{y}}(\mathbf{n}, \mathbf{m}, \mathbf{s}) \exp\left[j \mathbf{n} \mathbf{k}\_{\text{ox}} \mathbf{x} + j \mathbf{m} \mathbf{k}\_{\text{oy}} \mathbf{y} + \mathbf{s} \mathbf{z}\right] \mathbf{ds},\tag{34}$$

where the inverse Laplace transform is calculated according to the Salzer method [17, 18].


*Straight Rectangular Waveguide for Circular Dielectric Material in the Cross Section… DOI: http://dx.doi.org/10.5772/intechopen.104815*

## **Author details**

Zion Menachem Department of Electrical Engineering, Shamoon College of Engineering, Beer Sheva, Israel

\*Address all correspondence to: zionme@sce.ac.il; zionmm@gmail.com

© 2022 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.

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## *Edited by Francisco Bulnes*

Today, more sophisticated techniques are necessary for spectral analysis, reconstruction, restoration of signals, their digital and analogic processing, specialized signal diagnostics, and short intervals with occurrences that require greater speed and precision. As the frequency domain of the wavelet transform gets more detailed and considerably more specific in various applications, it necessitates specialized scholarly attention in its many forms and relationships with other transforms with special functions. For example, using the wavelet transform with special functions can prove valuable in creating and designing special signal filters or the interphase between reception-emission devices with specialized sensors for medical use. In quantum phenomena, its corresponding version of the wavelet transform is instrumental in the spectral study of particles and their correlation. Therefore, using specialists' and experts' views, this book delves into an exposition on spectral analysis, restoring, monitoring, and signal processing, as well as essential applications required in waveguides and for the improvement of medical images, proving the wavelet transform to be helpful in resolution analysis in time-frequency, with an emphasis on different methods of the calculus using FFT and DSTFT. This book has been divided into four sections covering all the abovementioned subjects.

Published in London, UK © 2022 IntechOpen © geralt / pixabay

Recent Advances in Wavelet Transforms and Their Applications

Recent Advances in

Wavelet Transforms

and Their Applications

*Edited by Francisco Bulnes*