**2. Time domain analysis**

Antennas intended for UWB systems need to possess superior pulse handling capabilities. Analysis of the transient response of the antenna is performed by direct time domain measurements [14, 15] or by a frequency domain measurement followed by Fourier Transformation [16, 17]. Frequency domain measurements take advantage of the high dynamic range and the standardized calibration of the vector network analyzer and is equally accurate to the direct time domain measurements.

In the latter method, the antenna is considered as a linear time invariant system described by its transfer function (gain and phase) and the associated impulse response [18, 19]. Transfer function and impulse response of UWB antennas are modeled as spatial vectors as the antenna characteristics depend on the signal propagation direction [16, 17]. **Figure 1** shows the model for the analysis which consists of a radio link made up of two antennas in free space under the approximation of far–field and line-of-sight propagation. In this model, the transmitting and receiving antennas are characterized by eliminating channel effects [20].

#### **2.1 Transient reception**

The time domain relation between the received voltage pulse *U* ! *Rx*ð Þ *ω*,*r*, *θ*, *φ* and the incident electric field pulse *E* ! *rad*ð Þ *ω*,*r*, *θ*, *φ* is [17],

**Figure 1.** *The UWB channel model.*

*Time Domain Performance Evaluation of UWB Antennas DOI: http://dx.doi.org/10.5772/intechopen.94546*

$$\frac{\overrightarrow{U}\_{\text{Rx}}(o\,,r,\theta,\,\rho)}{\sqrt{\overline{x}\_{\text{c}}}} = \overrightarrow{h}\_{\text{Rx}}(o\,,\theta,\,\rho)\frac{\overrightarrow{E}\_{\text{rad}}(o\,,r,\theta,\,\rho)}{\sqrt{\overline{x}\_{0}}}\tag{1}$$

In the time domain, the corresponding relation is,

$$\frac{\overrightarrow{u}\_{R\mathbf{x}}(t,r,\theta,\rho)}{\sqrt{\mathbf{z}\_{c}}} = \overrightarrow{h}\_{R\mathbf{x}}(t,\theta,\rho) \otimes \frac{\overrightarrow{e}\_{rad}(t,r,\theta,\rho)}{\sqrt{\mathbf{z}\_{0}}} \tag{2}$$

*zc* and *z*<sup>0</sup> are the antenna port and free–space characteristic impedance respectively, while ⊗ indicates convolution operation. If ð Þ *θ*, *φ* represents the elevation and azimuth angles, the receive antenna transfer function *h* ! *Rx*ð Þ *ω*, *θ*, *φ* and its impulse response *h* ! *Rx*ð Þ *t*, *θ*, *φ* are considered as a function of the angle of arrival of the received pulse.

An ideal receiving antenna should receive a voltage pulse of the same shape as the one incident on it from any direction. This means that it should have a Dirac– delta impulse response which is also be independent of the angle of arrival. In other words, the antenna transfer function should have a uniform amplitude and a linear phase response (or constant group delay). However, in practice, the receiving antenna transfer function is always band limited.

#### **2.2 Transient radiation**

The frequency domain relation between the transmitted electric field pulse and the applied voltage pulse is,

$$\frac{\overrightarrow{E}\_{\text{rad}}(o\rho, r, \theta, \varphi)}{\sqrt{\overline{z\_0}}} = \overrightarrow{h}\_{\text{Tx}}(o\rho, \theta, \varphi) \xrightarrow[r]{e^{-j\alpha r/c}} \frac{U\_{\text{Tx}}(o\rho, r, \theta, \varphi)}{\sqrt{\overline{z\_c}}} \tag{3}$$

and,

broadening of the radiated pulse. The same authors in [4] reports a much lesser pulse distortion for a tapered slot antenna operating in the 3.1–10.6 GHz UWB band. The present chapter intends to outline the design, simulation and measure-

Two monopole antennas with similar design topology is considered for the present analysis: the SQMA with a square radiator and RMA with a rectangular radiator. In both the antennas, the ground plane is optimally designed for a wide impedance bandwidth. The general practice to realize ultra wide-bandwidth in square/rectangular radiating elements is to modify the ground–radiator interface [5–10]. In SQMA, the inherent resonances of a square patch is matched over the wide band by incorporating cuts in the ground plane at specific locations [11]. The designs proposed in [11, 12] have been contemplated to design the RMA. In RMA, the PCB footprint is greatly reduced and radiation patterns in the 3.1–10.6 UWB is stable without much radiation squinting which is otherise observed in wideband antennas at higher frequencies primarily due to the current distribution on the antenna ground plane [13]. Incase of RMA, this aspect is taken into consideration in the design ensuring broadside radiation all through the band of operation. This is further confirmed from the spatio-temporal transfer characteristics and the transient response analysis, making it suitable for UWB applications.

Antennas intended for UWB systems need to possess superior pulse handling capabilities. Analysis of the transient response of the antenna is performed by direct time domain measurements [14, 15] or by a frequency domain measurement followed by Fourier Transformation [16, 17]. Frequency domain measurements take advantage of the high dynamic range and the standardized calibration of the vector network analyzer and is equally accurate to the direct time domain measurements. In the latter method, the antenna is considered as a linear time invariant system

described by its transfer function (gain and phase) and the associated impulse response [18, 19]. Transfer function and impulse response of UWB antennas are modeled as spatial vectors as the antenna characteristics depend on the signal propagation direction [16, 17]. **Figure 1** shows the model for the analysis which consists of a radio link made up of two antennas in free space under the approximation of far–field and line-of-sight propagation. In this model, the transmitting and receiving antennas are characterized by eliminating channel effects [20].

The time domain relation between the received voltage pulse *U*

*rad*ð Þ *ω*,*r*, *θ*, *φ* is [17],

!

!

*Rx*ð Þ *ω*,*r*, *θ*, *φ* and

ment steps followed to characterize UWB antennas in the time domain.

**2. Time domain analysis**

*Innovations in Ultra-WideBand Technologies*

**2.1 Transient reception**

**Figure 1.**

**110**

*The UWB channel model.*

the incident electric field pulse *E*

$$
\overrightarrow{h}\_{\text{Tx}}(o\rho,\theta,\varphi) = \frac{j\rho}{2\pi c} \,\overrightarrow{h}\_{\text{Rx}}(o\rho,\theta,\varphi) \tag{4}
$$

The time domain relation between the transmitted electric field pulse *e* ! *rad*ð Þ *t*,*r*, *θ*, *φ* and the applied voltage pulse *u* ! *Tx*ð Þ *t*,*r*, *θ*, *φ* at the transmitting antenna is,

$$\frac{\overrightarrow{\boldsymbol{e}}\_{\text{rad}}(\mathbf{t}, \mathbf{r}, \theta, \varphi)}{\sqrt{\mathbf{z}\_{0}}} = \frac{1}{r} \, \partial \left( t - \frac{r}{c} \right) \otimes \left[ \frac{1}{2\pi c} \, \frac{\partial}{\partial t} \overrightarrow{h}\_{\text{Rx}}(\mathbf{t}, \theta, \varphi) \right] \otimes \frac{u\_{\text{Tx}}(\mathbf{t}, \mathbf{r}, \theta, \varphi)}{\sqrt{\mathbf{z}\_{c}}} \tag{5}$$

where,

$$
\overrightarrow{h}\_{\text{Tx}}(\mathbf{t}, \theta, \rho) = \frac{1}{2\pi c} \frac{\partial}{\partial t} \overrightarrow{h}\_{R\text{x}}(\mathbf{t}, \theta, \rho) \tag{6}
$$

The convolution with the delta function in Eq. (5) represents the time delay attributed to the finite speed of light, denoted by *c*. Eq. (6) indicates that the transmit impulse response *h* ! *Tx*ð Þ *t*, *θ*, *φ* is a time derivative of the receive impulse response *h* ! *Rx*ð Þ *t*, *θ*, *φ* . As a result, an ideal antenna (with *h* ! *Rx*ð Þ¼ *t*, *θ*, *σ δ*ð Þ*t* ) will radiate an electric field pulse which would be a first–order time derivative of the input voltage pulse.

## **2.3 Model of transient transmission**

The full input-to-output characteristics, in both frequency and time domains is deduced from 1 and 3 as below,

$$\begin{aligned} \overrightarrow{\overline{U}}\_{\text{Rx}}(\alpha, r, \theta, \rho) &= \overrightarrow{A}\_{\text{Tx}, \text{Rx}} \frac{e^{-j\alpha r/c}}{r} \overrightarrow{h}\_{\text{Rx}}(\alpha, \theta, \rho) \\ &= \sqrt{\frac{2\alpha}{c}} \overrightarrow{A}\_{\text{Tx}, \text{Rx}} \frac{c}{2r\alpha} \frac{e^{-j\alpha r/c}}{2r\alpha} \sqrt{\frac{2\alpha}{c}} \overrightarrow{h}\_{\text{Rx}}(\alpha, \theta, \rho) \end{aligned} \tag{7}$$

where,

$$\begin{aligned} \stackrel{\rightarrow}{A}\_{\text{Tx},\text{Rx}} &= \frac{j\rho\iota}{2\pi c} \stackrel{\rightarrow}{h}\_{\text{Rx}}(\rho,\theta,\rho) \\\\ \stackrel{\rightarrow}{U}\_{\text{Br}\_2}(\rho,r,\theta,\rho) & \rightarrow & \rightarrow \end{aligned}$$

$$\mathcal{S}\_{\text{21}}(\boldsymbol{\omega}) = \frac{U\_{\text{Rx}}(\boldsymbol{\omega}, \boldsymbol{r}, \boldsymbol{\theta}, \boldsymbol{\varphi})}{U\_{\text{Tx}}(\boldsymbol{\omega}, \boldsymbol{r}, \boldsymbol{\theta}, \boldsymbol{\varphi})} = \vec{H}\_{\text{Tx}}(\boldsymbol{\omega}, \boldsymbol{\theta}, \boldsymbol{\varphi}) \, H\_{\text{Ch}}(\boldsymbol{\omega}) \, \vec{H}\_{\text{Rx}}(\boldsymbol{\omega}, \boldsymbol{\theta}, \boldsymbol{\varphi}) \tag{8}$$

where,

$$
\overrightarrow{H}\_{\text{Tx}}(\rho,\theta,\varphi) = \sqrt{\frac{2\alpha}{c}} \overrightarrow{A}\_{\text{Tx},\text{Rx}} \tag{9}
$$

$$H\_{Ch}(\rho) = \frac{c \ e^{-j\alpha r/c}}{2r\alpha} \tag{10}$$

deduced from Eq. (4). An IFFT operation on this would generate the pulse response

A scattering parameter *S*<sup>21</sup> measurement in the frequency domain can be used to deduce the transfer functions of both the transmitting and receiving antennas. Two identical standared horn antennas oriented in the bore sight direction is used, and considering Eq. (7), the corresponding transfer functions are found out from

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> *S*21ð Þ *ω*, *θ*, *φ HCh*ð Þ *ω*, *θ*, *φ*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> *S*21ð Þ *ω*, *θ*, *φ HCh*ð Þ *ω*, *θ*, *φ*

<sup>2</sup>*d<sup>ω</sup> exp* �*jω<sup>d</sup>*

Once the reference antenna (Tx Antenna) is characterized, transfer function of

*S*<sup>21</sup> is measured with frequency resolution 15.25 MHz and Eqs. (14)–(17) are used to compute the corresponding transfer functions. The data is further appended with zero padding for 0–2 GHz and 12–62.47 GHz range with 4096 points in the pass band. The conjugate of this signal is is then reflected to the negative frequencies to get a double sided spectrum of a real signal which is then transformed to the time

*c*

*S*21ð Þ *ω*, *θ*, *φ*

*m*�<sup>1</sup> � � (14)

ð Þ *m* (15)

� � (16)

*HTx*ð Þ *<sup>ω</sup> HCh*ð Þ *<sup>ω</sup>* (17)

*j* 2*π*

s

*HCh*ð Þ¼ *<sup>ω</sup> <sup>c</sup>*

*HAUT*ð Þ¼ *ω*, *θ*, *φ*

s

*ω c*

2*π j*

*c ω*

*h* !

*Rx*ð Þ *t*, *θ*, *φ* .

**Figure 2.**

**2.5 Measurement**

*(a) Input pulse (b) power spectral density.*

*Time Domain Performance Evaluation of UWB Antennas*

*DOI: http://dx.doi.org/10.5772/intechopen.94546*

*HTx*ð Þ¼ *ω*, *θ*, *φ*

*HRx*ð Þ¼ *ω*, *θ*, *φ*

where the free space transfer function is,

the AUT is found for multiple orientations using,

**2.6 Data processing, windowing**

domain using IFFT.

**113**

$$
\overrightarrow{H}\_{\text{Rx}}(\boldsymbol{\alpha}, \boldsymbol{\theta}, \boldsymbol{\varphi}) = \sqrt{\frac{2\boldsymbol{\alpha}}{c}} \overrightarrow{h}\_{\text{Rx}}(\boldsymbol{\alpha}, \boldsymbol{\theta}, \boldsymbol{\varphi}) \tag{11}
$$

In the time domain, 7 reads,

$$\frac{\overrightarrow{\boldsymbol{u}}\_{R\mathbf{x}}(\boldsymbol{t},\boldsymbol{r},\boldsymbol{\theta},\boldsymbol{\varphi})}{\boldsymbol{u}\_{\text{Tx}}(\boldsymbol{t},\boldsymbol{r},\boldsymbol{\theta},\boldsymbol{\varphi})} = \overrightarrow{\boldsymbol{h}}\_{\text{Tx}}(\boldsymbol{t},\boldsymbol{\theta},\boldsymbol{\varphi}) \otimes \boldsymbol{h}\_{\text{Ch}}(\mathbf{t}) \otimes \overrightarrow{\boldsymbol{h}}\_{R\mathbf{x}}(\boldsymbol{t},\boldsymbol{\theta},\boldsymbol{\varphi}) \tag{12}$$

#### **2.4 Simulation**

For the time domain characterization of the antennas, the fourth derivative of the Gaussian pulse given in Eq. (13) and shown in **Figure 2(a)** is chosen as the input pulse, as this pulse conforms to the FCC spectral mask as shown in **Figure 2(b)** when A = 0.333 and T = 0.175 nS.

$$s\_i(t) = A \cdot \left[ 3 - 6\left(\frac{4\pi}{T^2}\right)(t-\tau)^2 + \left(\frac{4\pi}{T^2}\right)^2(t-\tau)^4 \right] e^{-2\pi\left(\frac{4\pi}{T}\right)^2} \quad \left(\frac{V}{m}\right) \tag{13}$$

For the antennas discussed in the later sections of this chapter, the input voltage *uTx*ð Þ*t* (= *si*ð Þ*t* ) is specified in CST Microwave Studio and the radiated pulse *e* ! *rad*ð Þ *t*,*r*, *θ*, *φ* is calculated on a sphere of radius 25 cm. Fourier transforms of these two quantities are then calculated and the transfer function *H* ! *Rx*ð Þ *ω*, *θ*, *φ* is

*Time Domain Performance Evaluation of UWB Antennas DOI: http://dx.doi.org/10.5772/intechopen.94546*

#### **Figure 2.** *(a) Input pulse (b) power spectral density.*

deduced from Eq. (4). An IFFT operation on this would generate the pulse response *h* ! *Rx*ð Þ *t*, *θ*, *φ* .

## **2.5 Measurement**

**2.3 Model of transient transmission**

*Innovations in Ultra-WideBand Technologies*

*Rx*ð Þ *ω*,*r*, *θ*, *φ UTx*ð Þ *<sup>ω</sup>*,*r*, *<sup>θ</sup>*, *<sup>φ</sup>* <sup>¼</sup> *<sup>A</sup>*

! *Tx*,*Rx*

r

ffiffiffiffiffiffi 2*ω c*

*A* ! *Tx*,*Rx*

*Tx*,*Rx* <sup>¼</sup> *<sup>j</sup><sup>ω</sup>* 2*πc h* !

*Tx*ð Þ¼ *ω*, *θ*, *φ*

*Rx*ð Þ¼ *ω*, *θ*, *φ*

!

ð Þ *<sup>t</sup>* � *<sup>τ</sup>* <sup>2</sup> <sup>þ</sup>

two quantities are then calculated and the transfer function *H*

" #

*uTx*ð Þ*t* (= *si*ð Þ*t* ) is specified in CST Microwave Studio and the radiated pulse

!

¼

*A* !

*Rx*ð Þ *ω*,*r*, *θ*, *φ UTx*ð Þ *<sup>ω</sup>*,*r*, *<sup>θ</sup>*, *<sup>φ</sup>* <sup>¼</sup> *<sup>H</sup>*

> *H* !

*H* !

*Rx*ð Þ *t*,*r*, *θ*, *φ uTx*ð Þ *<sup>t</sup>*,*r*, *<sup>θ</sup>*, *<sup>φ</sup>* <sup>¼</sup> *<sup>h</sup>*

> *T*2 � �

deduced from 1 and 3 as below,

*U* !

*<sup>S</sup>*21ð Þ¼ *<sup>ω</sup> <sup>U</sup>*

In the time domain, 7 reads,

*u* ! !

where,

where,

**2.4 Simulation**

*e* !

**112**

A = 0.333 and T = 0.175 nS.

*si*ðÞ¼ *<sup>t</sup> <sup>A</sup>* � <sup>3</sup> � <sup>6</sup> <sup>4</sup>*<sup>π</sup>*

The full input-to-output characteristics, in both frequency and time domains is

*h* !

*c e*�*jωr=<sup>c</sup>* 2*rω*

*Rx*ð Þ *ω*, *θ*, *φ*

*Tx*ð Þ *ω*, *θ*, *φ HCh*ð Þ *ω H*

ffiffiffiffiffiffi 2*ω c*

*A* !

r

ffiffiffiffiffiffi 2*ω c*

*h* !

*Tx*ð Þ *t*, *θ*, *φ* ⊗ *hCh*ð Þ*t* ⊗ *h*

For the time domain characterization of the antennas, the fourth derivative of the Gaussian pulse given in Eq. (13) and shown in **Figure 2(a)** is chosen as the input pulse, as this pulse conforms to the FCC spectral mask as shown in **Figure 2(b)** when

> 4*π T*2 � �<sup>2</sup>

For the antennas discussed in the later sections of this chapter, the input voltage

*rad*ð Þ *t*,*r*, *θ*, *φ* is calculated on a sphere of radius 25 cm. Fourier transforms of these

ð Þ *<sup>t</sup>* � *<sup>τ</sup>* <sup>4</sup>

*e* �2*<sup>π</sup> <sup>t</sup>*�*<sup>τ</sup>* ð Þ *<sup>T</sup>*

!

*HCh*ð Þ¼ *<sup>ω</sup> c e*�*jωr=<sup>c</sup>*

r

*Rx*ð Þ *ω*, *θ*, *φ*

ffiffiffiffiffiffi 2*ω c*

*h* !

!

*Rx*ð Þ *ω*, *θ*, *φ* (8)

*Tx*,*Rx* (9)

<sup>2</sup>*r<sup>ω</sup>* (10)

*Rx*ð Þ *ω*, *θ*, *φ* (11)

*Rx*ð Þ *t*, *θ*, *φ* (12)

<sup>2</sup> *V m* � �

*Rx*ð Þ *ω*, *θ*, *φ* is

!

(13)

*Rx*ð Þ *ω*, *θ*, *φ*

(7)

r

*e*�*jωr=<sup>c</sup> r*

> A scattering parameter *S*<sup>21</sup> measurement in the frequency domain can be used to deduce the transfer functions of both the transmitting and receiving antennas. Two identical standared horn antennas oriented in the bore sight direction is used, and considering Eq. (7), the corresponding transfer functions are found out from

$$H\_{\rm Tx}(\boldsymbol{\alpha}, \boldsymbol{\theta}, \boldsymbol{\varphi}) = \sqrt{\frac{\boldsymbol{j}}{2\pi} \left(\frac{\boldsymbol{\alpha}}{c}\right)^2 \frac{\mathcal{S}\_{21}(\boldsymbol{\alpha}, \boldsymbol{\theta}, \boldsymbol{\varphi})}{H\_{\rm Ch}(\boldsymbol{\alpha}, \boldsymbol{\theta}, \boldsymbol{\varphi})}} \tag{14} \qquad \qquad \left(\boldsymbol{m}^{-1}\right) \tag{14}$$

$$H\_{\rm Rx}(\boldsymbol{\alpha},\boldsymbol{\theta},\boldsymbol{\varphi}) = \sqrt{\frac{2\pi}{j} \left(\frac{c}{a\boldsymbol{\nu}}\right)^2 \frac{\mathcal{S}\_{21}(\boldsymbol{\alpha},\boldsymbol{\theta},\boldsymbol{\varphi})}{H\_{\rm Ch}(\boldsymbol{\alpha},\boldsymbol{\theta},\boldsymbol{\varphi})}} \qquad (m) \tag{15}$$

where the free space transfer function is,

$$H\_{Ch}(\rho) = \frac{c}{2d\alpha} \exp\left(\frac{-j\alpha d}{c}\right) \tag{16}$$

Once the reference antenna (Tx Antenna) is characterized, transfer function of the AUT is found for multiple orientations using,

$$H\_{\rm AUT}(o,\theta,\varphi) = \frac{\mathbb{S}\_{21}(o,\theta,\varphi)}{H\_{\rm Tx}(o)H\_{\rm Ch}(o)}\tag{17}$$

#### **2.6 Data processing, windowing**

*S*<sup>21</sup> is measured with frequency resolution 15.25 MHz and Eqs. (14)–(17) are used to compute the corresponding transfer functions. The data is further appended with zero padding for 0–2 GHz and 12–62.47 GHz range with 4096 points in the pass band. The conjugate of this signal is is then reflected to the negative frequencies to get a double sided spectrum of a real signal which is then transformed to the time domain using IFFT.

Measurements using two identical wide band horns are shown in **Figure 3**. While **Figure 3(a)** represents the measured antenna transfer functions, **Figure 3(b)** shows the real and imaginary parts of the transfer functions. Transmitting and receiving antenna impulse responses are shown in **Figure 3(c)** which clearly indicates that *h* ! *Tx*ð Þ*t* is a derivative of *h* ! *Rx*ð Þ*t* .

#### **2.7 Time domain parameters**

The time domain evaluation of the antenna effects on the transmitted/received signals is carried out by analyzing the envelope of the analytic response for co– polarization. The real valued antenna's transient response is,

$$h\_n(t) = \Re \overline{h}\_n(t) \tag{18}$$

*FWHM*0*:*5ð Þ¼ *θ*, *φ t*<sup>2</sup> *<sup>p</sup>=*<sup>2</sup> � *t*<sup>1</sup>

*RINGING<sup>α</sup>* ¼ *t*<sup>2</sup> *<sup>α</sup>*�*<sup>p</sup>* � *t*<sup>1</sup>

from the peak value below a fraction *α* of the main peak.

*Time Domain Performance Evaluation of UWB Antennas*

*DOI: http://dx.doi.org/10.5772/intechopen.94546*

**2.8 Pulse distortion analysis**

!

analytic response *h*

� � � � *p=*2,*t*<sup>1</sup> <*t*<sup>2</sup>

The duration of the ringing is defined as the time until the envelope has fallen

� � � �

The lower bound for *α* is chosen according to the noise floor of the measurement. In order to compare the ringing of antennas with different gains under the condition of a constant noise floor, the fraction *α* is chosen to be 0.22 (�13 dB).

The antenna effects on the received signal is evaluated by convoluting the

For UWB systems, receivers are in general based on the pulse energy detection or correlation with the template waveform. Therefore, the pulse distortions can be

�<sup>∞</sup> *si*ð Þ� *<sup>t</sup> so*ð Þ *<sup>t</sup>* � *<sup>τ</sup> dt* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Ð <sup>þ</sup><sup>∞</sup> �<sup>∞</sup> <sup>∣</sup>*so*ð Þ*<sup>t</sup>* <sup>2</sup> <sup>∣</sup> *dt* <sup>q</sup> (23)

∣ *dt* �

where *τ* is the delay parameter varied in a sense to maximize the numerator. The fidelity parameter measurement is performed for different spatial orientations of the test antenna and is evaluated as the maximum of the cross–correlation function.

To demonstrate the necessity of the time domain analysis of UWB antennas, two

**Figure 4(a)** depicts the design of the SQMA. The overall length of the element determines the lower end of the operational band and the upper end by the feed gap '*d*'. It is the ground plane which determines the impedance bandwidth of a monopole antenna. In the present design, the cuts engraved in the ground plane match the multiple resonances, making the antenna operate over the specified ultra wide band. The SQMA has the following dimensions with units of lengths in *mm*: *s* = 2.3, *g* = 0.48, *l*\_*patch* = 16, *l*

The topology of the RMA comprises of a rectangular monopole of area *l*<sup>1</sup> � *b*<sup>1</sup> as shown in **Figure 4(b)**. This antenna is evolved from a basic printed monopole design shown in the same Figure. To facilitate ultra wide band performance, an impedance transformer is embedded in the CPW transmission line with increasing slot widths *g*1, *g*2, *g*<sup>3</sup> and *g*4. The special design of the transformer provide gradual transformation of the 50 Ω line impedance and at the same time, result in a new current path for

= 14.57, *b* = 13, *d* = 0.5, *w* = 1.4, *L* = 32, *W* = 30, *ε<sup>r</sup>* = 4.4, *tan* ð Þ*δ* = 0.02, *h* = 1.6.

Ð <sup>þ</sup><sup>∞</sup>

Ð <sup>þ</sup><sup>∞</sup> �<sup>∞</sup> <sup>∣</sup>*si*ð Þ*<sup>t</sup>* <sup>2</sup>

*<sup>n</sup>*ð Þ *t*, *θ*, *φ* with the input pulse *si*ð Þ*t* .

*so*ðÞ¼ *t h* !

examined by calculating fidelity factor which is defined as [21],

*F* ¼ *max*

It compares only shapes of the waveforms, not amplitudes.

**3. Performance comparison of UWB antennas**

topologies shown in **Figure 4** is considered.

**3.1 Antenna designs**

**115**

(20)

*<sup>p</sup>*,*t*<sup>1</sup> <sup>&</sup>lt;*t*<sup>2</sup> (21)

*<sup>n</sup>*ð Þ *t*, *θ*, *φ* ⊗ *si*ð Þ*t* (22)

The peak output voltage from an incident wave form depends on the peak value *p*ð Þ *θ*, *φ* of the antenna's transient response:

$$p(\theta,\varphi) = \max \quad \text{t} \; \left| \overrightarrow{h}\_n(t,\theta,\varphi) \right| \tag{19}$$

A measure for the linear distortion of the antenna is the envelope width, which is defined as the full width at half maximum (FWHM) of the magnitude of the transient response envelope.

#### **Figure 3.**

*(a) Transmitting and receiving transfer functions of identical wide band horn antennas (b) transmitting and receiving transfer functions after conjugate reflection (c) impulse response of the transmitting antenna and receiving antenna.*

*Time Domain Performance Evaluation of UWB Antennas DOI: http://dx.doi.org/10.5772/intechopen.94546*

$$\text{FWHM}\_{0.5}(\theta,\varphi) = \mathfrak{t}\_2 \big|\_{p/2} - \mathfrak{t}\_1 \big|\_{p/2, t\_1 < t\_2} \tag{20}$$

The duration of the ringing is defined as the time until the envelope has fallen from the peak value below a fraction *α* of the main peak.

$$R \text{INGING}\_a = t\_2|\_{a \cdot p} - t\_1|\_{p, t\_1 < t\_2} \tag{21}$$

The lower bound for *α* is chosen according to the noise floor of the measurement. In order to compare the ringing of antennas with different gains under the condition of a constant noise floor, the fraction *α* is chosen to be 0.22 (�13 dB).

## **2.8 Pulse distortion analysis**

Measurements using two identical wide band horns are shown in **Figure 3**. While **Figure 3(a)** represents the measured antenna transfer functions, **Figure 3(b)** shows the real and imaginary parts of the transfer functions. Transmitting and receiving antenna impulse responses are shown in **Figure 3(c)** which clearly

> ! *Rx*ð Þ*t* .

The time domain evaluation of the antenna effects on the transmitted/received signals is carried out by analyzing the envelope of the analytic response for co–

!

 

*<sup>n</sup>*ð Þ *t*, *θ*, *φ*

 

The peak output voltage from an incident wave form depends on the peak value

A measure for the linear distortion of the antenna is the envelope width, which is

*(a) Transmitting and receiving transfer functions of identical wide band horn antennas (b) transmitting and receiving transfer functions after conjugate reflection (c) impulse response of the transmitting antenna and*

*<sup>n</sup>*ð Þ*t* (18)

(19)

*hn*ðÞ¼ *t* ℜ*h*

*<sup>p</sup>*ð Þ¼ *<sup>θ</sup>*, *<sup>φ</sup> max t h*!

defined as the full width at half maximum (FWHM) of the magnitude of the

indicates that *h*

!

*Innovations in Ultra-WideBand Technologies*

**2.7 Time domain parameters**

transient response envelope.

**Figure 3.**

**114**

*receiving antenna.*

*p*ð Þ *θ*, *φ* of the antenna's transient response:

*Tx*ð Þ*t* is a derivative of *h*

polarization. The real valued antenna's transient response is,

The antenna effects on the received signal is evaluated by convoluting the analytic response *h* ! *<sup>n</sup>*ð Þ *t*, *θ*, *φ* with the input pulse *si*ð Þ*t* .

$$s\_o(t) = \overrightarrow{h}\_n(t, \theta, \rho) \otimes s\_i(t) \tag{22}$$

For UWB systems, receivers are in general based on the pulse energy detection or correlation with the template waveform. Therefore, the pulse distortions can be examined by calculating fidelity factor which is defined as [21],

$$F = \max \propto \frac{\int\_{-\infty}^{+\infty} \mathbf{s}\_i(t) \cdot \mathbf{s}\_o(t-\tau) \, dt}{\sqrt{\int\_{-\infty}^{+\infty} |\mathbf{s}\_i(t)|^2 |\, dt \cdot \int\_{-\infty}^{+\infty} |\mathbf{s}\_o(t)|^2 |\, dt}} \tag{23}$$

where *τ* is the delay parameter varied in a sense to maximize the numerator. The fidelity parameter measurement is performed for different spatial orientations of the test antenna and is evaluated as the maximum of the cross–correlation function. It compares only shapes of the waveforms, not amplitudes.

## **3. Performance comparison of UWB antennas**

To demonstrate the necessity of the time domain analysis of UWB antennas, two topologies shown in **Figure 4** is considered.

#### **3.1 Antenna designs**

**Figure 4(a)** depicts the design of the SQMA. The overall length of the element determines the lower end of the operational band and the upper end by the feed gap '*d*'. It is the ground plane which determines the impedance bandwidth of a monopole antenna. In the present design, the cuts engraved in the ground plane match the multiple resonances, making the antenna operate over the specified ultra wide band. The SQMA has the following dimensions with units of lengths in *mm*: *s* = 2.3, *g* = 0.48, *l*\_*patch* = 16, *l* = 14.57, *b* = 13, *d* = 0.5, *w* = 1.4, *L* = 32, *W* = 30, *ε<sup>r</sup>* = 4.4, *tan* ð Þ*δ* = 0.02, *h* = 1.6.

The topology of the RMA comprises of a rectangular monopole of area *l*<sup>1</sup> � *b*<sup>1</sup> as shown in **Figure 4(b)**. This antenna is evolved from a basic printed monopole design shown in the same Figure. To facilitate ultra wide band performance, an impedance transformer is embedded in the CPW transmission line with increasing slot widths *g*1, *g*2, *g*<sup>3</sup> and *g*4. The special design of the transformer provide gradual transformation of the 50 Ω line impedance and at the same time, result in a new current path for

13 GHz while that for the RMA are at 3.5 GHz, 5 GHz, 7 GHz and 11 GHz. Both

be effective from 8 GHz onwards. SQMA also shows pattern squinting as in **Figure 6(a)**. For the RMA, the pattern degradation at the higher frequencies are minimum; radiation is stable and directed towards the bore-sight at all frequencies. The cross–polar level exhibited by both the antennas remain better than �20 dB in the 3.1–10.6 GHz band. Both antennas have linear polarization, oriented in the Z direction. Peak gains of the antennas were measured by gain transfer method. Average value of the peak gain in the 3.1–10.6 GHz UWB is found to be 3.75 dBi for

3-D radiation patterns of the two antennas obtained from CST are shown in **Figure 6**. In SQMA, the pattern degradation at the higher frequencies were found to

As an UWB signal is transmitted or received, the pulse shape is altered. This distortion is reflected in the transfer function (in the frequency domain) or impulse response (in the time domain) of the antenna. The antenna transfer function, a complex quantity, would have a constant amplitude and linear phase response while the impulse response would be a delta function, in cases when the antenna has no

!

plane of the antennas are shown in **Figure 7(a)** (Using Eq. (3)). It is seen that

*Simulated radiation patterns at different frequencies for the the antennas (a) SQMA (b) RMA.*

!

*Rx*ð Þ *ω*, *θ*, *φ* and *h*

*Rx*ð Þ *ω*, *θ*, *φ* simulated in CST for the x-y

*Tx*ð Þ *ω*, *θ*, *φ* differ only in their relative

antennas cover the 3.1-10.6 GHz UWB.

*DOI: http://dx.doi.org/10.5772/intechopen.94546*

*Time Domain Performance Evaluation of UWB Antennas*

SQMA and 2.6 dBi for the RMA.

effect on the pulse transmitted.

*3.3.1 Results of simulation in CST*

the intensity plots for *h*

**Figure 6.**

**117**

Magnitude of the transfer function *h*

!

**3.3 Performance evaluation in the time domain**

**Figure 4.** *Design and parameters of (a) SQMA (b) RMA.*

radiation. The coupling of the guided waves to the radiator is strongly dependent on parameter *d* and the resonances in the structure can be matched by suitably choosing *d*. For the RMA, the geometric parameters are: *s* = 2.3, *g* = 0.28, *d* = 4.25, *l*<sup>1</sup> = 9, *b*<sup>1</sup> = 10, *lg* = 17.75, *l*<sup>2</sup> = 6, *b*<sup>2</sup> = 7.3, *g*<sup>1</sup> = 0.45, *g*<sup>2</sup> = 1.35, *g*<sup>3</sup> = 2, *g*<sup>4</sup> = 2.7, *t* = 1, *sf* = 2, *l <sup>f</sup>* = 3, *L* = 30, *W* = 12, *ε<sup>r</sup>* = 4.4, *tan* ð Þ*δ* = 0.02, *h* = 1.6. Unit of all lengths are in *mm*.

## **3.2 Performance evaluation in the frequency domain**

Measured and simulated return losses of the two antennas are shown in **Figure 5**. The prominent resonances in the SQMA are at 3.8 GHz, 8 GHz and

**Figure 5.** *Measured and simulated S*11ð Þ *dB of the (a) SQMA and (b) RMA.*

*Time Domain Performance Evaluation of UWB Antennas DOI: http://dx.doi.org/10.5772/intechopen.94546*

13 GHz while that for the RMA are at 3.5 GHz, 5 GHz, 7 GHz and 11 GHz. Both antennas cover the 3.1-10.6 GHz UWB.

3-D radiation patterns of the two antennas obtained from CST are shown in **Figure 6**. In SQMA, the pattern degradation at the higher frequencies were found to be effective from 8 GHz onwards. SQMA also shows pattern squinting as in **Figure 6(a)**. For the RMA, the pattern degradation at the higher frequencies are minimum; radiation is stable and directed towards the bore-sight at all frequencies. The cross–polar level exhibited by both the antennas remain better than �20 dB in the 3.1–10.6 GHz band. Both antennas have linear polarization, oriented in the Z direction. Peak gains of the antennas were measured by gain transfer method. Average value of the peak gain in the 3.1–10.6 GHz UWB is found to be 3.75 dBi for SQMA and 2.6 dBi for the RMA.

## **3.3 Performance evaluation in the time domain**

As an UWB signal is transmitted or received, the pulse shape is altered. This distortion is reflected in the transfer function (in the frequency domain) or impulse response (in the time domain) of the antenna. The antenna transfer function, a complex quantity, would have a constant amplitude and linear phase response while the impulse response would be a delta function, in cases when the antenna has no effect on the pulse transmitted.
