## 6 Results and Discussions
### Time-Domain Analysis  
The federal communication commission's (FCC) specifications for designing ultra-wideband (UWB) antennas necessitate additional performance parameters beyond those required for standard antennas. The UWB provides an unlicensed 7.5 GHz frequency band  
Figure 21: Gain versus frequency curves of the (**a**) UWB and (**b**) dual-band notch antennas.  
Figure 20: Simulated and measured radiation patterns of the UWB notch antenna. (**a**) 4.2 GHz, (**b**) 7 GHz, and (**c**) 9.5 GHz, and (**d**) the measurement setup.  
ranging from 3.1 to 10.6 GHz for short-range connectivity. A UWB transmits and receives signals using short pulses for this intent. Due to the obvious transmission link and distance, signal distortion and dispersion are possible during communication. The transfer function, phase response, group delay, and fidelity factors need to be examined for the linear phase between the input and output signals. To investigate the aforementioned metrics, two identically designed antennas are aligned face-to-face and side-to-side, situated at a distance of 100 mm apart. The transfer function curves of both the UWB and the UWB band notch are depicted in Figure 22. The transfer function/isolation describes the amount of correlation between the elements. The UWB and UWB band notch antennas have isolation values of less than \(-40\) dB and \(-35\) dB throughout the antennas' operating frequency in both the face-to-face and side-to-side arrangements. These values indicate that the antennas have negligible correlation effects. The phase response curves indicate linearity in the transmitted and received signals. The phase responses of the projected UWB and UWB band notch are depicted in Figure 23. It can be witnessed that there is a negligible variation in the peaks of the phase response curve except for the frequency band, which is filtered out. The phase response curve signifies the linear phase between the input and output signals.  
A further important time-domain parameter is the group delay, which is calculated quantitatively as the negative derivative of the phase variation in accordance with the angular frequency. As the signal propagates, it experiences amplitude and phase disturbance. The input signal may have numerous frequency components. For such signals, the group delay signifies the phase linearity of the system, and it is described by Equation (11). The acceptable value of the group delay is less than 1 ns.  
\[\tau_{g}(\omega)=-\frac{d\varphi(\omega)}{d\omega} \tag{11}\]  
Figure 23: Phase responses of the antennas: (**a**) UWB, and (**b**) UWB band notch.  
Figure 22: The transfer functions of the antennas: (**a**) UWB, and (**b**) UWB band notch.  
The group delays of the projected antennas are depicted in Figure 24, and they are less than 1 ns in both configurations--side-to-side and face-to-face--except for the notched frequency bands. The transmitted and received pulse shapes are analyzed using another significant metric, the fidelity factor (FF). This parameter signifies the amount of pulse shape distortion, not the amplitude of the pulse. The fidelity factors for the projected antennas are computed in both the face-to-face and side-to-side orientations. The input and out signals are normalized to ensure that the signals' shapes are examined, and they are depicted in Figure 25. The similarity between the input and output signals is computed using Equations (12)-(14), and they are tabulated in Table 4.  
\[T_{s}^{n}=\frac{T_{s}(t)}{\sqrt{\int_{-\infty}^{\infty}\left|T_{s}(t)\right|^{2 }}dt}\qquad, \tag{12}\]  
\[R_{s}^{n}=\frac{R_{s}(t)}{\sqrt{\int_{-\infty}^{\infty}\left|R_{s}(t)\right|^{2 }}dt},\text{ and } \tag{13}\]  
\[\text{FF}=\max\int_{-\infty}^{\infty}T_{s}^{n}(t)\ \ R_{s}^{n}(t+\tau)dt. \tag{14}\]  
Figure 24: The group delay of the antennas: (**a**) UWB, and (**b**) UWB band notch.  
Figure 25: Normalized inputs and outputs of the signals: (**a**) UWB, and (**b**) UWB band notch.