## 5 Time Domain Performance  
A super wide-band antenna which was used to send very short pulses was greatly affected by dispersion. The pulses sent from the transmitting port will never be the same at the receiving port, but the pulses should be recognized by the receiver. For this reason, time-domain analyses are important to calculate the distortion. To this end, two identical antennas were set up with two different orientations (side by side and face to face). In the far-field region, one of the antennas transmits and other receives the signal. An important feature of the antenna is FF (fidelity factor), which compares the transmitted and received signals between two identical antennas. The transmitted and received pulses are normalized, as shown in Equations (1) and (2), respectively. The normalization of the signals was done to compare only the pulse shapes, and not the magnitudes of the signals, because the receiving signal was expected to be much lower than the transmitted one. Between these two signals, cross-correlation was done at every point in time. When overlaps occurred between both pulses, a maximum value of correlation was obtained. Equation (3) was used to calculate the fidelity factor [30].  
\[T_{s}^{n}=\frac{T_{s}(t)}{\sqrt{\int\limits_{-\infty}^{\infty}\big{|}T_{s}(t) \big{|}^{2}dt}} \tag{1}\]  
\[R_{s}^{n}=\frac{R_{s}(t)}{\sqrt{\int\limits_{-\infty}^{\infty}\big{|}R_{s}(t) \big{|}^{2}dt}} \tag{2}\]  
\[FF=\max\int\limits_{-\infty}^{\infty}T_{s}^{n}(t)R_{s}^{n}(t+\tau)dt \tag{3}\]  
The normalized values of the input and received signal for the side by side and face to face configurations are shown in Figure 7a. The calculated fidelity factor by face to face alignment is 86.7% and for side by side alignment is 82.3%. The second feature of the antenna is to find the group delay, also known as  
Figure 6: (**a**) Measurment setup for E-plane (**b**) measurement setup for H-plane.  
the time delay of the system. This is used to find the phase distortion of the system, which is defined with the help of the following equation [20].  
\[GD=-\frac{d\varphi}{d\omega} \tag{4}\]  
where \"\(\varphi\)\" is the phase of \(\mathbb{S}_{21}\) (transfer function). The group delay of the proposed work for side by side and face to face orientation is shown in Figure 7b, along with the proposed orientation setup. This figure shows that at some frequencies, there are some variations in the group delay, but the values are still less than 1.4 ns for the whole proposed band. The acceptable value of the group delay is up to 3.8 ns [31].  
The third feature of the antenna is the transfer function where two antennas are connected at the two ports of the vector network analyzer and check the phase of \(\mathbb{S}_{21}\) in degree. The phase of \(|\mathbb{S}_{21}|\) for two different orientations is shown in Figure 8, which is almost linear and in constant variation, showing that the proposed antenna has less distortion for both configurations (side by side and face to face).  
Figure 8: Variation of phase (\(\mathbb{S}_{21}\)) in degree for two different configurations.  
Figure 7: (**a**) Input and received signals for face to face and side by side configuration (**b**) Measured group delay (ns) with two different orientations.