## 4 Numerical Results
### 5G Antenna Design

Table 5: Numerical results obtained by Wilcoxon signed-rank test (values below significant level are indicated in bold). In this work, we present the optimization problem of an aperture-coupled E-shaped antenna operating in the n258 (center frequency: 26.4 GHz) and n260 (center frequency: 38.5 GHz) 5G NR frequency bands is investigated. The antenna is designed on a multi-layer \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Nominal Values [101]** & **UC1 26.4 GHz** & **UC2 38.5 GHz** \\ \hline \(W_{antenna}\) & 0.422\(\lambda_{1}\) & 3.23 & 2.21 \\ \hline \(L_{antenna}\) & 0.317\(\lambda_{1}\) & 2.42 & 1.66 \\ \hline \(W_{feed}\) & Feed line width for 50 \(\Omega\) & 0.59 & 0.59 \\ \hline \(L_{feed}\) & 0.739\(\lambda_{2}\) & 4.57 & 3.13 \\ \hline \(L_{aperture}\) & 0.148\(\lambda_{1}\) & 1.13 & 0.78 \\ \hline \(W_{aperture}\) & 0.016\(\lambda_{1}\) & 0.12 & 0.08 \\ \hline \(O_{aperture}\) & 0.211\(\lambda_{2}\) & 1.30 & 0.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Extracted parameter values as they obtained from the design method presented in [101], \(\lambda_{1}\) is the wavelength of the antenna dielectric, and \(\lambda_{2}\) is the wavelength of the feed dielectric (computed values are expressed in mm). Figure 7: Perspective views and computed results of the reflection coefficient as a function of frequency for the two different use cases of an aperture-coupled patch antenna in the millimeter-wave frequency bands based on the design method described in [101] (**a**) use case 1 designed in the n258 frequency band (center frequency: 26.4 GHz), (**b**) use case 2 designed in the n260 frequency band (center frequency: 38.5 GHz), and (**c**) reflection coefficient (magnitude of \(S_{11}\)) as a function of frequency for the two different use cases of an aperture-coupled patch antenna in the millimeter-wave frequency bands. approach (from bottom to top: feed-line, substrate, ground plane, substrate, E-shaped antenna) and optimized (the objects included in the optimization process is the E-shaped antenna and the feed line). It consists of a double dielectric substrate layer; the bottom one consists of Arlon 25N (\(\epsilon_{r}=3.38\), \(tan\delta=0.0025\)) with 0.762 mm thickness and the top one consists of Taconic (\(\epsilon_{r}=2.21\), \(tan\delta=0.001\)) with 1.58 mm thickness. An aperture is placed between the bottom and top substrate layers to feed the E-shaped antenna. Moreover, a modified ground plane is utilized to enhance the characteristics of the antenna. Figure 8 illustrates the proposed antenna geometry indicating the required design parameters. To solve the design problem of the aperture-coupled E-shaped patch antenna, an optimization technique is required. The proposed antenna consists of 15 different design parameters. We address this problem by applying the Whale Optimization Algorithm (WOA) [35] as the optimization method. The selected algorithm is incorporated with the HFSS electromagnetic (EM) solver. The integration of the WOA source code with the EM solver is achieved using an HFSS MATLAB API wrapper. The optimization problem can be defined by the following objective function  
\[ \begin{array}{l}F(\vec{y})=max(S_{11}^{26.4 GHz}(\vec{y}),S_{11}^{38.5 GHz}( \vec{y}))+\Psi\times max(0,S_{11}^{26.4 GHz}(\vec{y})-L_{dB})\\ +\Psi\times max(0,S_{11}^{18.5 GHz}(\vec{y})-L_{dB})\end{array}  \tag{16}\]  
where \(\vec{y}\) is the vector containing the antenna geometry variables for each solution, \(S_{11}^{26.4 GHz}\) and \(S_{11}^{38.5 GHz}\) are the desired \(S_{11}\) magnitudes, \(L_{dB}\) is the \(S_{11}\) limit in dB, and \(\Psi\) is a penalty number (10\({}^{+10}\)). In our case, we set \(L_{dB}=-10\) dB. Table 7 summarizes the best values of the design parameters obtained by the WOA. Figure 8: Design of the aperture-coupled E-shaped antenna (**a**) patch antenna, (**b**) modified ground plane, (**c**) feed line, and (**d**) perspective view of the proposed antenna.