## 4 Numerical Results
### Linear Antenna Array Design
Taking into consideration the above formulation, one of the most common objectives in antenna design is to suppress the SLL of the linear array. This objective can be carried out by finding the optimum positions and phases of the elements in the array. Therefore, this type of problem can be recast as an optimization one by minimizing the following objective function  
\[F(\vec{y},\vec{\phi})=\max_{\theta\in T}\{AF_{dB}(\vec{y},\vec{\phi},\theta)\} \tag{12}\]  
where \(T\) is the set of theta angle values that fall outside the angular range of the main lobe of the linear array. Figure 5: Schematic diagram of the geometry in a linear antenna array. Generally speaking, the formulation of the directivity of the linear array in (8) can be expressed as [91]  
\[D=\frac{\left(\sum\limits_{m=-M}^{M}A_{m}\right)^{2}}{\sum\limits_{m=-M}^{M}\sum \limits_{n=-M}^{M}A_{n}A_{m}e^{i\left(\phi_{m}-\phi_{n}\right)}\frac{\sin\left[ \frac{2\pi}{\lambda}(y_{m}-y_{n})\right]}{\frac{2\pi}{\lambda}(y_{m}-y_{n})}} \tag{13}\]  
For a uniformly excited linear array, (13) can be simplified to  
\[D=\frac{4M^{2}}{\sum\limits_{m=-M}^{M}\sum\limits_{n=-M}^{M}e^{i\left(\phi_{m }-\phi_{n}\right)}\frac{\sin\left[\frac{2\pi}{\lambda}(y_{m}-y_{n})\right]}{ \frac{2\pi}{\lambda}(y_{m}-y_{n})}} \tag{14}\]  
In this work, we investigate the case of a 50-element linear array as a position-phase synthesis of uniformly excited arrays. We set \(d_{\text{min}}=0.5\lambda\) and \(d_{\text{max}}=\lambda\), where \(d_{\text{min}}\) is the minimum and \(d_{\text{max}}\) is the maximum allowable inter-element distance, accordingly. The number of unknowns in our case is \(2M\). The array phases vary from \(0^{\circ}\) to \(360^{\circ}\). The optimization problem is highly dimensional with \(D=100\) decision variables. The population size \(NP\) is also set to \(100\), whereas the maximum number of iterations \(MaxIt\) is set to \(1000\). Finally, all algorithms are applied for 50 independent trials. The corresponding equally spaced uniform array of 50 elements exhibits an SLL value of \(-18.65\) dB and directivity of \(16.99\) dB. Table 3 lists the comparative results of the 50-element linear array design case. From the presented results, we can conclude that GWO achieves the highest score compared to all other algorithms. PSO obtained the second best array design. Figure 5(a) illustrates the distribution of the results in percentiles (boxplot). From the obtained results, we can deduce that the GWO achieves the best 50% percentile (median value) of the cost function compared to the rest of the algorithms values. The worst result is obtained by the SSA. It is also noteworthy that the upper and lower values of the PSO distribution are similar to the values of the SSA distribution. Figure 5(b) depicts the average convergence rate of the algorithms under study. We can derive that GWO and WOA algorithms obtain the fastest convergence, whereas SSA and PSO algorithms converge more slowly. The radiation patterns of the best-obtained array design are portrayed in Figure 5(c). From the presented graph, we can conclude that the best array design is found by applying the GWO algorithm. The peak SLL value of the best linear array design is \(-24.22\) dB and the directivity is \(17.85\) dB. Therefore, the obtained results by the optimization process exhibit an improvement both in the SLL and directivity values of about \(6\) dB and \(1\) dB, accordingly, when compared to the values of the corresponding uniformly excited linear array. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Algorithm** & **Best** & **Worst** & **Mean** & **Median** & **Std. Dev. ** \\ \hline GWO & \(-\)**24.22** & \(-\)**20.65** & \(-\)**23.16** & \(-\)**23.19** & \(0.69\) \\ \hline SSA & \(-\)23.62 & \(-\)14.30 & \(-\)17.34 & \(-\)15.41 & \(3.13\) \\ \hline WOA & \(-\)21.86 & \(-\)17.23 & \(-\