# Optimal Pattern Synthesis of Linear Array Antennas Using the Nonlinear Chaotic Grey Wolf Algorithm
## 6 Data Analysis
### Array Factor of a 20-Element Uniform Linear Array Antenna with a Lower Notch  
In the second example, an LAA with 20 elements was considered to decrease the SLL in the sidelobe regions of \(\theta=\left[0^{{}^{\circ}},82^{{}^{\circ}}\right]\cup\left[98^{{}^{\circ}},180^{ {}^{\circ}}\right]\) with lower notches in the specific directions of \(\theta=\left[50^{{}^{\circ}},60^{{}^{\circ}}\right]\cup\left[120^{{}^{\circ}}, 130^{{}^{\circ}}\right]\). The NCGWO algorithm was executed with a population size of 30 and at most 500 iterations, consistent with previous studies [36]. Figure (a)a depicts the LAA pattern synthesized using the NCGWO algorithm and compares it with the results of the GWO, IGWO, GOA, and SMO algorithms. Figure (b)b illustrates the iteration curves that correspond to the GWO, IGWO, and NCGWO algorithms. Although all three algorithms obtain the optimal value within 500 generations, a discernible disparity is evident in the figure. Specifically, the NCGWO algorithm demonstrates a more favorable optimal value than GWO and IGWO. Figure 12 illustrates the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Optimized Amplitude Current** & **Peak SLL (dB)** & **Near SLL (dB)** & **Execution Time (s)** \\ \hline GWO & 0.9531, 0.9452, 0.5549, 0.4634, 0.5083 & \(-16.8570\) & \(-31.8400\) & 3.1800 \\ \hline IGWO & 1.0000, 0.8376, 0.6409, 0.4349, 0.4768 & \(-20.2761\) & \(-31.9129\) & 2.4520 \\ \hline PSO & 0.9242, 0.7997, 0.6030, 0.3814, 0.4766 & \(-19.4041\) & \(-32.4200\) & 2.9300 \\ \hline MA & 1.0000, 0.8585, 0.6383, 0.4253, 0.5070 & \(-20.0367\) & \(-32.4500\) & 9.3700 \\ \hline NCGWO & 1.0000, 0.8645, 0.6455, 0.4105, 0.3999 & \(-22.4420\) & \(-36.4027\) & 0.8290 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results for a 10-element linear array antenna. Figure 10: 3D radiation patterns of a 10-element linear array antenna SLL near the main lobe minimization: (**a**) before and (**b**) after. Figure 9: (**a**) Radiation patterns of a 10-element linear array antenna SLL; (**b**) iteration curve. 3D radiation patterns of a 20-element LAA before and after the NCGWO optimization. NCGWO significantly reduces the PSLL and the notch. Table 5 shows the optimization results. The results demonstrate that the NCGWO algorithm achieved a PSLL of \(-\)28.1781 dB, and the PSLL of the notch was \(-\)64.3883 dB. Compared with the reported results of the GWO, IGWO, GOA, and SMO algorithms [36], the PSLL of the NCGWO algorithm was 1.2140, 1.1637, 4.0781, and 0.4781 dB lower, respectively, and the PSLL of the notch was 1.7452, 2.2447, 7.6883, and 3.1883 dB lower, respectively. Previous studies [35] did not provide information on the running time, and the present study independently executed the algorithm 50 times and calculated the average. Table 5 shows that the NCGWO algorithm requires less time than other algorithms. Figure 11: (a) Radiation patterns of the SLL of a 20-element linear array antenna with a lower notch; (b) iteration curve. Figure 12: 3D radiation patterns of the SLL of a 20-element linear array antenna with a lower notch: (a) before and (b) after. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multirow{2}{*}{**Optimized Amplitude Current**} & **Peak SLL** & **Notch** & **Execution Time** \\  & & **(dB)** & **Depth (dB)** & **(s)** \\ \hline GWO & 0.8785, 0.9057, 0.8858, 0.7978, 0.6418 & \multirow{2}{*}{26.9677} & \multirow{2}{*}{62.6431} & \multirow{2}{*}{3.075} \\  & 0.5180, 0.3663, 0.3101, 0.1407, 0.0939 & & & \\ \hline IGWO & 0.9795, 0.9716, 0.9386, 0.7924, 0.7202 & & & \\  & 0.5613, 0.5115, 0.3950, 0.2175, 0.1190 & & & \\ \hline SMO & 1.0000, 0.9990, 1.0000, 0.8360, 0.6430 & & & \\  & 0.6540, 0.4770, 0.5970, 0.2580, 0.2150 & & & \\ \hline GOA & 1.0000, 0.9860, 0.9900, 0.7960, 0.7360 & & & \\  & 0.5630, 0.5270, 0.4470, 0.2430, 0.1510 & & & \\ \hline NCGWO & 1.0000, 0.9636, 0.9417, 0.7958, 0.7000 & & & \\  & 0.5752, 0.4474, 0.3626, 0.1699, 0.1008 & & & \\ \hline \hline \end{tabular}
\end{table}