# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 4 Results and Discussion
### Antenna Position Optimization
#### 4.1.4 Design 4: Side Lobe Average Power Minimization Along with Null Placement of a 20-Element LAA  
The deep null placement, along with the SLL reduction, is presented in this design by controlling the position of the antenna elements. The fitness function used in this design is provided in Equation (30), where the first part deals with SLL reduction, while the latter is \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{**SLL (dB)**} & \multirow{2}{*}{**FNBW (Degree)**} \\ \cline{2-2} \cline{5-10}  & **I** & **II** & **III** & **IV** & **V** & **VI** & **VII** & **VIII** \\ \hline ULA & \(-13.18\) & \(-17.76\) & \(-20.43\) & \(-22.54\) & \(-23.63\) & \(-24.84\) & \(-25.33\) & \(-25.77\) & \(11.40\) \\ \hline ALO & \(-22.43\) & \(-24.35\) & \(-26.00\) & \(-27.68\) & \(-28.38\) & \(-29.71\) & \(-32.17\) & \(-34.24\) & \(19.60\) \\ \hline AOA & \(-26.61\) & \(-26.02\) & \(-39.36\) & \(-21.89\) & \(-20.95\) & \(-25.14\) & \(-24.49\) & \(-32.23\) & \(19.40\) \\ \hline MAOA & \(-23.44\) & \(-27.27\) & \(-23.68\) & \(-25.10\) & \(-22.19\) & \(-26.04\) & \(-27.26\) & \(-31.07\) & \(17.00\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Positive half SLL and FNBW of design 3. Figure 8: Array pattern of a 20-element LAA for design 3. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{**Optimized Element Positions (\(\times\lambda\))**} \\ \cline{2-10}  & **I** & **II** & **III** & **IV** & **V** & **VI** & **VII** & **IX** & **X** \\ \hline ULA & 0.2500 & 0.7500 & 1.2500 & 1.7500 & 2.2500 & 2.7500 & 3.2500 & 3.7500 & 4.2500 & 4.7500 \\ \hline ALO & 0.1315 & 0.3750 & 0.7360 & 0.9940 & 1.1966 & 1.7861 & 1.7908 & 3.4498 & 2.4173 & 2.8025 \\ \hline AOA & 0.1331 & 0.3750 & 0.7898 & 0.8750 & 1.4217 & 2.3874 & 1.6250 & 3.7500 & 2.1250 & 3.0014 \\ \hline MAOA & 0.1595 & 0.5210 & 0.7873 & 1.1636 & 1.4770 & 1.8623 & 2.2809 & 2.7105 & 3.3559 & 4.0624 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimized positions for positive half elements of design 3.  
responsible for placing the nulls at appropriate locations as defined by \(\theta_{k}\). Here, the region of the side lobes is considered as \([0^{\circ},82^{\circ}]\) and \([98^{\circ},180^{\circ}]\), while the nulls are required at \(64^{\circ},76^{\circ},104^{\circ}\), and \(116^{\circ}\). \[\text{fitness function}=\sum_{i}\frac{1}{\Delta\theta_{1}}\!\!\int_{\theta_{ \mathrm{B}_{1}}}^{\theta_{\mathrm{B}_{1}}}|\mathrm{AF}(\theta)|^{2}\mathrm{d }\theta+\sum_{k}|\mathrm{AF}(\theta_{k})|^{2} \tag{30}\]  
The optimized antenna positions obtained after simulation are mentioned in Table 5, while the array pattern is outlined in Figure 9. The null depth level in decibels and FNBW in degrees are outlined in Table 6. This clearly indicates that maximum null depths of \(-64.21\) dB and \(-89.00\) dB are attained with MAOA in comparison to ALO, AOA, and DA-optimized antenna arrays. The beamwidth with MAOA is increased by \(4.20\) dB from ULA but is still less broadening in comparison to other applied algorithms. 4.1.5. Design 5: Side Lobe Average Power Minimization Along with FNBW Constraint of a 20-Element LAA  
This multi-objective function is handled by optimizing the position of the antenna elements using MAOA. The fitness function for this design is provided in Equation (31), where SLL reduction is governed by the first part, whereas the second part focuses on maintaining the constant beamwidth offered by the antenna array. In this design, the side lobe region is decided as [0\({}^{\circ}\), 82\({}^{\circ}\)] and [98\({}^{\circ}\), 180\({}^{\circ}\)].