# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 4 Results and Discussion
### Circular Antenna Array  
CAA is one of the most commonly used antenna array configurations in radar and mobile communication applications because of its adaptive beamforming and direction-finding capabilities. In this configuration, the elements are placed in a circular configuration as shown in Figure 20.  
Figure 19: Convergence profile of design 10. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{8}{c}{**Normalized Excitation Current Amplitudes**} & \multicolumn{2}{c}{**FNBW**} \\ \cline{2-11}  & **I** & **II** & **III** & **IV** & **V** & **VI** & **VII** & **VIII** & **IX** & **X** & **(Degree)** \\ \hline ULA & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 11.40 \\ \hline ALO & 0.7626 & 0.7626 & 0.7626 & 0.7626 & 0.7628 & 0.8348 & 0.9262 & 0.9500 & 1.0000 & 0.7927 & 10.80 \\ \hline DA & 0.7783 & 0.7783 & 0.7783 & 0.7783 & 0.7783 & 0.7783 & 0.7783 & 0.8976 & 1.0000 & 0.7783 & 11.20 \\ \hline STO & 0.7693 & 0.7693 & 0.7693 & 0.7693 & 0.7746 & 0.8257 & 0.9455 & 0.9385 & 1.0000 & 0.8182 & 10.80 \\ \hline MAOA & 0.7800 & 0.7800 & 0.7802 & 0.7871 & 0.8037 & 0.7803 & 0.9953 & 1.0000 & 0.9901 & 0.8250 & 10.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Optimized current amplitude of positive half elements and FNBW for design 10. Figure 18: Array pattern of a 20-element LAA for design 10. The array factor of CAA is given by  
\[AF(\Phi,I,d)=\sum_{i=1}^{N}I_{n}*exp(j[ka\sin\theta.\cos\left(\Phi-\Phi_{n}\right) +\alpha_{n}]) \tag{32}\]  
\[ka=\frac{2\pi a}{\lambda}=\frac{\sum_{i=1}^{N}d_{i}}{\lambda}\text{ \emph{and} }\Phi_{n}=\frac{2\pi\sum_{i=1}^{N}d_{i}}{ka}=2\pi\left(\frac{n}{N}\right)\]  
where \(I_{n}\) denotes the excitation current of the n-th element in the antenna array. The distance between the i-th and (i + 1)-th elements is represented by \(d_{i}\), and \(a\) represents the radius of the circle. The wave number k is defined as \(k=\frac{2\pi}{\lambda}\), where \(\lambda\) is the wavelength of operation. The elevation angle, measured from the z-axis, is denoted by \(\theta\), while \(\Phi\) represents the azimuth angle measured from the x-axis. The angular position of the n-th element in the x-y plane is given by \(\Phi_{n}\), and its phase excitation is \(\alpha_{n}\). The total number of elements in the CAA is denoted by N.  
Design 11: SLL Reduction with FNBW Constraint of a 14-Element CAA  
The fitness function for this multi-objective goal is given in (33), in which the FNBW constraint is handled by the first term, while the second is responsible for the reduction in SLL of the CAA,  
\[fitness=w_{1}\times\{|AF(\theta_{mu1})|+|AF(\theta_{mu2})|\}+w_{2}\times\{| AF(\theta_{msk1})|+|AF(\theta_{msk2})|\} \tag{33}\]  
\(\theta_{mu1}\) and \(\theta_{mu2}\) represents the lower and upper nulls of the main beam, respectively, while \(\theta_{msk1}\) and \(\theta_{msk2}\) denotes the angle of the maximum side lobe in the lower region \((-\pi\) to \(\theta_{mu1})\) and upper region \((\theta_{mu2}\) to \(\pi)\) respectively. \(w_{1}\) and \(w_{2}\) are the weighting factors for the above two objectives, which add more flexibility in tuning the two constraints. In this case, their values are \(w_{1}=0.1\) and \(w_{2}=0.9\). This design highlights the synthesis of 14-element CAA for minimizing the SLL along with constant beamwidth. Here, the parameter selected for optimization is the excitation amplitude of the individual antenna elements. The goal here is to obtain a radiation pattern having SLL less than the uniform CAA, which is \(-5.50\) dB, along with a constant beamwidth of \(39.26^{\circ}\), which is the beamwidth of the uniform circular array (UCA). MAOA is applied to estimate the non-uniform amplitudes of the CAA for which the simulation is performed with 15 independent runs having 1000 iterations in each run. The lower bound limit is 0.1, while the upper bound has a value of 1. The obtained radiation pattern  
Figure 20: Geometry of a circular antenna array having N antenna elements.