# MAOA: A Swift and Effective Optimization Algorithm for Linear Antenna Array Design
## 2 Related Works  
### Linear Antenna Array  
The geometric configuration of the LAA with 2N elements is placed symmetrically on a horizontal line along the x-axis and is shown in Figure 1. The placement of the antenna is made in such a manner that each side of the origin has N elements contributing to an even number of antenna elements in an array. The radiation mechanism of the antenna array is governed by the theory of pattern multiplication. According to this theory, the far-field radiation pattern of an antenna array is computed by the multiplication of the array factor with the radiation pattern of a single antenna, also called the element pattern. Mathematically, it can be expressed as shown in Equation (1).  
\[\begin{split}\text{Total radiation pattern = Array factor }\times\text{Element pattern}\\ \text{RP (\emptyset) = AF(\emptyset) \times EP(\emptyset)}\end{split} \tag{1}\]EP(\(\theta\)) is decided by the nature of the single antenna element selected for making an antenna array. For a dipole antenna, its value is \(\sin(\theta)\), while for an isotropic antenna, it is taken to be unity. The array factor AF(\(\theta\)) can be modified by optimizing the values of the excitation current amplitude, its phase, and the position of the antenna elements in an array, which in turn influence the radiation characteristics of the antenna array. The mathematical relation of an array factor in the azimuth plane is expressed in Equation (2) as follows:  
\[\mathrm{AF}(\theta)=2\sum_{n=1}^{N}\mathrm{I_{n}cos}\{\mathrm{kx_{n}cos}\left( \theta\right)+\mathcal{O}_{n}\} \tag{2}\]  
where \(\mathrm{I_{n}}\) and \(\mathcal{O}_{n}\) are the excitation current amplitude and phase of the n-th element, respectively, while \(\mathrm{x_{n}}\) is the position of n-th element from the origin. The wave number is given by \(\mathrm{k}=2\pi/\lambda\), while the azimuth angle is denoted by \(\theta\). In this work, the amplitude and position are the two parameters considered for optimization.  
When the spacing of the antenna elements is optimized, the amplitude of the current and phase are assumed to be constant, i.e., \(\mathrm{I_{n}}=1\) and \(\mathcal{O}_{n}=0\). Using these two values, Equation (2) is modified and expressed in Equation (3) as follows:  
\[\mathrm{AF}(\theta)=2\sum_{n=1}^{N}\cos\{\mathrm{kx_{n}cos}(\theta)\} \tag{3}\]  
Spacing between the antenna elements highly affects the radiation characteristics of the antenna array. If they are placed very close to each other, it will result in a mutual coupling effect, and if placed too far, it may lead to the formation of grating lobes. To avoid these undesired conditions, the following criteria for interelement spacing must be satisfied as mentioned in Equation (4):  
\[\frac{\lambda}{4}<\left|x_{n+1}-x_{n}\right|<\frac{\lambda}{2} \tag{4}\]  
where \(x_{n}\) and \(x_{n+1}\) denote the positions from the origin of the \(n\)-th and \((n+1)\)-th antenna elements.  
When the excitation current amplitude is optimized, the spacing between the antenna elements and phase of the current is assumed to be constant, i.e., \(x_{n}=\frac{\lambda}{2}\) and \(\mathcal{O}_{n}=0\). Using these two values, Equation (2) is modified and expressed in Equation (5) as follows:  
Figure 1: LAA with 2N elements distributed horizontally on the x-axis.  
\[\begin{split}\text{AF}(\theta)=2\sum\limits_{n=1}^{N}\text{I}_{n} \text{cos}\Big{\{}\frac{2\pi}{\Lambda}\times\frac{\lambda}{2}\text{cos}(\theta )\Big{\}}\\ \text{AF}(\theta)=2\sum\limits_{n=1}^{N}\text{I}_{n}\text{cos} \{\pi\times\text{cos}(\theta)\}\end{split} \tag{5}\]