# Optimal Design of Aperiodic Reconfigurable Antenna Array Suitable for Broadcasting Applications
## 4 Optimization Results  
As mentioned earlier, CST in conjunction with PSOvm is used to find the optimal values for all the parameters specified in Section 2 and Table 1. In this paper, PSOvm employs swarms of 20 particles (\(N=20\)). This size was selected after several optimization trials and has shown promising results. As in every optimization problem, a fitness function, usually to be minimized, must be defined in order to guide the optimization algorithm towards the desired results. The expression of the fitness function used here is as follows:  
\[\begin{split}& Fit=-FG-min(FG-20,G_{min})+max(SLL_{1},-20)+max(SLL_{2},-20) \\ &+\sum_{n=1}^{8}max(SWR_{n},1.2),\end{split} \tag{12}\]  
where \(FG\) is the array gain (in dBi) at \(\theta=93^{\circ}\) which ensures gain maximization and the desired main lobe tilting by \(3^{\circ}\), \(G_{min}\) is the minimum gain inside the service area (from \(\theta=93^{\circ}\) to \(\theta=143^{\circ}\)), \(SLL_{1}\) represents the \(SLL\) inside the angular sector from \(\theta=0^{\circ}\) to \(\theta=80^{\circ}\), \(SLL_{2}\) is the \(SLL\) inside the sector defined from \(\theta=150^{\circ}\) to \(\theta=360^{\circ}\) and finally \(SWR_{n}\) represents the \(SWR\) at the input port of the \(n\)th element. The 2 nd term in the fitness function ensures that the desired null level of \(-20\) dB is achieved inside the service area, while the 3 rd and the 4 th terms set the \(SLL\) goal of \(-20\) dB for both the angular sectors outside the service area. Finally, the 5 th term ensures that all the array elements exhibit an \(SWR\leq\)1.2 at their inputs (on the transmission lines edges). The angular sectors from \(\theta=80^{\circ}\) to \(\theta=93^{\circ}\) and from \(\theta=143^{\circ}\) to \(\theta=150^{\circ}\) serve as transition areas and therefore the radiation pattern inside those areas is not taken into account. The use of \(min\) and \(max\) functions in (12) is to help the optimization algorithm in satisfying all the requirements by ignoring any improvement in \(G_{min}\), \(SLL_{1}\), \(SLL_{2}\) and \(SWR_{n}\) when the desired values of them are reached. Finally, it must be stressed that the optimization algorithm is tasked to achieve 13 objectives in total, \(FG\) maximization and main lobe tilting by \(3^{\circ}\), null level \(-20\) dB, \(SLL\) reduction in two sectors \(SLL_{1}\) and \(SLL_{2}\), and \(8\)\(SWR\) values by optimizing 26 parameters (described in Section 2) thus, solving a highly complex multi-objective and multi-variable problem.  
The dimensions and the feeding weights of the optimized structure are shown in Tables 2 and 3, respectively. Table 4 shows the electromagnetic characteristics of the optimized antenna in the central frequency 698 MHz and Table 5 the input impedances of the array elements. It can be seen that all the requirements have been satisfied. A high \(FG\) of 13.9 dBi, main lobe tilting by \(3^{\circ}\) and null level of \(-20\) dB have been achieved inside the service area. The \(SLL\) requirement inside the angular sector from \(\theta=0^{\circ}\) to \(\theta=80^{\circ}\) is satisfied achieving \(-20.8\) dB, as well as, the \(SLL\) value inside the second angular sector from \(\theta=150^{\circ}\) to \(\theta=360^{\circ}\) achieving an \(SLL\) of \(-21\) dB. Finally, the \(SWR\) for all the array elements is below or equal to \(1.2\) (S-parameter \(\leq-21\) dB) making the \(SWR\) optimization completely successful. Consequently, PSOvm is a capable optimization algorithm and antennas subject to many requirements can be designed. The normalized radiation pattern of the optimized structure is illustrated in Figure 3. Figure 4 shows the S-parameters of all the array elements. It can be seen that all S-parameters are below \(-21\) dB thus achieving the SWR objectives. The optimized antenna exhibits an axial ratio (AR) greater than \(22\) dB (linear polarization) inside the service area (\(\theta=93^{\circ}\) to \(