# Optimal Design of Switchable Wearable Antenna Array for Wireless Sensor Networks
## 4 Discussion  
In our investigation, we used four optimization approaches (min-max, nadir point distance maximization, utopia point distance minimization, and full Pareto-like) to the antenna array performance optimization, as described in Section 2.2. Out of the four approaches, we obtained the best results using the min-max version of the optimization algorithm. In Figure 15 we show the result of the elementary-antenna placement optimization. The initial and optimized (final) positions are presented for comparison. It can be observed that the initial and final positions differ significantly, but the bilateral symmetry of placement is maintained. It is worth noting that the symmetry, although intuitively correct, was not a-priori forced by the algorithm. Under the circumstances, the fact that our stochastic optimization converged to the symmetry can be regarded as a manifestation of its proper operation. The resultant optimized array has switchable directivity with 15 possible patterns following from all possible combinations of phase shift values (0 or \(\alpha_{const}=117.2\)) fed to the elementary antennas. All 15 possible patterns are illustrated in Figure 16. It can be concluded that even for such a small four-element array, the variability of the null and lobe positions of the patterns offer a possibility to improve the signal-to-interference ratio significantly, provided that the source of the interference and the signal are angularly spaced by 45\({}^{\circ}\) or in most cases even only ca 30\({}^{\circ}\) (see Figure 16). \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Variant of the Optimization Algorithm**} & **Objective Function Value** & **Number of Iterations** \\ \hline Min-max & \(\mathrm{f_{m}}=1.96\) & 109 \\ \hline Nadir point distance maximization & \(\mathrm{f_{r}}=1.87\) & 145 \\ \hline Utopia point distance minimization & \(\mathrm{f_{u}}=1.62\) & 71 \\ \hline Pareto-like optimization & \(\mathrm{f_{p}}=1.76\) & 56 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of optimization algorithm variants in terms of performance. Figure 15: Elementary-antenna placement optimization. The initial and optimized (final) positions are shown. \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Algorithm**} & **Number of Iterations** & **Objective Function Value** \\ \hline ESTra & 109 & 1.96 \\ \hline Nelder-Mead simplex (Matlab function _fminsearch_) & 445 & 1.47 \\ \hline Interior-point (Matlab function _fmincon_) & 628 & 1.70 \\ \hline Powell & 415 & 1.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of convergence and objective function value for several optimization algorithms. Example radiation patterns of the antenna array before and after the optimization process with min-max version of the algorithm are presented in Figure 17. For the initial antenna configuration, the worst case was with the useful signal located at \(\varphi_{s}\) = 90\({}^{\circ}\) and the interference located at \(\varphi_{i}\) = 0\({}^{\circ}\). The phasing vector [90\({}^{\circ}\) 0\({}^{\circ}\) 0\({}^{\circ}\) 90\({}^{\circ}\)] provided the SIR value equal to 0.84. For the optimized array, the worst case was for the signal located at \(\varphi_{s}\) = 90\({}^{\circ}\) and interference located at \(\varphi_{i}\) = 315\({}^{\circ}\). In this case the phasing vector [117.2\({}^{\circ}\) 117.2\({}^{\circ}\) 117.2\({}^{\circ}\) 0\({}^{\circ}\)] yields the value of SIR = 1.96. Figure 18 presents the radiation patterns of the optimized antenna array for the worst combination of signal and interference angles after optimization, obtained with the simplified model, cylindrical model, and heterogeneous model. The angular distributions of antenna gain obtained with the FDTD and simplified model used in the optimization process are similar, but the values of gain are different. In the case of min-max optimization, the algorithm required 109 iterations to find the final solution, and the number of calls to the radiation pattern computations was equal to 405,480.