# Optimization of a Coherent Dual-Beam Array Feed Network for Aperiodic Concentric Ring Antennas
## 3 Numerical Results
The global stochastic algorithm was run in 20 independent trials for every dual-beam pattern. For this case, the directions of maximum radiation correspond to a slice of the total multi-beam pattern for values of \(\varphi\) (\(\varphi\)-cutting patterns), where the optimized dual-beam array factors are predominately a function of the \(\theta\) angle. Figure 4 depicts the performance of the normalized radiation patterns for the two-beam configuration under study (Figure 3) explained in detail in this same section. In these numeric examples, the target of maximum radiation in this multi-beam scenario is steered towards the azimuth plane and directed at \(45^{\circ}\), \(95^{\circ}\), and \(245^{\circ}\) for beam number 1. The beam number 2 is set in \(345^{\circ}\), \(145^{\circ}\), and \(295^{\circ}\). The optimization process utilized to improve the radius of every ring \(\overline{r}=[r_{1},r_{2},\ldots,r_{5}]\) on each dual-beam example allows us to obtain and demonstrate a satisfactory performance regarding isolation level and directivity. Concerning the near-optimal ring spacings (radii), we found that the optimization process in these multi-beam cases achieves the following: \([1.29\lambda,1.81\lambda,2.45\lambda,2.96\lambda,4.73\lambda]\) for Figure 4a, \([0.98\lambda,1.61\lambda,2.24\lambda,2.98\lambda,4.26\lambda]\) for Figure 4b, and \([1.45\lambda,2.14\lambda,2.66\lambda,3.28\lambda,4.85\lambda]\) for Figure 4c, and the aperiodic apertures are illustrated in Figure 5.  
Figure 3: Schematic of the dual-beam array feed network subsystem and the aperiodic antenna array aperture. Figure 4: Normalized patterns in dB generated by the two-beam feeding network with a non-uniform CCRA optimized using differential evolution. Furthermore, Table 1 presents the precise numerical values of sidelobe level and directivity retrieved and depicted in each numeric example from Figure 4 with the proposed C-BFN configuration. This table condenses the information from the worst numerical values of SLL and D achieved through the aperiodic aperture via C-BFN on the multi-beam scenario for each numeric example and contrasts the results of the previously mentioned parameters with a uniform aperture (Uniform) with a equivalent coherent feeding network. In addition, Table 1 collects the mean outcomes of the improvement obtained through the aperiodic aperture employing the proposed feeding subsystem on each beam for all of the optimized examples. In each numerical example, the aperiodic layout is straightforwardly set out in comparison with the equivalent uniform layout. Additionally from Table 1, we can observe that the performance obtained by the worst case of the normalized two-beam patterns generated from the proposed coherent beamforming network outperforms the case of a multiple radiation patterns under the condition of an uniform concentric ring array while using the same feeding system. The general performance shows that a C-BFN configured on a non-uniformly ring-spaced concentric ring array notoriously improves in identical conditions to an uniformly ring-spaced concentric ring array fed by the similar coherent feeding system, in terms of sidelobe level and directivity. The improvement in the shown examples is approximately to \(-5.6\) dB of the sidelobe level and \(3.0\) dB of the directivity for beam number \(1\) and \(-2.5\) and \(1.3\) dB for the same parameters for beam number \(2\). The improvements (expected values) for each beam are presented in Table 1.  
Figure 5: Aperiodic layouts of the concentric ring arrays in the numeric examples while optimizing antenna rings (\(\bar{r}\)). As was demonstrated, the uniform circular concentric geometry fed by a C-BFN maintains the usual nature of circular antenna arrays, retaining inherent high directivity while registering high sidelobe levels over the range of interest.