# Optimization of a Coherent Dual-Beam Array Feed Network for Aperiodic Concentric Ring Antennas
## 3 Numerical Results  
In this section, the simulation setup is described alongside the numerical results obtained from the two-beam C-BFN configuration for applications in ACRAs. One main objective under this scenario is to estimate the dual-beam pattern generated by the antenna system (i.e., the beamforming network and the non-uniform antenna array), as well as the performance in a comparative study with a uniform layout scenario in order to compare the results. Following this, we made a comparison with the periodic antenna element distribution using the proposed array feeding network and taking the worst solutions from the aperiodic array. Thus, we evaluate the feasible improvements of non-uniform over the uniform aperture on a CCRA fed by a coherent network with steering and multi-beam capabilities. Additionally, we focused on this from an optimization synthesis problem point of view. The complete multi-beam system is optimized under a fair simulation setup with multiple trials by a global optimizer. The intention behind this consists of, using the obtained behavior of the dual-beam radiation patterns analyzed, showing and exhibiting the complexity reduction and performance improvements of using a two-beam coherent feed system with an ACRA. Figure 3 illustrates the general representation of the antenna setup utilized. The system shows a CCRA integrated by non-uniform spaced rings. Specifically, without loss of generality, this diagram represents an ACRA constituted by radiant elements (92 in total) located within five different circular rings (\(N_{r}=5\)). The number of radiant elements inside each circular ring in the array is constituted, respectively, by \([N_{1},N_{2},N_{3},N_{4},N_{5}]=[6,12,18,25,31]\). This radiant element distribution corresponds to \(N_{e}=2\pi r_{n}/d_{\min}\mid r_{n}=n\lambda/2\). In the numerical examples, in order to attenuate any achievable mutual coupling interaction among antenna elements in the array, the spacing between adjacent radiant  
elements in each circular ring is set at a minimum of \(d=0.5\lambda\) and each circular ring is set to be located by the same minimum distance of \(0.5\lambda\) with isophoric feeding (a single excitation level) and \(r_{1}\geq 0.5\lambda\). As depicted in the bottom part of Figure 3, the feeding subsystem is a C-BFN based on \(Q\) blocks of one layer, which in total integrates the complete coherent network subsystem. In this considered configuration, the \(Q\) required C-BFN blocks per ring is equal to the number of circular rings \(N_{r}\), i.e., one C-BFN block for every circular ring within the aperiodic array. In this case, the designed configuration for feeding the aperiodic concentric array is constituted by five C-BFN blocks. For the particular case of this proposal, the (\(N\)) output ports of each block are interconnected straight to a radiant element in every circular ring, as shown in Figure 3. Furthermore, at the ends of each C-BFN block, the indices denote the amount of output port signals (top of the individual C-BFN blocks) and the input port signals (bottom part of every C-BFN block). In addition, the proposed C-BFN allows for controlling a couple of independent beams simultaneously in the alternation of the feeding ports in each block, as depicted in the lower part of Figure 3. Thus, the number of beams or signals applied is represented by the indices at the feeding ports, i.e. for the conformation of one of the two beams, each index number belongs to a group of complex inputs. For the numerical experiments, the DE algorithm was programmed as well as used to optimize the aperiodic antenna system as well as the feeding network. The evolutionary algorithm is set up to initialize with a population of 120 individuals, a crossover rate \(\text{CR}=0.9\), a factor of mutation fixed in \(F=0.5\), and a maximum number of generations/iterations set in \(G_{\text{max}}=1000\). A random selection, one difference vector, and binomial crossover (DE/rand/1/bin) comprise the DE scheme utilized to optimize the multi-beam radiation pattern.