# Generation of Sparse Antennas and Scatterers Based on Optimal Current Grid Approximation
## 3 Verification of the Modeling Results for Scattering Plate Using MoM with PBF Based on WG Model
Based on the graph showing the relationship between RCS\({}_{\text{max}}\) and GEET (for example, Figure 6), a suitable GEET value is chosen to create the final sparse (FS) structure (in this study, use GEET = 40%). Subsequently, the original WG structure is sparsified while systematically varying the excitation direction in increments \(\Delta\theta\), \(\Delta\varphi\) according to \(\theta\) and \(\varphi\) based on the OCGA and GEET, as outlined in the algorithm in Figure 4.  
When considering all excitation directions of the incident wave (\(\theta_{\text{inc}}\) = 0-359\({}^{\circ}\) and \(\varphi_{\text{inc}}\) = 0-359\({}^{\circ}\)) with \(\Delta\theta\) = \(\Delta\varphi\) = 1\({}^{\circ}\), a total of 360 \(\times\) 360 = 129,600 sparse structures would be generated. However, the considered plate structure is symmetrical (specifically, the scattering characteristics when the plane is excited from the front are identical to those when excited from the back). Therefore, this study focuses only on \(\theta_{\text{inc}}\) = 0-180\({}^{\circ}\) and \(\varphi_{\text{inc}}\) = 0-180\({}^{\circ}\) (i.e., exciting the front face of the plate). This results in 181 \(\times\) 181 = 32,761 unique sparse structures. This approach significantly reduces the computational time and memory required for generating scattering structures (approximately by about 4 times). However, during the simulation, it became apparent that the number of structures (32,761) was still quite large, leading to significant computational time. This is because we not only aim to create sparse structures for each direction of the incident wave but also need to determine the scattering characteristics of each sparse structure. The structure to be sparse  
Determine the \(\varphi_{\text{mc}}\) and \(\theta_{\text{mc}}\) that obtains the highest BSCS magnitude  
Select appropriate GEET to sparse structure in the region of incident  
plane wave, based on RCS\({}_{\text{max}}\) dependence on GEETs  
Figure 10: The OCGA algorithm for creating final sparse scatterers when the incident wave is excited in region direction. On each sparse structure obtained for different incident wave directions, we determined which segments remained and which were discarded. The characteristics of the remained segments were saved in a matrix called \"segments remained\" (\(M_{\rm r}\)) with dimensions \(N_{\rm a}\times N_{\rm s}{}^{\rm max}\) (where \(N_{\rm a}\) is the number of excitation directions, and \(N_{\rm s}{}^{\rm max}\) is the number of segments of the sparse structure with maximum remaining segment in the region direction). Subsequently, the indices of the segments for each sparse structure were saved in a matrix called \"index segments remained\" (\(M_{\rm ir}\)). We calculated the total number of different segments (\(N_{\rm t}\)) in the \(M_{\rm ir}\) matrix. We then determined the number of repetitions of each segment (\(R_{k}\)) in the \(M_{\rm ir}\) matrix and compared \(R_{k}\) for each \(k\)-th index segment with a predefined threshold value (\(T\)). Any index segment with \(R_{k}\) less than \(T\) was discarded. The result was a collection of indices corresponding to segments that exceed the threshold, which was saved in a matrix called \"final segments\" (\(M_{\rm fs}\)) with dimensions \(1\times N_{\rm fs}\) where \(N_{\rm fs}\) is the number of segments in the final sparse structure. Using the indices of these segments, we constructed the final sparse structure. Finally, based on the indices of the segments in the final sparse structure, we determined the impedance matrix of the final structure (\(\mathbf{Z}_{\rm fs}\)) by retaining the corresponding rows and columns in \(\mathbf{Z}\). We also calculated the excitation vector for the final structure (\(\mathbf{v}_{\rm fs}\)). The scattering characteristics of the structure can then be determined by solving the linear algebraic equations to find \(I_{\rm fs}\). It is important to consider the threshold value \(T\) when creating the final structure.