# 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
Even after sparsing the structure, there are still quite a few segments to consider. Therefore, recalculating \(\mathbf{Z}_{\mathrm{s}}\) and \(\mathbf{v}_{\mathrm{s}}\) using this approach significantly increases the computational cost. The second approach leverages a significant advantage related to the formation of the initial impedance matrix (\(\mathbf{Z}\)) using MoM with PBF. When an \(i\)-th segment is removed from the structure, it is sufficient to eliminate the corresponding \(i\)-th row and \(i\)-th column in matrix \(\mathbf{Z}\). In other words, the existence of each individual segment is independent of the others, which contrasts with the use of basis functions constructed from points along the wire (e.g., triangular functions). Therefore, this approach is considered more optimal in this context. Furthermore, this approach not only accelerates \(\mathbf{Z}_{\mathrm{s}}\) computation but also facilitates the rapid identification of the \(\mathbf{v}_{\mathrm{s}}\) element by removing the \(i\)-th elements (corresponding  
Figure 5: The OCGA algorithm for creating sparse scatterers when the incident wave is excited in a fixed direction. to the index \(i\) of the removed segment) from \(\mathbf{v}\) (the excitation vector for the initial WG). As a result, the computational cost of analyzing the scattering characteristics of the sparse structure is significantly reduced. Next, we analyzed the results of using OCGA to sparse the considered WG structure. In previous publications [66; 67], the scattering characteristics of the rectangular plate, as represented by the radar cross section (RCS), are typically examined when the incident wave is directed orthogonally to the plate (\(\theta_{\mathrm{inc}}=\varphi_{\mathrm{inc}}=90^{\circ}\)). Therefore, in this section, we first investigated the variation in the RCS maximum magnitude (\(\mathrm{RCS}_{\mathrm{max}}\)) (when \(\theta_{\mathrm{scat}}=\varphi_{\mathrm{scat}}=90^{\circ}\)) for the plate when the GEET value changed (Figure 6). From Figure 6, it can be observed that when GEET is set to 0-34%, the \(\mathrm{RCS}_{\mathrm{max}}\) value remains largely unchanged. This is because, in addition to the horizontal segments, which do not contribute to the scattering field, the \(\left\lvert\mspace{-1.0 mu }\right\rvert\) of the vertical segments is generally greater than GEET (exceeding 34% of \(\left\lvert\mspace{-1.0 mu }\right\rvert_{\mathrm{max}}\)). Specifically, as seen in Figure 6, \(\left\lvert\mspace{-1.0 mu }\right\rvert_{\mathrm{min}}\) on segment EF is 0.44 mA, which is still greater than approximately 34% of \(\left\lvert\mspace{-1.0 mu }\right\rvert_{\mathrm{max}}\) on segment AB (1.23 \(\times\) 0.34 = 0.41 mA). When GEET increases to 34-55%, \(\mathrm{RCS}_{\mathrm{max}}\) decreases sharply. This can be explained by the fact that \(\left\lvert\mspace{-1.0 mu }\right\rvert\) on the vertical wires in the middle of the plate is smaller than the GEET value and leads to the removal of the wires. Furthermore, from Figure 6, it can be noted that \(\left\lvert\mspace{-1.0 mu }\right\rvert\) of the vertical wires in the middle of the plate is approximately equal, so even a slight increase in GEET within the range of 34-55% results in the removal of a significant number of vertical segments in that region. Unlike the case when GEET = 0-34%, the removal of a large number of horizontal wires does not affect the scattering properties. However, the sudden decrease in the number of mid-plate segments (where \(\left\lvert\mspace{-1.0 mu }\right\rvert\) is not small) in this case leads to a sharp decrease in the scattering field, resulting in a rapid decrease in \(\mathrm{RCS}_{\mathrm{max}}\) in this range (Figure 6). When GEET reaches 45%, \(\mathrm{RCS}_{\mathrm{max}}\) decreases significantly (down to 120 m\({}^{2}\)) and then increases again when GEET is at 46% (up to 240 m\({}^{2}\)). This phenomenon can depend on the phase of the current in the removed wires.