# 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
If small values of \(\Delta\theta\) and \(\Delta\varphi\) are used, the number of repeated segments will be significantly higher compared to the cases when larger \(\Delta\theta\) and \(\Delta\varphi\) are used. Consequently, the choice of \(T\) for generating the final structure with smaller \(\Delta\theta\) and \(\Delta\varphi\) will be higher than those when using larger \(\Delta\theta\) and \(\Delta\varphi\). Using this algorithm, we obtained the final structure with \(T\) = 50 (FS50), 100 (FS100), 200 (FS200), and 300 (FS300), as illustrated in Figure 11. Additionally, we considered the scattering characteristics of the structure that retained the most segments after sparsing the original WG scatterer during the change in the wave excitation direction. The reason for investigating this structure is that it may exhibit scattering properties similar to those of the original structure. After calculations, we found that the structure with the incident wave directed at \(\theta_{\rm inc}=\varphi_{\rm inc}=90^{\circ}\) (S40) retained the most segments after the excitation direction change. The obtained BSCSs for the final structures and S40 in the xOy and yOz planes were then compared with those for the original WG structure (Figure 12). Additionally, the deviations in the maximum magnitude of BSCSs and the maximum BSCS deviation between the results are summarized in Table 5.  
Figure 11: The MATLAB models of FS50 (**a**), FS100 (**b**), FS200 (**c**), and FS300 (**d**). Figure 12 and Table 5 demonstrate that the BSCS for FS50 in the xOy and yOz planes matches very well with that of the original WG structure. Specifically, in the xOy plane, the BSCS matches well in the range of \(\varphi\) = 20-160\({}^{\circ}\), while in the yOz plane, there is almost no deviation between the obtained results. The structures FS100, FS200, and FS300 also achieve good similarity with the WG structure, although not as good as FS50. The S40 structure shows the greatest deviation in the BSCS compared to the WG structure, with a clear difference observed in the yOz plane. However, in the range of \(\varphi\) = 0-20\({}^{\circ}\) in the xOy plane, the S40 BSCS aligns better with that of the WG structure than FS50. Overall, the scattering characteristics of the FS structures align well with those of the original WG structure, and this alignment decreases as the threshold \(T\) value increases. Figure 8:0 The obtained BSCS for original and sparse structures in xOy (**a**) and yOz (**b**) planes. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Structure**} & \multicolumn{2}{c}{**BSCS\({}_{\text{max}}\)**} & \multicolumn{2}{c}{**Maximum BSCS Deviation**} \\ \cline{2-4}  & **xOy and yOz** & **xOy** & **yOz** \\ \hline S40 & 1.39 & 2.45 & 17.1 \\ FS50 & 0.000187 & 3.11 & 0.25 \\ FS100 & 0.194 & 4.77 & 6.46 \\ FS200 & 0.479 & 1.554 & 8.89 \\ FS300 & 0.479 & 1.554 & 13.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: BSCS (dBsm) deviations for S40, FS50, FS100, FS200, and FS300. All these observations demonstrate the effectiveness of the OCGA method in creating sparse structures when the specific direction of incoming waves is not known. The OCGA for Creating Sparse Scatterers Excited in Specific Region Through the Scattering Plate Example  
In practice, scattering structures are often studied when excited by incoming waves directed at a specific region. Therefore, in this section, we discuss generating sparse structures in this context. More specifically, using GEET = 40%, we created sparse scatterers while varying the incoming wave direction in ranges of \(\theta\) = 60-120\({}^{\circ}\) and \(\varphi\) = 60-120\({}^{\circ}\). The analysis steps were carried out as described in the algorithm in Figure 9, with \(\Delta\theta\) = \(\Delta\varphi\) = 1\({}^{\circ}\). At this point, the BSCS characteristics were determined over the entire region corresponding to the incident wave angles (with BSCS include of 61 \(\times\) 61 = 3721 values).