# Extended Smoothing Methods for Sparse Test Data Based on Zero-Padding
## 4 Numerical Verification
### Angular Spectrum Changes of 4 \(\times\) 4 Two-Dimensional Discrete Data  
The corresponding angular spectrums of two-dimensional discrete data before and after smoothing are given below, as shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{x}\) & \(\mathbf{y}\) & \(\mathbf{0}\) & \(\mathbf{0.1}\) & \(\mathbf{0.2}\) & \(\mathbf{0.3}\) \\ \hline
0 & 1.093266 & \(-1.21412\) & \(-1.19775\) & \(-0.76967\) & \(-1.10933\) & \(-1.08906\) \\
0.05 & 1.410773 & \(0.026433\) & \(-1.56284\) & \(-1.20641\) & \(-0.33273\) & \(-0.67326\) & \(-0.80885\) & \(0.559588\) \\
0.1 & 1.109273 & \(-0.04438\) & \(-1.1135\) & \(-0.56615\) & \(0.371379\) & \(0.244235\) & \(0.032557\) & \(0.766005\) \\
0.15 & 0.027023 & \(-0.45354\) & \(-0.70917\) & \(-0.43589\) & \(0.05199\) & \(0.314429\) & \(0.351934\) & \(0.296774\) \\
0.2 & \(-0.86365\) & \(-0.55925\) & \(-0.00685\) & \(-0.10807\) & \(-0.22558\) & \(0.287471\) & \(0.552527\) & \(-0.16371\) \\
0.25 & \(-0.70265\) & \(0.102538\) & \(1.161923\) & \(1.009139\) & \(0.579476\) & \(0.970416\) & \(1.107161\) & \(0.063816\) \\
0.3 & 0.077359 & 0.742037 & \(1.53263\) & \(1.477426\) & \(1.117356\) & \(1.171941\) & \(1.10061\) & \(0.436552\) \\
0.35 & 0.6811 & 0.582514 & 0.308256 & 0.238614 & 0.194752 & \(-0.01727\) & \(-0.05362\) & \(0.32663\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Regenerated two-dimensional discrete data. Figure 4: Original and regenerated two-dimensional data: (**a**) Original two-dimensional discrete data; (**b**) regenerated two-dimensional discrete data. Through the analysis of the angular spectrum change in Figure 5, it can be found that the variation of each region is consistent with the variation law shown in Figure 3. Tables 3 and 4 list the angular spectrums before and after smoothing. Now we analyze the variation of the angular spectrum in region A in line with Section 3.2. According to Table 3, the original angular spectrum value at \((1,1)\) in region A is \(0.017045\), and the regenerated angular spectrum value listed in Table 4 at the same position is \(0.03409\). This is in accordance with the principle that the angular spectrum value of the region A\({}^{\prime}\) is twice that of the corresponding region A. It can be seen from Table 3 that the original angular spectrum at (3,2) in region B is \(0.01527\). It can be seen from Table 4 that the position is divided into two positions after smoothing, which are (3,2) and (7,2), respectively. The generated angular spectrum value is \(0.01527\), which is the same as that of the corresponding region B. It can be seen from Table 3 that the original angular spectrum at (3,3) in region C is \(-0.00131\), and it can be seen from Table 4 that the smoothed position is divided into four regions, which are (3,3), (7,3), (3,7), and (7,7), respectively, and the generated angular \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{k_{x}}\) & \(\mathbf{k_{y}}\) & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
1 & 0.03409 & 0.018455 & 0.021149 & 0 & 0 & 0 & 0.021149 & 0.018455 \\
2 & \(-0.02872\) & 0.081582 & 0.042617 & 0 & 0 & 0.042617 & 0.018458 \\
3 & \(-0.06751\) & 0.01527 & \(-0.00066\) & 0 & 0 & \(-0.00066\) & 0.01527 \\
4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
7 & \(-0.06751\) & 0.01527 & \(-0.00066\) & 0 & 0 & \(-0.00066\) & 0.01527 \\
8 & \(-0.02872\) & 0.018458 & 0.042617 & 0 & 0 & 0.042617 & 0.081582 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Regenerated angular spectrum value after smoothing. Figure 5: Angular spectrums before and after smoothing: (**a**) Original angular spectrum; (**b**) regenerated angular spectrum. spectrum value is \(-0.00066\). This conforms to the principle that the value of the angular spectrum of region C' is reduced to half of the value of the angular spectrum of region C.  
It can be seen from Table 4 that the region D' is the zero-padded region, and the generated angular spectrum values are all zeros.