## 4 Simulation Analysis and Verification
This will directly affect the determination of the interferogram sampling step, which is expected to have a significant impact on the number of sampling points and spectral recovery. Figure 7 shows the real spectral distribution of each pixel after considering the diffraction effect. On this basis, we simulated and analyzed the interference data sampled under the two conditions, either considering the diffraction effect or not. (2) Simulation of the interferogram diffraction effect  
Under the same maximum optical path difference of the interferometer, the sampling step of the optical path difference for each pixel with and without considering the diffraction effect is shown in Table 4.  
The interference data obtained using our simulation are arranged by pixel, as shown in Figures 8 and 9. Figure 8 represents the sampled interferogram corresponding to the case without considering the diffraction effect, and Figure 9 represents the sampled interferogram corresponding to the case considering the diffraction effect. Figure 10 shows the interferogram of the spectrum corresponding to the fifth pixel. It can be observed that the interferogram sampling step after considering the diffraction effect is smaller, with more sampling points and richer details. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Bandwidth for Each Pixel (cm\({}^{-1}\))**} & \multicolumn{2}{c}{**Ratio**} \\ \cline{2-4}
**Pixel** & \multicolumn{2}{c}{**Considering the Diffraction Effect**} & \multicolumn{2}{c}{**Without Considering the Diffraction Effect**} \\ \cline{2-4}
**Number** & **Considering the Diffraction Effect** & **the Diffraction Effect** & **Effect/Without Considering the Diffraction Effect** \\ \hline
1 & 13.1354 & 4.8029 & 2.7349 \\
2 & 17.9101 & 4.7926 & 3.7370 \\
3 & 21.4248 & 4.7823 & 4.4800 \\
4 & 21.3788 & 4.7720 & 4.4800 \\
5 & 21.3329 & 4.7618 & 4.4800 \\
6 & 21.2871 & 4.7516 & 4.4800 \\
7 & 21.2415 & 4.7414 & 4.4800 \\
8 & 21.1960 & 4.7312 & 4.4800 \\
9 & 17.6710 & 4.7211 & 3.7430 \\
10 & 12.9322 & 4.7110 & 2.7451 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Bandwidth comparison for each pixel. Figure 7: Real distribution, arranged by pixel. Figure 8: Interferogram without considering the diffraction effect. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**Maximum OPD Sampling Step (cm)**} & \multicolumn{2}{c}{**Ratio**} \\ \cline{2-4}
**Pixel** & \multirow{2}{*}{\begin{tabular}{c} **Considering the** \\ **Diffraction Effect** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Without Considering** \\ **the Diffraction Effect** \\ \end{tabular} } & \multicolumn{2}{c}{
\begin{tabular}{c} **Without Considering** \\ **Diffraction Effect**/**Considering** \\ **the Diffraction Effect** \\ \end{tabular} } \\ \hline
1 & - & - & - \\
2 & - & - & - \\
3 & 0.0233 & 0.1046 & 4.4945 \\
4 & 0.0234 & 0.1048 & 4.4792 \\
5 & 0.0234 & 0.1050 & 4.4836 \\
6 & 0.0234 & 0.1052 & 4.4880 \\
7 & 0.0235 & 0.1054 & 4.4923 \\
8 & 0.0236 & 0.1057 & 4.4815 \\
9 & - & - & - \\
10 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of OPD sampling step for each pixel. Figure 9: Interferogram considering the diffraction effect. (3) Simulation of the diffraction effect on the spectral recovery process  
According to the theoretical analysis based on BPS-FTS, we performed spectral recovery of interferograms sampled at two different optical path difference sampling steps with and without diffraction effects. In this process, as the interferogram sampling points obtained according to the bandpass sampling theory are too discrete, the theoretically recovered spectrogram should also consist of some discrete sampling points on the input spectrum. We displayed the spectral shape of the recovered spectrum smoothly, so that the consistency between the recovered spectrum and the input spectrum can be accurately judged. In this paper, the interferometric data are zero-filled in the interferometric dimension during the recovery process.