## 4 Simulation Analysis and Verification
This process corresponds to the data interpolation of the recovered spectral dimensions without changing spectral information. Accordingly, we performed a spectral recovery simulation on the above two interferograms. The recovered spectra of each pixel with and without considering the diffraction effect are shown in Figures 11 and 12, respectively. Figure 11: Recovered spectra of each pixel, considering the diffraction effect. Figure 10: Interferogram of the spectrum corresponding to the fifth pixel. Insets show isolated interferogram spectrums. We took the recovered spectra of the middle six pixels in both cases and normalized them with the input spectrogram; the comparison is shown in Figure 13. The black line in Figure 13 represents the real input spectrum after considering the diffraction effect, the red line represents the recovered spectrum without considering the diffraction effect, the blue line represents the recovered spectrum after considering the diffraction effect, and the green line represents the curve of the normalized energy collected by each pixel when the system is stationary with 0 optical path difference, such as for a grating spectrometer. It can be observed that if the diffraction effect is not considered, the recovered spectral curve will undergo serious aliasing; hence, only a lower spectral resolution can be achieved. In the case where the diffraction effect is considered, the recovered spectrum is fundamentally the same as the input spectral curve with more obvious details. This is also consistent with our theoretical analysis in Section 4, and the green marker line trend is also consistent with the spectrum trend. To quantitatively evaluate the recovered spectral accuracy with and without the diffraction effect, a method is used to calculate the relative deviation of the recovered spectrum from the input spectrum, and the spectral correlation  
Figure 12: Recovered spectra of each pixel, without considering the diffraction effect. Figure 13: Comparison of recovered and input spectra. used to calculate the similarity between the recovered and input spectra. Equation (22) details the relative deviation calculation formula as follows [22]:  
\[R_{error}=\frac{1}{N}\sum\limits_{n=1}^{N}\frac{|S_{n}^{\prime}-S_{n}|}{S_{n}} \times 100\% \tag{22}\]  
where \(S_{n}^{\prime}\) is the recovery spectrum, \(S_{n}\) is the standard spectrum, and \(N\) is the number of spectral bands. The SCM method [23; 24] is a derivative of the Pearson's correlation coefficient and is defined as follows:  
\[R_{S,S^{\prime}}=\frac{\sum\limits_{n=1}^{N}{(S_{n}^{\prime}-\mu_{S_{n}^{ \prime}})(S_{n}-\mu_{S_{n}})}}{\sqrt{\sum\limits_{n=1}^{N}{(S_{n}^{\prime}-\mu_ {S_{n}^{\prime}})^{2}\sum\limits_{n=1}^{N}{(S_{n}-\mu_{S_{n}})^{2}}}}} \tag{23}\]  
where \(S_{n}^{\prime}\) is the recovered spectrum, \(\mu_{S_{n}^{\prime}}\) is the mean value of the recovered spectrum, \(S_{n}\) is the input spectrum, and \(\mu_{S_{n}}\) is the mean value of the input spectrum. The results of the calculations are listed in Table 5.  
From these calculation results, we can observe that the relative deviation of the recovered spectrum considering the diffraction effect is smaller than the relative deviation of the recovered spectrum without considering the diffraction effect. Moreover, the spectral similarity between the recovered spectrum considering the diffraction effect and the input spectrum is significantly greater than that without considering the diffraction effect, and its similarity to the input spectrum is close to 1. This demonstrates that in BPS-FTS, the influence of the diffraction effect must be considered.