# Suitable Integral Sampling for Bandpass-Sampling Time-Modulated Fourier Transform Spectroscopy
## 4 Simulation Analysis and Verification  
To assess the influence of integration sampling time on the interferogram and the resulting reconstructed spectrum, we performed the simulation using Matlab, its version number is '9.8.0.1873465 (R2020a) Update 8'. We used the \"MATLAB and Simulink Student Suite\" and \"MATLAB and Simulink Student Suite\" toolboxes. In the simulation, we refer to the standard solar spectrum and randomly select a part as the input spectrum of any pixel for analysis to ensure the randomness of the simulation analysis. This particular spectrum spanned the wavenumber range from 4875.77 cm\({}^{-1}\) to 4884.81 cm\({}^{-1}\), and its spectral profile is presented in Figure 4. In this context, in accordance with Equation (2), \(m\) can assume any positive integer value up to and including 540. Without loss of generality, the simulation results of other pixels are similar to this pixel. Figure 4: Input spectrum. The simulations in this paper are all based on this input spectrum. (1) SMTF analysis  
Without loss of generality, we randomly selected \(m=50\) to investigate the connection between integration sampling time and SMTF under the condition \(k=0\), resulting in the adjusted SMTF expression given in Equation (14):  
\[\text{SMTF}=\text{sinc}\bigg{\{}\pi\
u r\bigg{[}\frac{(1-k)m}{2v_{\text{max}}}+ \frac{k(m-1)}{2v_{\text{min}}}\bigg{]}\bigg{\}}=\text{sinc}\bigg{(}\pi\
u r \frac{50}{2v_{\text{max}}}\bigg{)}. \tag{14}\]  
SMTF curves were then generated for each wavenumber \(\
u\) within the range \(0\leq r\leq 1\), which are shown in Figure 5.  
As the range of the spectrum was extremely narrow, the SMTF could be approximated using Equation (15):  
\[\text{SMTF}=\text{sinc}\bigg{(}\pi\
u r\frac{50}{2v_{\text{max}}}\bigg{)} \approx\text{sinc}(25\pi\
u). \tag{15}\]  
According to the analysis in Section 3, it is apparent that, within the range of \(0\leq r\leq 1\), the SMTF is shown in Figure 6. When Equation (13) is satisfied, the spectrum can be accurately restored. We marked the range of \(r\) that satisfy this criterion in Figure 6.  
Figure 5: The SMTF for each wavenumber at different integration times. Figure 6: SMTF values for different integration times. In this case, \(k=0\) and \(m=50\). This figure shows the SMTF curve. The blue curve in the figure represents the values of SMTF within the range where \(0\leq r\leq 1\), while the red curve represents the values of SMTF within the blue curve that satisfy Equation (13). From Equation (9) and Figures 5 and 6, it is evident that SMTF presented zero crossings with a period of \(T\approx 0.04\). This implies that significant distortion occurs in the spectrum when \(r\) is an integer multiple of \(0.04\). In the corresponding period, some values cover the tolerance requirements of the restoration spectrum. However, as the integration time increases, because the value range that satisfies the condition of Equation (13) becomes smaller and smaller, the values of \(r\) that can meet the tolerance of the restoration spectrum accuracy become smaller and smaller. Starting from the 18 th interval, there will be no \(r\) that can meet the restored spectrum tolerance requirements. In order to verify the above conclusion, we conducted a simulation analysis of the interferogram and the reconstructed spectrum with \(r=0.8=20T\). Figure 7 presents the interferogram, with a maximum OPD of \(2.5\) cm and a sampling interval of \(0.0051\) cm. The data in the figure indicate that, at this point, the interferogram error was at its maximum. Figure 8 displays the reconstructed spectrum. During spectrum reconstruction, zero-padding was applied to the interferogram to ensure the smooth display of the reconstructed spectrum curve, facilitating accurate assessment of the similarity between the reconstructed spectrum and the input spectrum. This zero-padding corresponded to data interpolation in the dimension of the reconstructed spectrum and preserved the information of the spectrum.