# Suitable Integral Sampling for Bandpass-Sampling Time-Modulated Fourier Transform Spectroscopy
## 4 Simulation Analysis and Verification
The reconstructed spectrum clearly exhibited significant aliasing, leading to substantial distortion when compared to the shape of the input spectrum. To quantitatively analyze the accuracy of the reconstructed spectrum, we employed the spectral relative deviation method to calculate the relative deviation between the reconstructed spectrum and the input spectrum. Additionally, the SCM method was used to assess the similarity between the reconstructed spectrum and the input spectrum, with the aim of quantitatively evaluating the accuracy of the restored spectrum under different sampling integration times [14]. Equation (16) provides the formula for calculating the relative deviation:  
\[R_{error}=\frac{1}{N}\sum_{n=1}^{N}\frac{\left|S_{n}^{\prime}-S_{n}\right|}{S _{n}}\times 100\%, \tag{16}\]  
where \(S_{n}^{\prime}\) is the reconstructed spectrum, \(S_{n}\) is the standard spectrum, and \(N\) is the number of segments of the spectrum. Figure 4:0 Comparison between the interferogram at \(r=0.8\) and an ideal interferogram. The black curve represents the ideal interferogram after normalization, while the green curve represents the interferogram normalized at \(r=0.8\). SCM is derived from the Pearson correlation coefficient, the definition of which is given in Equation (17):  
Figure 4:1  
where \(S^{\prime}_{n}\) is the reconstructed spectrum, \(\mu_{S^{\prime}_{n}}\) is the mean of the reconstructed spectrum, \(S_{n}\) is the input spectrum, and \(\mu_{S_{n}}\) is the mean of the input spectrum. To mitigate the impact of the Gibbs effect at both ends of the spectrum on the accuracy of the reconstructed spectrum, we excluded a small portion of the data (those with relatively minor amplitudes). Calculations using Equations (16) and (17) revealed that, when \(r=0.8\), the spectral relative deviation reached 39.29%, while the SCM was only 63.76%. These results demonstrate that the accuracy of the reconstructed spectrum is notably poor in this instance. Based on the earlier analysis, it was deduced that, within the range of \(0\leq r\leq 1\), the reconstructed spectrum exhibited significant distortion with a periodicity of \(T\approx 0.04\), encompassing a total of \(N=\frac{1}{7}=25\) cycles. This is corroborated by the simulation results of spectral relative deviation and SCM, depicted in Figures 9 and 10. We also found that under the \(r\) value marked in Figure 6, the relative spectral deviation and SCM of the corresponding restored spectrum were within the tolerance range. This also proves that the criterion of Equation (13) is more applicable to BPS-FTS than Equation (11). As \(r\) increased, the spectral relative deviation generally increased, the SCM decreased, and the likelihood of reconstructed spectra distortions also increased. (2) Simulation Analysis of Interferograms and Reconstructed Spectra when Equation (13) is Met. To assess the error in the reconstructed spectra when Equation (13) is met, we arbitrarily selected two values of \(r\); namely, \(r=0.055\) and \(r=0.34\). In these two instances, we conducted simulations for interferogram sampling and the reconstruction of the spectrum based on the input spectrum. The maximum OPD of the BPS-FTS, as well as the values of \(m\) and \(k\), remained consistent with the parameters mentioned earlier, ensuring the same sampling interval as previously described. For these two distinct \(r\) values, we derived the corresponding sampled interferograms and compared them with the ideal interferogram. Figure 4:2 Reconstructed spectrum at \(r=0.8\). The black curve represents the input spectrum, while the green curve represents the normalized reconstructed spectrum at \(r=0.8\). The results are depicted in Figure 4:3 Notably, these figures reflect the \(r\) values, and the normalized sampled interference data aligned closely with the ideal interference data.