# Suitable Integral Sampling for Bandpass-Sampling Time-Modulated Fourier Transform Spectroscopy
## 5 Conclusions  
The significant advantage of BPS-FTS is its ability to considerably increase the sampling interval, thereby reducing the number of required sampling points. In this paper, we conducted a comprehensive theoretical study and simulation analysis to investigate the impact of the rectangular sampling step size on the reconstructed spectrum, thus further leveraging this advantage. This research was conducted with the aim of offering valuable theoretical and technical insights for the practical implementation of this technology. In the future, we will conduct further experimental verification.  
Both the theoretical analysis and simulation verification results revealed that the actual interferogram under rectangular sampling is influenced by the SMTF of FTS. In the reconstructed spectrum, this influence is evident as the product of the actual spectrum and a sinc function, which varies with different values of \(m\). The integration time has varying effects on the reconstructed spectrum. It was observed that, when the condition shown in Equation (12) is met, the rectangular sampling step allows for accurate spectrum reconstruction within an acceptable error range. At this time, SMTF can reach 0.9. However, outside of this range, there are also some values that can meet the accuracy requirements of the constructed spectrum.  
Our simulation analysis also revealed that, due to the inherent under-sampling characteristics of bandpass sampling, using the average value \(\overline{\text{SMTF}}\) as the MTF criterion (specifically, Equation (13)) provides a more suitable range of integration sampling times that meet the tolerance requirements of the reconstruction spectrum. Therefore, in the practical implementation of systems, Equation (13) can be considered the fundamental design and analysis criterion for integration sampling in BPS-FTS.  
Our theoretical analysis further indicated that changes in integration time can cause periodic and significant distortion in the reconstructed spectrum, with a period of \(T\), which is shown in Equation (9). This distortion is influenced by the values of \(m\) and \(k\) and becomes more pronounced as the integration time increases. It can even make the relative spectral deviation reach 39.29%, and the SCM is reduced to 63.67%. Consequently, in BPS-FTS, while it is possible to extend the sampling interval, the integration time for sampling cannot be arbitrarily increased based solely on the increased sampling interval; instead, it must adhere to the constraints mentioned earlier. This also confirms the correctness of Equation (13) from another aspect.  
Figure 12: Reconstructed spectra. The black curve represents the actual input spectrum, the blue curve represents the reconstructed spectrum for \(r=0.055\), and the red curve represents the reconstructed spectrum for \(r=0.34\).  
**Author Contributions:** Conceptualization, X.C., Q.L. and Z.T.; methodology, X.C. and Q.L.; software, X.C.; investigation, Y.T., P.H., J.W., N.Z., W.L., Y.L., J.S. ([PERSON]) and J.S. ([PERSON]); resources, Y.L.; data curation, X.C. and Q.L.; writing--original draft preparation, X.C.; writing--review and editing, X.C. and Q.L.; supervision, Q.L., Y.L. and Z.T.; project administration, Q.L. and Y.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA28050401) and the National Key Research and Development Program of China, 2022 YFB3903000 (2022 YFB3903001).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data are contained within the article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.