## 5 Conclusions  
Compared with the traditional Fourier transform spectroscopy technique, BPS-FTS can significantly increase the sampling step and thus reduce the number of sampling points, which leads to a reduction in difficulty for FTS implementation. However, to date, although many studies exist, all of them solely focus on the introduction of technical principles, and there is a distinct lack of analysis of the various influencing factors in the technology implementation process, thus increasing the difficulty with which the technology can be implemented quickly. To address this problem, we focused on theoretical research and simulation analysis of the optical system's diffraction effect.  
Through theoretical analysis and simulation, it can be observed that owing to the influence of the diffraction effect, the narrowband spectral bandwidth corresponding to each sampling pixel is 4.48 times that of the spectral bandwidth without considering the diffraction effect. As a result, the interferogram sampling step, when each pixel does not consider the diffraction effect, is approximately 4.5 times that of the interferogram sampling step when the diffraction effect is considered. Therefore, for each pixel, if the actual interferogram is sampled according to the ideal interferogram sampling step, this leads to interferogram under-sampling, which, in turn, leads to spectral aliasing of the recovered spectrum, resulting in a serious distortion of the spectrum. The above conclusion is also verified by the recovered spectrum simulation analysis. The sampling interferogram-recovered spectrum considering the influence of the diffraction effect is fundamentally consistent with the input spectrum, the relative deviation between them is 1.30%, and the spectral similarity is 0.9669. However, without considering the diffraction effect, the  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Considering the Diffraction Effect** & **Without Considering the Diffraction Effect** \\ \hline R error & 1.30\% & 6.23\% \\ SCM & 0.9669 & \(-\)0.0056 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Calculation results of spectral accuracy evaluation criteria.  
sampling interferogram-recovered spectrum is far from the input spectrum: the relative deviation of the two is 6.23%, and the spectral similarity is only \(-0.0056\). Therefore, in the technical realization process of BPS-FTS, the influence of the diffraction effect of the optical system must be fully considered. In the future, we will conduct further theoretical analysis and research on other errors, such as non-uniform sampling and noise, hoping to promote the implementation and popularization of this technology.  
In conclusion, aiming at the phenomenon by which the real narrowband spectral boundary is difficult to define due to the diffraction effect in BPS-FTS, we deduced the real narrowband spectral boundary and solved the problems of interferogram under-sampling and restoring spectral distortion caused by the inaccurate spectral boundary. These results provide effective theoretical and technical support for the implementation of this technology.  
Conceptualization, X.C., Q.L., B.X. and Z.T.; methodology, X.C. and Q.L.; software, X.C., Q.L. and Y.T.; investigation, Y.T., J.W., N.Z., W.L., Y.L. and J.S.; resources, Y.L.; data curation, X.C., Q.L. and B.X.; writing--original draft preparation, X.C., Q.L. and B.X.; writing--review and editing, X.C., Q.L. and B.X.; supervision, B.X., Q.L., Y.L. and Z.T.; project administration, Q.L., B.X. and Y.L. All authors have read and agreed to the published version of the manuscript.  
Not applicable.  
The authors declare no conflict of interest.