## 2 Introduction to Bandpass Sampling Theory and BPS-FTS
### Bps-Fts  
According to the analysis in Section 2, it can be observed that to fully exploit the advantages of bandpass sampling theory and reduce the number of interferogram sampling points, it is first necessary to subdivide the broadband spectrum into narrowband spectra. According to spectral imaging theory, a grating is an effective element capable of subdividing a broadband spectrum into a continuous narrowband spectrum. If a plane reflection grating (or a transmission grating) is set in the outgoing optical path of a [PERSON] interferometer, then the parallel interfering beam from the outgoing optical path of the interferometer will be diffracted by the grating after entering the plane reflection grating,forming a continuous monochromatic interfering beam arranged by wavelength, after which it will enter the focusing lens and finally interfere at the image plane. If a line array sensor is used, then a continuous narrowband interferogram can be obtained. The basic schematic is shown in Figure 1.  
When the moving mirror moves at a constant velocity back and forth, each sensor pixel outputs a complete interferometric signal corresponding to a narrowband spectrum during a one-way constant linear motion. According to the bandpass sampling theory, as the spectral range corresponding to the interferogram obtained by each pixel decreases exponentially, the interferogram sampling step also increases exponentially. If we perform Fourier transformation on the sampled interferometric signal, we can recover a series of narrowband spectra. By arranging all pixel recovery spectra by wavenumber and superimposing their intensities, the complete spectrum can be obtained. Assuming that the number of pixels along the grating dispersion direction of the line array sensor is \(M\) and the corresponding pixel size is \(P\), then the sensor length along the grating dispersion direction is \(L_{D}=M\cdot P\). Let the spectrum detected by the system be \(spe(\lambda)\) and its effective band range be \(\lambda_{\text{min}}\leq\lambda\leq\lambda_{\text{max}}\). The corresponding effective wavenumber range is \(\
u_{\text{min}}\leq\
u\leq\
u_{\text{max}}\), where \(\
u_{\text{max}}=1/\lambda_{\text{min}}\), \(\
u_{\text{min}}=1/\lambda_{\text{max}}\). Ideally, we typically consider the grating dispersion to be linear, and the wavelength of the dispersion spectrum is uniformly distributed on the sensor image plane. As the sensor is rectangularly sampled, it will divide the broadband spectral signal into \(M\) segments of discrete narrowband spectral signals at equal wavelength intervals, and for the \(i\)th pixel, the wavelength of the narrowband spectral signal covered is shown in Equation (6):  
\[\left\{ \begin{array}{l}\lambda_{\text{min}}^{i}=\lambda_{\text{min}}+(i-1) \cdot\Delta\lambda\\ \lambda_{\text{max}}^{i}=\lambda_{\text{min}}+i\cdot\Delta\lambda\end{array} \right. \tag{6}\]  
where \(\Delta\lambda=(\lambda_{\text{max}}-\lambda_{\text{min}})/M\). However, for a practical optical system, the image formed by a point object is typically a diffraction pattern determined by the system aperture diaphragm [11; 12; 13], which leads to crosstalk and aliasing of the spectra between adjacent pixels. In this case, the actual  
Figure 1: Basic schematic of Fourier transform spectroscopy with bandpass sampling (BPS-FTS). narrowband spectral signal wavelength range corresponding to each pixel is larger than the wavelength range determined by Equation (6). If we perform bandpass sampling on the interferogram of the actual narrowband spectrum according to the spectral range determined by Equation (6), the sampling step at this point does not meet the actual narrowband spectrum bandpass sampling condition, and the spectrum recovered from the sampled interferogram will produce spectral aliasing, resulting in the distortion of the recovered spectrum. However, the existing literature does not include a theoretical analysis of the influence of the diffraction effect.