# Suitable Integral Sampling for Bandpass-Sampling Time-Modulated Fourier Transform Spectroscopy
## 2 Principle of BPS-FTS  
Figure 1 shows the BPS-FTS scheme based on Michelson moving mirror FTS [14], which is commonly referred to as time-modulated FTS. In this spectrometer, a grating is introduced into the interference path, dividing the broad spectrum into a sequence of narrow spectra. This allows for bandpass sampling of each interferogram corresponding to each narrowband spectrum, resulting in a substantial decrease in the number of sampling points for each interferogram. Simultaneously, bandpass sampling can widen the OPD sampling interval and reduce the sampling frequency. While the moving mirror undergoes a uniform reciprocating motion, each pixel of the linear array sensor produces a complete interference signal corresponding to a narrowband spectrum during one-way linear motion. After applying Fourier transforms to the sampled interference signals from each pixel, a series of reconstructed narrowband spectra can be generated. These reconstructed narrowband spectra can be organized in wavenumber order, following which their intensities can be summed to obtain the spectrum for the entire range of the spectrum. Figure 2a illustrates the operating principle of BPS-FTS. Figure 2b illustrates the calculation steps of the reconstructed spectrum for each pixel. Figure 1: Schematic diagram of the principle of BPS-FTS. Figure 2: _Cont._  
Ideally, the diffraction effect of the grating disperses the entire spectrum detected by the sensor into a continuous distribution of monochromatic light. This monochromatic light is systematically sampled at uniform wavelength intervals by a linear array sensor, where each pixel corresponds to a specific narrow segment of the spectrum. Assuming that the range of the spectrum for pixel \(j\), represented in wavenumbers, is denoted by \(\upsilon_{\max}^{j}\sim\upsilon_{\min}^{j}\), we can apply the fundamental theory of FTS to express the interference signal received by the receiver. Figure 2: (**a**) Operating principle of BPS-FTS, the blue arrow represents that the OPD is constantly changing. (**b**) The calculation steps of the reconstructed spectrum for each pixel. pixel \(j\), as indicated in Equation (1), with \(B(\
u)\) representing the incident light intensity and \(x\) representing OPD. As \(x\) continuously varies within a specified range, Equation (1) allows us to generate the complete interferogram curve within a maximum OPD range. \[l^{j}(x)=\frac{1}{2}\int\limits_{\
u_{\min}}^{\
u_{\max}^{j}}B(\
u)[1+\cos(2 \pi\
u x)]d\
u. \tag{1}\]  
In conventional FTS, the interferogram is typically sampled following the Nyquist sampling theorem, where the minimum frequency of the measured signal is considered to be zero. However, in practical applications, the measurable spectrum is usually limited to a finite bandwidth. Bandpass sampling theory allows the interferogram to be sampled beyond the constraints of the Nyquist sampling theorem, ensuring that the spectrum is reconstructed without introducing aliasing. Assuming incident light with a spectrum range of \(\
u_{\min}\) to \(\
u_{\max}\) (herein, \(\
u_{\min}=1/\lambda_{\max}\) and \(\
u_{\max}=1/\lambda_{\min}\)), the sampling interval \(\Delta x\) for the interferogram can be determined using Equation (2) [9; 10; 14], where \(m\) is any positive integer less than or equal to \(\
u_{\max}/(\
u_{\max}-\
u_{\min})\). When \(m=1\), this condition corresponds to the Nyquist sampling condition, which has been widely studied. Therefore, the theoretical analysis in this paper is limited to positive integers for \(m\) within the range \([2,m_{\max}]\). \[\frac{m-1}{2\
u_{\min}}\leq\Delta x\leq\frac{m}{2\
u_{\max}}. \tag{2}\]  
Ideally, the sampled interferogram of a BPS-FTS system can be understood as the convolution of an ideal interferogram with a one-dimensional comb function. However, in practical BPS-FTS systems, in order to ensure that the sampled interferogram and the reconstructed spectrum satisfy the SNR requirement, each sampling point of the interferogram needs to be held for a short duration.