# Signal-to-Noise Ratio Analysis of Bandpass Sampling Time-Modulated Fourier Transform Spectroscopy
## 2 Modeling Analysis of SNR  
### Modeling of Monochromatic Light Interferogram Based on Energy Transfer  
BPS-FTS is an innovative fusion of dispersion and interference technology based on traditional FTS. In BPS-FTS, each pixel of the linear array detector collects the interferogram of the target, samples the interferogram through the bandpass sampling theory, and performs Fourier transform on the sampled interferogram to obtain a series of narrowband spectra. The reconstructed spectra of all pixels are arranged by wave number, and the intensities at the same wave number are superimposed to obtain the final reconstructed spectrum of the complete spectrum. The basic principle is shown in Figure 1.  
According to the empirical formula \(error=\frac{4}{SNR\cdot v_{\max}}\) (where \(v_{\max}\) is the max wave number) of sampling accuracy proposed by [PERSON], because \(\Delta x=\frac{1}{a}\cdot\frac{1}{2v_{\max}}\) (where \(\Delta x\) is the sampling interval and \(a\) is any positive number greater than or equal to 1), the sampling error tolerance \(error=\frac{4}{SNR\cdot v_{\max}}=\frac{8a}{SNR}\Delta x\) can be obtained. It can be seen that the sampling error tolerance is proportional to the sampling interval. The larger the sampling interval, the larger the sampling error tolerance. Therefore, under the same resolution conditions, BPS-FTS can reduce the accuracy and stability requirements for the interferometer moving parts. According to the SNR equation, to analyze and evaluate the BPS-FTS SNR, we first need to obtain the energy and noise of the interferogram finally collected by the system. According to the analysis of the BPS-FTS signal transmission link, the energy of the target needs to be transmitted through the optical system, finally reaches the detector, and is converted from an optical signal to an electrical signal. This transmission process follows the radiation energy transfer theory and can be quantitatively described. Therefore, we analyze the radiation energy transfer process of BPS-FTS and establish an energy model of monochromatic light interferograms to provide model support for the subsequent extended spectrum SNR analysis. In FTS, the spectrum is usually described by wave number \(\
u\) (the inverse of the wavelength). Assume that \(\tau\) is the transmittance of the optical system, \(D_{p}\) is the diameter of the primary mirror of the optical system, \(D_{s}\) is the diameter of the secondary mirror of the optical system, and \(\varepsilon=D_{s}^{2}/D_{p}^{2}\) is defined to represent the shading ratio of the optical system. \(F=f/D_{p}\) is the system \(F\) number given by the focal length \(f\), \(A_{d}\) is the detector pixel size, and \(I(x,\
u)\) is the spectral radiance at the detector image plane. Therefore, the  
Figure 1: The Basic Principle of Michelson Time-Modulated BPS-FTS. A grating (either a plane reflection grating or a transmission grating) is set in the output light path of the Michelson interferometer. Then, after the parallel interference light in the output light path of the interferometer enters the grating, it is diffracted by the grating to form continuous monochromatic interference light arranged by wavelength, then enter the convergent lens, and finally interfere at the image plane. By using a linear array detector, a continuous, narrow spectral interferogram can be obtained. -luminous flux \(\Phi_{d}(x,\
u)\) corresponding to the detector can be approximately written as follows [17; 18]:  
\[\Phi_{d}(x,\
u)=\frac{A_{d}\pi(1-\varepsilon)}{4F^{2}}\tau I(x,\
u) \tag{1}\]  
In order for the detector to receive enough energy, the interferogram usually requires a certain integration time \(\Delta t\) during the sampling process. At this time, the energy corresponding to the interferogram is the integral of the energy within the OPD range corresponding to the monochromatic light \(\
u\) within the time \(\Delta t\).