# Signal-to-Noise Ratio Analysis of Bandpass Sampling Time-Modulated Fourier Transform Spectroscopy
## 4 Simulation Analysis and Verification  
Given the importance of signal-to-noise ratio analysis, in order to verify the analysis conclusions in Sections 2 and 3, we use the typical solar altitude angle and target reflectivity as the input conditions of MODTRAN(PcModWin 4.0 v3r1 Version 1.2) and simulate the spectral range of 3985 cm\({}^{-1}\)-5015 cm\({}^{-1}\). When the solar altitude angle is 60\({}^{\circ}\) and the target reflectivity is 0.2, the corresponding spectral radiance curve (as shown in Figure 2) is obtained. According to the SNR formula, when the maximum wave number of the spectrum is determined, no matter how the spectrum range changes, for FTS, the average SNR of the reconstructed spectrum will not change. Therefore, in order to facilitate simulation and comparative analysis, we select a spectrum with a wave number range of 3995 cm\({}^{-1}\)\(\sim\)4009 cm\({}^{-1}\) (as shown in Figure 3) based on Figure 2 as the simulation input spectrum for analysis and verification. This setting has no effect on the analysis conclusions.  
Based on the theoretical model of SNR established in the previous studies, the instrument system parameters are first determined in the simulation process, and then the SNR of the interferogram and reconstruction spectrum under different parameters are simulated and calculated, respectively, through the established model. The instrument system parameters used in the simulation are shown in Table 1.  
For the detector parameter setting, we refer to the parameters of HAMAMATSU's G11478-512 WB detector (Hamamatsu, Japan) for optimization and adjustment. The detector is a linear array detector with a single pixel size of 25 \(\upmu\)m \(\times\) 250 \(\upmu\)m and a total of 512 pixels. When designing the parameters, we merge the pixels, merging every 10 pixels into 1 pixel. The size of the merged pixel is 250 \(\upmu\)m \(\times\) 250 \(\upmu\)m, so that the pixel size matches  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Spectral range & 3995 cm\({}^{-1}\)-4009 cm\({}^{-1}\) \\ Spectral resolution & 0.02 cm\({}^{-1}\) \\ Optical system shading ratio & 1:9 \\ F number & 2.63 \\ Average transmittance of an optical system & 0.5 \\ Average quantum efficiency of detector & 0.7 \\ Reflectance of a beam splitter & 0.5 \\ Transmittance of a beam splitter & 0.5 \\ Speed of moving lens & 0.25 cm/s \\ Maximum OPD & 2.5 cm \\ Pixel area & 250 \(\upmu\)m \(\times\) 250 \(\upmu\)m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
Figure 3: Input spectrum. Part of the spectral radiance curve obtained by MODTRAN in Figure 2.  
Figure 2: Spectral radiance curve obtained by MODTRAN.  
the spot shape. The number of merged pixels is 51. Assuming that the selected spectrum is received by the \(j\)th pixel after passing through the optical system, and assuming that the area occupied by the on-axis object point on the image plane is basically the same as the area of a merged pixel of the detector, it can be obtained that the luminous flux corresponding to the detector pixel \(j\) is consistent with Equation (1), and the SNR under the set parameters can be simulated and analyzed based on it.  
According to the aforementioned SNR model, the SSNR of BPS-FTS is not only affected by the ratio \(r_{B}\), but also by the value of \(m\). Therefore, it is necessary to simulate and analyze the SNR under different parameter conditions. For a given spectral range, according to the basic properties of bandpass sampling, we calculated that \(m\) can be any positive integer between 1 and 286. When m = 1, bandpass sampling is equivalent to Nyquist sampling, and BPS-FTS is equivalent to FTS. Both the average SNR of the interferogram and the average SNR of the reconstructed spectrum of FTS and BPS-FTS are consistent.