# Signal-to-Noise Ratio Analysis of Bandpass Sampling Time-Modulated Fourier Transform Spectroscopy
## 4 Simulation Analysis and Verification
### Simulation Analysis of Signal-to-Noise Ratio for Different m Values  
In BPS-FTS, the value of \(m\) is an extremely important parameter, and its impact on the SNR is also a very important issue. Through simulation analysis of the impact of the values of \(m\) and \(r_{B}\) on the BPS-FTS SNR, we obtained the results of the average SNR of the BPS-FTS and FTS interferogram when different values of \(m\) and \(r_{B}\) are taken, as shown in Figure 8. Because \(m\) can take many values, in order to more intuitively show the comparison effect of the average SNR of the interferogram under different parameters, we only select a few \(m\) values for display. Figure 9 shows the average SNR of the reconstructed spectra of FTS and BPS-FTS under different \(m\) and different \(r_{B}\).  
Figure 6: The number of electrons in FTS.  
Figure 7: The number of electrons in BPS-FTS.  
In order to more clearly show the difference among the curves under different \(m\), in Figure 10, we partially magnify the average SNR value of the reconstructed spectrum at \(r_{B}=0.5\) in Figure 9.  
In order to more vividly demonstrate the influence of \(m\) on the average SSNR of the interferogram and the reconstructed spectrum, without loss of generality, we still take the case of \(r_{B}=0.5\) for simulation. The simulation curves of the average SNR of the interferogram and the reconstructed spectrum at different \(m\) are shown in Figures 11 and 12, respectively. From the figures, we can see that at this time, the average SNR of the interferogram increases with the increase in \(m\), but the average SNR of the reconstructed spectrum maintains an oscillating equilibrium state, basically maintained at around 1342. This is consistent with the calculated values in Section 4.1 and the results shown in Figure 5, which also verifies our conclusion in Section 3.  
Figure 8: Comparison curves of the average SNR of the interferogram between FTS and BPS-FTS under different \(m\) and \(r_{B}\). It can be seen that when \(m\) is a fixed value, the average SNR of the interferogram increases with the increase in \(r_{B}\). When \(r_{B}\) is a fixed value, as the value of \(m\) increases, the average SNR of the interferogram also increases. When \(m>1\), the average SNR of the interferogram is greater than the average SNR of the interferogram when \(m=1\) (FTS), which is consistent with our previous analysis conclusions.  
Figure 9: Comparison curves of the average SNR of the reconstructed spectra between FTS and BPS-FTS under different \(m\) and \(r_{B}\). It can be seen that when \(m\) is a fixed value, the average SNR of the reconstructed spectrum increases with the increase in \(r_{B}\); when \(r_{B}\) is the same, as \(m\) increases, the average SNR of the spectrum remains basically unchanged. Among them, when \(m=1\) is the case of FTS, which also shows that when the integration time increases proportionally with the increase in the sampling interval of OPD (that is, when \(r_{B}\) is the same), BPS-FTS and FTS can achieve basically the same SNR.