# Signal-to-Noise Ratio Analysis of Bandpass Sampling Time-Modulated Fourier Transform Spectroscopy
## 3 Correlation Analysis Between Interferogram and Reconstructed Spectrum Signal-to-Noise Ratio
Therefore, Equation (23) can be rewritten as follows:  
\[K\cdot I_{ave}^{2}\approx\frac{1}{K}\cdot V\cdot B_{ave}^{2} \tag{26}\]  
According to the basic definition of SNR, the average SNR of the interferogram can be obtained, which is the ratio of the average energy of the interferogram to the average energy of the noise, that is:  
\[SNR_{I_{ave}}=I_{ave}/n_{I_{ave}}=\sqrt{\frac{V}{K}}\cdot\frac{B_{ave}}{n_{B_{ave }}} \tag{27}\]  
Similarly, the average SNR of the reconstructed spectrum is the ratio of the average energy of the reconstructed spectrum to the average energy of the noise as follows:  
\[SNR_{B_{ave}}=B_{ave}/n_{B_{ave}} \tag{28}\]Combining Equations (27) and (28), we can obtain that there is the following corresponding relationship between the average SNR of the interferogram and the average SNR of the reconstructed spectrum:  
\[SNR_{Bev}=\sqrt{\frac{K}{V}}SNR_{lev} \tag{29}\]  
It can be seen from Equation (29) that the average SNR of the reconstructed spectrum is proportional to the average SNR of the interferogram, proportional to the square root of the number of sampling points of the interferogram, and inversely proportional to the square root of the number of spectral bands of the reconstructed spectrum. That is, when the average SNR of the interferogram remains unchanged, the SNR of the reconstructed spectrum increases with the increase in the number of sampling points of the interferogram and decreases with the increase in the number of spectral segments of the reconstructed spectrum signal. This is also the basis for the subsequent derivation of the average SNR of the reconstructed spectrum in this paper. According to Equations (12), (20) and (29), the average SNR of the reconstructed spectrum of BPS-FTS is as follows:  
\[ \begin{array}{l}SNR_{BPS-B_{max}}=\sqrt{\frac{iB}{V}}SNR_{I_{BPS}}\\ =\sqrt{\frac{iB}{V}}\cdot\frac{1}{iB}\cdot\sqrt{\frac{A_{d}\pi(1-\epsilon)}{4F^ {2}}}\cdot\frac{1}{2s}\cdot\tau\cdot\eta\cdot\frac{1}{hcv}\cdot RT\cdot\frac{1} {\
u_{\max}}\cdot\Delta v\cdot\\ \frac{iB-iB}{\sum\limits_{iB=1}^{iB}\sqrt{\frac{iB-\
u}{V}}B(\
u_{ib})\cdot r_ {B}\cdot m\cdot\left[1+\cos(2\pi\
u_{ip}x_{ib})\text{sinc}(\frac{\pi\
u_{ip}r _{B}\cdot m}{2\
u_{\max}})\right]}\\ =\sqrt{\frac{m}{4L\
u_{\max}V}}\cdot\frac{A_{d}\pi(1-\epsilon)}{4F^{2}}\cdot \frac{1}{2s}\cdot\tau\cdot\eta\cdot\frac{1}{hcv}\cdot RT\cdot\frac{1}{\
u_{\max }}\cdot\Delta v\cdot\\ \frac{iB-4L\
u_{\max}/m}{\sum\limits_{ib=1}^{iB-\
u}\sqrt{\sum\limits_{ib=1}^{ iB-\
u}B(\
u_{ib})\cdot r_{B}\cdot m\cdot\left[1+\cos(2\pi\
u_{ip}x_{ib})\text{ sinc}(\frac{\pi\
u_{ip}r_{B}\cdot m}{2\
u_{\max}})\right]}}\end{array}  \tag{30}\]  
Combining Equation (30) to analyze the average SNR of the reconstructed spectrum of BPS-FTS, the following conclusions can be obtained:  
1. When \(m\) is a fixed value, as \(r_{B}\) increases, the average SNR of the reconstructed spectrum also increases. 2. When \(r_{B}\) is a fixed value, as \(m\) increases, the average SNR of the interferogram will fluctuate slightly, causing the average SNR of the reconstructed spectrum to also fluctuate slightly, but has overall consistency. 3. When \(r_{B}\cdot m\) is a fixed value, the average SNR of the reconstructed spectrum also decreases (the number of sampling points of the corresponding interferogram decreases). Therefore, under the same integration time, when \(m>1\), the average SNR of the reconstructed spectrum of BPS-FTS is lower than that of FTS. At this time, it is necessary to make a comprehensive consideration between low sampling rate and high signal-to-noise ratio and select the FTS solution as needed.