# Signal-to-Noise Ratio Analysis of Bandpass Sampling Time-Modulated Fourier Transform Spectroscopy
## 2 Modeling Analysis of SNR
### Signal-to-Noise Ratio Modeling of BPS-FTS
\begin{array}{l}e_{B\_x}(x_{ib})=\frac{A_{d}\pi(1-\epsilon)}{4F^{2}}\cdot \frac{1}{2s}\cdot\tau\cdot\eta\cdot\frac{1}{hcv}\cdot RT\cdot\\ \sum\limits_{i=1}^{iv=V}\left\{\left[\frac{r_{B}\cdot m}{\
u_{\max}}\cdot B(v_ {iv})+\frac{r_{B}\cdot m}{\
u_{\max}}\cdot B(v_{iv})\cdot\cos(2\pi\
u_{iv}x_{ ib})\text{sinc}(\frac{\pi\
u_{iv}r_{B}\cdot m}{2\
u_{\max}})\right]\cdot\Delta \upsilon\right\}\end{array}  \tag{17}\]  
Therefore, according to Equations (7) and (17), the photon noise at each \(x_{ib}\) of its interferogram can be expressed as follows:  
\[\begin{array}{l}n_{B\_x}(x_{ib})=\sqrt{e_{B\_x}(x_{ib})}\\ =\sqrt{ \begin{array}{l}\frac{A_{d}\pi(1-\epsilon)}{4F^{2}}\cdot\frac{1}{2s} \cdot\tau\cdot\eta\cdot\frac{1}{hcv}\cdot RT\cdot\\ \sum\limits_{i=1}^{iv=V}\left\{\left[\frac{r_{B}\cdot m}{\
u_{\max}}\cdot B(v_ {iv})+\frac{r_{B}\cdot m}{\
u_{\max}}\cdot B(v_{iv})\cdot\cos(2\pi\
u_{iv}x_{ ib})\text{sinc}(\frac{\pi\
u_{iv}r_{B}\cdot m}{2\
u_{\max}})\right]\cdot\Delta \upsilon\right\}\end{array} } \tag{18}\]Then, the corresponding SNR of the interferogram is as follows:  
\[SNR_{BPS}(x_{ib})=\sqrt{ \begin{array}{c}\frac{A_{d}\pi(1- \epsilon)}{4F^{2}}\cdot\frac{1}{2s}\cdot\tau\cdot\eta\cdot\frac{1}{hcv}\cdot RT \cdot\frac{1}{v_{\max}}\cdot\Delta v\cdot\\ \sum\limits_{iv=1}^{m}r_{B}\cdot m\cdot B(v_{iv})\cdot\left[1+\cos(2\pi v_{iv} x_{ib})\text{sinc}(\frac{\pi v_{iv}r_{B}\cdot m}{2v_{\max}})\right]\end{array} } \tag{19}\]  
According to Equations (12) and (19), the average SNR of the interferogram at this time is as follows:  
\[SNR_{BPS}=mean[SNR_{BPS}(x_{ib})] \tag{20}\] \[=\frac{1}{iB}\cdot\sum\limits_{ib=1}^{ib=iB}\sqrt{\frac{A_{d}\pi( 1-\epsilon)}{4F^{2}}\cdot\frac{1}{2s}\cdot\tau\cdot\eta\cdot\frac{1}{hcv} \cdot RT\cdot\frac{1}{v_{\max}}\cdot\Delta v\cdot r_{B}\cdot m}\] \[=\frac{m}{4 Lv_{\max}}\cdot\sqrt{\frac{A_{d}\pi(1-\epsilon)}{4F^{ 2}}\cdot\frac{1}{2s}\cdot\tau\cdot\eta\cdot\frac{1}{hcv}\cdot RT\cdot\frac{1} {v_{\max}}\cdot\Delta v\cdot r_{B}\cdot m}.\] \[\sum\limits_{ib=1}^{iB=4 Lv_{\max}/m}\sqrt{\sum\limits_{ib=1}^{iB =V}B(v_{iv})\cdot\left[1+\cos(2\pi v_{iv}x_{ib})\text{sinc}(\frac{\pi v_{iv}r _{B}\cdot m}{2v_{\max}})\right]}\]  
where mean [] represents the average value of the sequence value. According to Equation (15), for BPS-FTS, its interferogram sampling integration time \(\Delta t_{B}\) is proportional to \(r_{B}\cdot m\). Since BPS-FTS has many parameters, the following conclusions can be drawn after a specific analysis based on the corresponding relationship between the parameters:  
1. When \(m\) is a fixed value, as \(r_{B}\) increases, it can be seen that the average SNR of the interferogram is also getting higher and higher. 2. When \(r_{B}\) is a fixed value, it can be seen from Equation (20) that with the increase in \(m\), the sampling interval of the interferogram increases, and the average signal-to-noise ratio of the interferogram also increases. Compared with FTS under Nyquist sampling conditions, when the OPD change ratio is the same, the average signal-to-noise ratio of the interference signal of BPS-FTS is higher than that of FTS under Nyquist sampling conditions. 3. When \(r_{B}\cdot m\) is a fixed value, the average signal-to-noise ratio of the BPS-FTS interferogram is consistent with that of the traditional FTS. (It is worth noting that in the actual sampling process, the change in \(m\) will cause the position of the interferogram sampling point to change. Since the interferogram is an oscillating signal, when \(m\) is taken at different values, the actual calculated average signal-to-noise ratio of the interference signal will have slight fluctuations.)