# 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
In fact, for FTS, within a certain integration time, the change in time corresponds to the change in OPD, and the change in OPD will cause the change in the interferogram. Therefore, the actual corresponding energy should be based on the change in OPD within the time \(\Delta t\), and the changing interferogram should be integrated within this change. In FTS, if the moving speed of the moving lens is \(s\), then the change in OPD within the integration time \(\Delta t\) is \(\Delta x=2s\Delta t\), and the corresponding energy within this integration time is as follows:  
\[P_{d}(x,\
u)=\frac{A_{d}\pi(1-\varepsilon)}{4F^{2}}\cdot\tau\cdot\frac{1}{2s} \cdot\int\limits_{x-s\Delta t}^{x+s\Delta t}I(x,\
u)dx \tag{2}\]  
For monochromatic light, the corresponding single photon energy is \(E_{1}=hc/\lambda=hcv\), from which we can get the number of photons reaching the detector at \(x\) (OPD) within the integration time \(\Delta t\):  
\[ \begin{split} e_{p}(x,\
u)&=P_{d}(x,\
u)/E_{1}\\ &=\frac{1}{hcv}\cdot\frac{A_{d}\pi(1-\varepsilon)}{4F^{2}}\cdot \tau\cdot\frac{1}{2s}\cdot\int\limits_{x-s\Delta t}^{x+s\Delta t}I(x,\
u)dx \end{split}  \tag{3}\]  
According to the photoelectric conversion effect of the detector, the detector converts the received photons into electrons at a certain efficiency. Assuming that the average quantum efficiency of the detector in the response band is \(\eta\), then for FTS, within the integration time \(\Delta t\), for monochromatic light with a wave number of \(\
u\), at the OPD of \(x\), the number of electrons corresponding to the target light signal received by the detector is as follows:  
\[ \begin{split}& e_{s}(x,\
u)=e_{p}(x,\
u)\cdot\eta\\ &=\frac{1}{hcv}\cdot\frac{A_{d}\pi(1-\varepsilon)}{4F^{2}}\cdot \tau\cdot\eta\cdot\frac{1}{2s}\cdot\int\limits_{x-s\Delta t}^{x+s\Delta t}I(x, \
u)dx\end{split}  \tag{4}\]  
According to the basic principle of FTS, monochromatic light with a wave number \(\
u\) is divided into two coherent beams with equal energy after entering the interferometer. At this time, the spectral radiance modulated by interference at the image plane [19; 20; 21; 22; 23; 24] is as follows:  
\[I(x,\
u)=2 RTB(\
u)[1+cos(2\pi\
u x)] \tag{5}\]  
where R is the reflectance of the beam splitter and T is the transmittance. Substituting Equation (5) into Equation (4), the energy of the interferogram expressed in electrons is as follows:  
\[ \begin{split}& e_{s}(x,\
u)=\frac{1}{hcv}\cdot\frac{A_{s}\pi(1- \varepsilon)}{4F^{2}}\cdot\tau\cdot\eta\cdot\frac{1}{2s}\cdot\int\limits_{x-s \Delta t}^{x+s\Delta t}2 RTB(\
u)[1+cos(2\pi\
u x)]dx\\ &=\frac{1}{hcv}\cdot\frac{A_{d}\pi(1-\varepsilon)}{4F^{2}}\cdot \tau\cdot\eta\cdot\frac{1}{2s}.\\ &[2 RTB(\
u)\cdot 2s\Delta t+2s\Delta t\cdot 2 RTB(\
u)\cdot\cos(2\pi \
u x)\text{sinc}(2\pi\
u s\Delta t)]\end{split}  \tag{6}\]  
For FTS, its noise mainly includes shot noise, detector noise, and digital quantization noise [6]. Among them, shot noise is directly related to the incident light signal. It is usually generated by the photons radiated by the target object, so it is also called photon noise. Its change obeys Poisson distribution [6; 18]. Therefore, even under a stable light source, photon noise still exists, and its energy depends on the number of photons and the randomness of reaching the detector. This is also the main source of noise in FTS. The detector noise is determined by the detector's own characteristics and the use environment. Moreover, the distribution on each channel of the FTS is independent and the same and has nothing to do with the incident light power and the sampling method. Compared with shot noise (photon noise) [6; 17; 18], the impact of detector noise on the SNR in FTS is very small and can be ignored. Digital quantization noise can also be avoided by selecting high-precision, low-noise ADC devices, optimizing PCB layout and circuit design, and optimizing signal processing algorithms (such as digital filtering, error correction, etc.) [6]. Therefore, this paper only considers the impact of photon noise.