# Suitable Integral Sampling for Bandpass-Sampling Time-Modulated Fourier Transform Spectroscopy
## 3 Analyzing Suitable Sampling Conditions for BPS-FTS
\tag{7}\]  
The SMTF at this point is  
\[\text{SMTF}=\text{sinc}\bigg{\{}\pi\
u r\bigg{[}\frac{(1-k)m}{2i_{\max}}+\frac{ k(m-1)}{2i_{\min}}\bigg{]}\bigg{\}}. \tag{8}\]  
From Equation (8), it is evident that, for any wavenumber \(\
u\) within the range \([\
u_{\min},\
u_{\max}]\), the SMTF associated with the integrated sampled interferogram is influenced by the values of \(m\), \(k\), and \(r\). Different settings for these three parameters introduce variations in the SMTF, resulting in differences in the interferogram, which in turn has an impact on the reconstructed spectrum. It can be observed, from Equation (8), that when \(r=p\times\frac{1-(1-k)m}{v\left[\frac{(1-k)m}{2i_{\max}}+\frac{k(m-1)}{2i_{\min }}\right]}=p\times\frac{2\
u_{\max}\
u_{\min}}{v\left[(1-k)m\
u_{\min}+k(m-1) \
u_{\max}\right]}\) (where \(p\) is a non-negative integer), then \(SMTF=0\). In this scenario, regardless of the variation in the OPD, a precise interferogram corresponding to the wavenumber spectrum cannot be obtained, rendering the reconstruction of the spectrum infeasible. In other words, if interferograms are obtained for different \(r\) values within \(0\leq r\leq 1\) and used for spectrum reconstruction, the resulting spectrum presents maximum distortion with a period of \(T\), as shown in Equation (9). \[T=\frac{2\
u_{\max}\
u_{\min}}{v\left[(1-k)m\
u_{\min}+k(m-1)\
u_{\max}\right]} \tag{9}\]  
Within \(0\leq r\leq 1\), there are \(N=\frac{1}{T}\) cycles, and when \(m\) and \(k\) are fixed, the SMTF range shifts away from its peak position and decreases rapidly as \(r\) increases. The range that meets the error tolerance for spectrum reconstruction also significantly contracts. By combining Equation (4) with Equation (8), we can determine the range of \(r\), as described in Equation (10):  
\[r\leq\frac{1}{2\
u}\cdot\frac{1}{\frac{(1-k)m}{i_{\max}}+\frac{k(m-1)}{i_{ \min}}}. \tag{10}\]  
Considering that the wavenumber range \(\
u\) is constrained within \([\
u_{\min},\
u_{\max}]\), we can establish a further constraint on the values of \(r\) to ensure that interferograms corresponding to all wavenumbers within the entire range of the spectrum can be accurately reconstructed within the defined error tolerance. This constraint is expressed in Equation (11):  
\[r\leq\frac{1}{2\
u_{\max}}\cdot\frac{1}{\frac{(1-k)m}{i_{\max}}+\frac{k(m-1)}{ i_{\min}}}=\frac{1}{2}\cdot\frac{\
u_{\min}}{(1-k)m\
u_{\min}+k(m-1)\
u_{ \max}}. \tag{11}\]  
Based on bandpass sampling theory, each positive integer \(m\) has an associated interval that meets the SMTF criterion. For varying values of \(k\) and \(m\), the corresponding values of \(r\) will fall within distinct intervals that satisfy the SMTF criterion constraint. When we set \(r_{\lim}=\frac{1}{2}\cdot\frac{\
u_{\min}}{(1-k)m\
u_{\min}+k(m-1)\
u_{\max}}\), accurate spectrum reconstruction becomes possible if the rectangular sampling width remains within the range of \(2s\Delta t\leq r_{\lim}\Delta x\), as shown in Equation (12):  
\[2s\Delta t\leq\frac{1}{2}\cdot\frac{\
u_{\min}}{(1-k)m\
u_{\min}+k(m-1)\
u_{ \max}}\Delta x(0\leq k\leq 1). \tag{12}\]  
However, exceeding this range with the rectangular sampling width might result in the sampled interferogram failing to meet the SMTF constraint, leading to distorted reconstructed spectra. It is important to note that the aforementioned analysis explores sufficient conditions for approximate and accurate spectrum reconstruction under rectangular sampling conditions, but these cannot be considered as necessary conditions. Further analysis revealed that, when the data are under-sampled, employing the average SMTF instead of the maximum SMTF provides a more effective means of evaluating sampling quality [31].