# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 2 Fundamentals of Fourier Basis Pursuit
### Short-Time Fourier Transform Based on Fourier Basis Pursuit  
The STFT is a common technique for analyzing non-stationary signals [8,18]. By dividing a signal into a series of short time intervals and performing an FT on each segment, the STFT enables the study of a signal's time-frequency characteristics. The STFT is expressed as:  
\[STFT\{x\}[m,k]=X_{m}[k]=\sum_{n=0}^{N-1}x[n+m]w[n]\cdot e^{-j\frac{2\pi}{N}kn \leavevmode\
obreak\ \leavevmode\
obreak\ FBP\leavevmode\
obreak\  
where \(x(t)\) represents the phase at time \(t\); \(a\), \(b\), and \(c\) are the initial phase, initial frequency offset, and frequency drift, respectively; \(f_{k}\) is the frequency of the periodic term; \(\varphi_{k}\) is the initial phase of the periodic term; \(\varepsilon(t)\) represents the fitting error.  
The spectral analysis model (SAM) is a clock prediction method based on the FT [6, 7, 8]. This prediction method models the clock bias using Equation (13) and extrapolates the model to obtain future clock bias values. Usually, the parameter \(p\) and \(f_{k}\) are obtained from spectral analysis, and the parameters \(A_{k}\) and \(\varphi_{k}\), are estimated by LSM. In clock prediction, the SAM prediction model, which considers the period term, has better prediction accuracy than the quadratic polynomial model (QP), which does not consider the period term. However, since the frequency and amplitude of the periodic term vary over time, errors will occur at the boundaries of the data when modeling clock bias data using the SAM, resulting in a difference between the predicted periodic term and the actual periodic term.  
This paper addresses the problem that the LSM-based SAM cannot effectively fit the periodic term at the boundary of the data and proposes a clock bias prediction method based on FBPBPF, that is, the FBP prediction method [21, 26]. In this method, FBP spectral is used instead of FT to perform spectral analysis on the clock bias data, and FBPBPF is used to extract the periodic term of the clock bias to obtain a new clock bias model as follows:  
\[x_{FBP}(t)=a+bt+\frac{1}{2}ct^{2}+2 Real\left(\sum_{k=1}^{p}y_{k}(t)\right) \tag{14}\]  
where \(y_{k}\) represents the time series obtained by filtering the clock bias data within a specified frequency band using the FBPBPF. \(Real\) represents the real part operation because the data is transformed into the complex domain after FBP. Assuming that the starting time of the prediction is \(t_{m}\) and the step length of the prediction is \(t_{n}\), the clock bias at \(t_{m}+t_{n}\) can be expressed as:  
\[x_{FBP}(t_{m}+t_{n})=a+bt+\frac{1}{2}ct^{2}+2 Real\left(\sum_{k=1}^{p}y_{k}(t_{ m})e^{2\pi if_{k}t_{n}}\right) \tag{15}\]  
In summary, the frequency and amplitude of the periodic term calculated by the SAM using LSM are the result of global analysis and cannot give the frequency and amplitude changes in the time dimension. The SAM prediction model cannot ensure the fitting accuracy of the whole and the boundary at the same time. The FBP prediction method uses FBPBPF to effectively extract the period terms at the data boundary and then uses Equation (15) to predict the boundary. Compared with SAM, it can effectively improve the short-term accuracy of clock prediction.