# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 2 Fundamentals of Fourier Basis Pursuit
### Fourier Basis Pursuit  
The sparse representation of signals has received extensive attention and research in recent years and has applications in many areas, such as signal separation, denoising, coding, tidal forecasting, and extraction of polar motion signals [21; 22; 23; 24; 25; 26]. This paper applies it to the analysis and extraction of periodic terms in satellite clock bias. Sparse representation mainly involves sparse representation algorithms and atomic dictionaries. Scholars have proposed many sparse representation algorithms, including the matching pursuit algorithm [27], the basis pursuit algorithm [28], and the frame method [29]. Scholars have proposed many atomic dictionaries to address the different characteristics of signals, such as the Fourier dictionary, warplets dictionaries [30], Chirplet dictionaries [31], etc. [PERSON] [32] and [PERSON] [28] proved that the basis pursuit algorithm gives the most sparse results, and the Fourier dictionary is the simplest and most basic dictionary. Based on these two points, this paper proposes the Fourier basis pursuit spectrum and the Fourier basis pursuit bandpass filter methods. Based on Equation (2), the matrix \(\Phi\) is constructed as follows:  
\[\Phi= \begin{bmatrix}e^{j\frac{2\pi}{N}\cdot K\cdot 0}/N\\ e^{j\frac{2\pi}{N}\cdot K\cdot 1}/N\\ \vdots\\ e^{j\frac{2\pi}{N}\cdot K\cdot(N-1)}/N\end{bmatrix} _{N\times M} \tag{9}\]  
where \(K=[0,\ 1,\dots,\ M-1]\). \(M\) is the number of atoms in the dictionary, and \(N\) is the length of the time series. Each column of the matrix \(\Phi\) represents the atom of the Fourier dictionary. The dictionary is overcomplete for \(M>N\), in which case, the number of atoms is greater than the length of the time series [28]. Each atom of the Fourier dictionary demonstrates different frequency characteristics. Thus, Equation (2) can be expressed as \(x=\Phi\cdot X\), in which \(x\) is the original signal and \(X\) is the corresponding coefficient vector of the Fourier dictionary. The principle of basis pursuit is to find a representation of the signal whose coefficients have the minimum L1-norm [32]. Formally, one solves the problem if:  
\[X^{*}=\underset{X\in R^{p}}{argmin}\|X\|_{1}\ \text{{subject to}}\ \Phi\cdot X=x \tag{10}\]  
The basis pursuit minimization of Equation (10) involves convex optimization that can be reformulated as a linear programming problem [32]. Through global optimization, basis pursuit can obtain a better resolution than matching pursuit and can obtain a better result than the method of frames [28; 33]. The matrix \(\Phi\) is the Fourier dictionary and the solution \(X^{*}\) of Equation (10) is obtained according to the L1-norm. Then, the solution \(X^{*}\) can be defined as the FBP spectrum. If\(X^{*}\) takes the corresponding angular frequency range \([\omega_{1},\omega_{2}]\), the corresponding filtering results are obtained, and the FBPBPF method can be defined as:  
\[y=\Phi\cdot X^{*}{}_{[\omega_{1},\omega_{2}]} \tag{11}\]  
where in Equation (11), \(y\) is the corresponding filtering result with angular frequency range \([\omega_{1},\omega_{2}]\). Here, the difference \(\omega_{2}-\omega_{1}\) should exceed the minimum frequency resolution, which primarily depends on the number M of atoms in the dictionary rather than the length \(N\) of the time series. This section presents a comprehensive development of the fundamental principles underpinning the FBP spectrum and the FBPBPF method, both of which are grounded in the Fourier dictionary and basis pursuit. When compared to the traditional WFT, the FBP spectrum, coupled with basis pursuit, delivers superior spectral resolution and optimal amplitude allocation at fixed frequencies. This approach effectively mitigates spectral tailing and leakage caused by finite time series lengths [34].