# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter  
[PERSON]  
1 State Key Laboratory of Geodesy and Earth's Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China; [EMAIL_ADDRESS] (C.S. ); [EMAIL_ADDRESS] (L.L. ); [EMAIL_ADDRESS] (H.H.) [PERSON]  
1 State Key Laboratory of Geodesy and Earth's Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China; [EMAIL_ADDRESS] (C.S. ); [EMAIL_ADDRESS] (L.L. ); [EMAIL_ADDRESS] (H.H.) [PERSON]  
1 State Key Laboratory of Geodesy and Earth's Dynamics, Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences, Wuhan 430077, China; [EMAIL_ADDRESS] (C.S. ); [EMAIL_ADDRESS] (L.L. ); [EMAIL_ADDRESS] (H.H.) [PERSON]  
3 Fourth Institute of Oceanography, Ministry of Natural Resources, Beihai 536000, China; [EMAIL_ADDRESS] (H.  
evaluation of spaceborne atomic clocks, the fitting accuracy of the atomic clock noise coefficient, and the prediction of satellite clocks [2; 6; 7; 8]. To achieve this, some tools are used to detect these periodic terms, such as Allan variance, Fourier transform (FT) and short-time Fourier transform (STFT) [4; 9; 10]. Several signal processing methods, such as the least squares method (LSM), digital filter, empirical mode decomposition (EMD), and wavelet transform, have been applied to extract the periodic terms from the clock bias [7; 8; 10; 11; 12; 13]. The LSM is a traditional regression technique that models harmonic terms with constant amplitude and frequency by means of least-squares fitting. However, in practical scenarios, the satellite clock bias is non-stationary and there are systematic errors between the time-varying periodic terms and the model [8]. The digital filters based on the FT, such as finite impulse response (FIR) and infinite impulse response (IIR) filters, can reliably extract time-varying signals by selectively suppressing or preserving certain frequency components. However, sidelobes, noise components, and trend residuals in the clock bias may cause spectral peak shifts or introduce spurious frequency elements [14]. Narrow frequency bands may lead to incomplete extraction of periodic signals, while broader bands may introduce noise components and residual trends. In addition, the wavelet transform is particularly suited for non-stationary signals, but the selection of the optimal wavelet basis function often requires specialized knowledge and lacks standardization [15]. EMD has the problem of mode mixing [16]. Finally, all of these methods are susceptible to boundary effects, estimations at the beginning and end of a time series tend to be less accurate, with errors intensifying closer to the boundaries. Such distortions can compromise subsequent analyses. To suppress sidelobe artifacts in spectral analysis and accurately extract periodic terms in satellite clock bias, this paper proposes the Fourier basis pursuit (FBP) spectrum and the Fourier basis pursuit bandpass filter (FBPBPF) methods. The FBP method employs L1-norm minimization to constrain the Fourier coefficients to obtain the FBP spectrum, thereby improving spectral resolution, suppressing sidelobes, and enhancing the main lobe. Further, the FBPBPF based on the FBP spectrum can substantially reduce boundary effects and maximally suppress noise interference. Studies using simulated and real data indicate that the FBP suppresses spectral leakage more effectively than standard windowed Fourier transform (WFT) approaches, such as rectangular, Hanning, and Blackman windows. The FBPBPF also shows significant improvements in accuracy over traditional signal extraction methods such as LSM, FIR, and IIR, particularly at the boundaries. The remaining sections of this paper are organized as follows: Section 2 introduces the theoretical background of FBP. Section 3 discusses the modeling of the satellite clock bias and its prediction method.