# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 5 Conclusions  
This study analyzed the impact of periodic terms in satellite clock bias on the performance analysis of atomic clocks, highlighted the importance of extracting these terms, and introduced the FBP spectrum and the FBPBPF-based method for extracting periodic terms in clock bias. Analysis of simulated satellite clock bias data demonstrated that the FBPBPF achieves superior noise suppression and accuracy in extracting time-varying periodic terms compared to the LSM, FIR, and IIR methods. Specifically, compared to the LSM, FIR, and IIR, the FBPBPF reduces the entire extraction error by an average of 6.81%, 19.96%, and 26.55%, respectively, and reduces the extraction error at the boundaries by an average of 11.47%, 63.67%, and 60.63%, respectively, demonstrating a significant advantage in mitigating boundary effects. Compared to the SAM, the FBP prediction method significantly improves prediction accuracy across all time intervals, as verified using WHU satellite clock bias data. For predictions over 6, 12, 18, and 24 h, the average RMS errors of the FBP prediction method are reduced by 15.85%, 11.04%, 6.45%, and 4.01%, respectively.  
The time-frequency characteristics of the satellite clock bias were analyzed using FBP-based STFT. Since the FBP spectrum is superior to the rectangular, Hanning and Blackman windows in terms of reducing spectral leakage, in addition to the 24 h, 12 h and 6 h periodic terms commonly found in BDS satellite clock bias, periodic terms of 8 h, 20 h and 28 h periodic terms were additionally found for certain satellites. Through a comparative analysis of the lag 1 ACF noise type before and after the correction of the periodic term, it was identified that the periodic term is the main factor driving the abnormal fluctuations in the noise index. A comparative analysis of the frequency stability before and after the periodic term correction revealed that abnormal peaks and oscillations in the original frequency stability curve were effectively suppressed after the correction, confirming the accuracy and effectiveness of the FBPBPF in extracting periodic terms from the clock bias. After the periodic term correction, the satellite clock frequency stability  
Figure 10: The lag 1 ACF noise type before and after periodic term correction. \(\tau\) is the averaging time interval. The blue line represents the lag 1 ACF of the original clock bias and the red line represents the lag 1 ACF after periodic term correction using FBPBPF.  
improved significantly, with enhancements of 42.7%, 46.6%, and 38.5% at 10,000 s, 20,000 s, and 40,000 s averaging times, respectively. These improvements help to accurately analyze the performance of the spaceborne atomic clock.  
Conceptualization, C.S. and G.W.; Data curation, C.S.; Formal analysis, C.S.; Funding acquisition, G.W.; Investigation, G.W.; Methodology, C.S. and G.W.; Project administration, G.W.; Resources, C.S.; Software, W.S.; Supervision, G.W. and L.L.; Validation, C.S., G.W. and L.L.; Visualization, D.R. and H.H.; Writing--original draft, C.S.; Writing--review and editing, C.S., D.R. and H.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Project ID 42074011) and the Key Projects Deployed by the Major Scientific and Technological Mission Bureau of the Chinese Academy of Sciences (Project ID T24Y6303).  
The data supporting this research can be found at [[ftp://igs.gnsswhu.cn/pub/gnss/products/mgex/](ftp://igs.gnsswhu.cn/pub/gnss/products/mgex/)]([ftp://igs.gnsswhu.cn/pub/gnss/products/mgex/](ftp://igs.gnsswhu.cn/pub/gnss/products/mgex/)), accessed on 1 August 2024.  
We are grateful to the GNSS Research Center of Wuhan University for providing the BDS clock data.  
The authors declare no conflicts of interest.