# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 4 Experimental Analysis
### Time-Frequency Analysis of the BDS Satellite Clocks  
Precision clock bias data provided by WHU ([[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) was used in this study. The data cover the period from 5 February 2023 to 4 March 2023, totaling 28 days. During satellite operations, clock bias data often contain outliers and clock jumps due to space environment disturbances and intentional control. This study employs the median absolute deviation method to detect outliers in the clock bias data [8] and then uses the method in [37] to detect clock jumps. The quadratic polynomial fitting method is used to remove the trend term to obtain the clock bias residual data. To accurately observe the time-frequency characteristics of the clock bias residual data, we analyze the periodic terms using the FBP-based STFT in both the time and frequency domains. To evaluate the effectiveness of this method in enhancing frequency resolution and reducing sidelobe effects, we performed comparative experiments using the rectangular, Hanning, and Blackman windows as window functions of the STFT [18]. By generating time-frequency spectrograms of the C01 and C22, we can observe how the frequency and energy distribution of the periodic terms vary over different time intervals, as shown in Figure 6. Based on the results of the time-frequency analysis, the main periods of the clock bias terms are identified and summarized in Table 2.  
As shown in Table 2, IGSO and GEO satellite clocks, represented by C01, C07, and C40, have 24 h periodic terms. MEO satellite clocks, represented by C12, C22, and C45, generally have 6 h and 12 h periodic terms. Additionally, the frequency and amplitude of these periodic terms are not constant but vary over time; as shown in Figure 6, the frequency of the C01 24 h periodic term and the amplitude of the C22 12 h periodic term fluctuate within 28 days. Figure 6a shows that the STFT based on a rectangular window exhibits significant spectral leakage, with large sidelobes causing false periodic terms and reducing the accuracy of periodic term identification. Figure 6a also shows that the use of Hanning and Blackman windows in the STFT effectively suppresses sidelobes, but this also broadens the main lobe, which reduces the frequency resolution in the time-frequency spectrogram. Compared to the rectangular, Hanning, and Blackman windows, the FBP significantly enhances the frequency resolution and sidelobe suppression in the STFT, thereby improving the clarity of the time-frequency spectrogram. For example, as shown in Figure 6b, the 20 h periodic term for C22 is clearly identified in the STFT based on FBP, whi \begin{table}
\begin{tabular}{c c c c c} \hline
**PRN** & **Main Period (h)** & **System** & **Clock Type** & **Satellite Type** \\ \hline C01 & 6, 8, 12, 24 & BDS-2 & RB & GEO \\ C07 & 24 & BDS-2 & RB & IGSO \\ C12 & 12 & BDS-2 & RB & MEO \\ C22 & 12, 20 & BDS-3 & RB & MEO \\ C40 & 8, 12, 24 & BDS-3 & H & IGSO \\ C45 & 6, 24, 28 & BDS-3 & H & MEO \\ \hline \end{tabular}
\end{table} 
Table 2: Six BDS satellites of different types with significant periodicity ([[http://www.csno-tarc.cn/](http://www.csno-tarc.cn/)]([http://www.csno-tarc.cn/](http://www.csno-tarc.cn/)), accessed on 1 August 2024). Figure 6: Time-frequency spectrum of the BDS satellite clock bias. (**a**) the spectrum of C01; (**b**) the spectrum of C22. In the experiment, the window width and the overlap ratio of the STFT are kept constant for all methods. The amplitude distortion introduced by the Hanning and Blackman windows is compensated by the corresponding window function recovery factors. Each row represents the experimental results for the same satellite clock, and each column represents a different spectral analysis method. To facilitate intuitive analysis of periodic phenomena in the time series, the frequency axis is converted to a period axis.