# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 4 Experimental Analysis
### Simulation
Consequently, window functions cannot entirely eliminate peak frequency shifts due to noise or spectral leakage from data truncation. As shown in Figure 2a, compared to the windowed FT, the FBP spectrum concentrates energy in the main lobe while significantly reducing the spectral leakage. It should be noted that the small peaks in the FBP spectrum can be considered as spectral leakage since the modeled quasi-periodic signals yield both large and small peaks. However, this leakage is limited in range, unlike the broader leakage observed in the windowed FT. Although noise interference causes some peak frequency shifts in the periodic terms, the FBPBPF method, based on the FBP spectrum, allows for selective extraction of spectral components within a given frequency range without requiring exact peak frequencies. In this experiment, FBPBPF primarily extracts the spectrum of the main lobe, as shown in the black box in Figure 2a. While sidelobe exclusion may introduce minor extraction errors, the main lobe in the FBP spectrum is more concentrated than in the WFT, with notably less spectral leakage. This improvement reduces the influence of noise and data truncation on the spectral analysis, resulting in higher extraction accuracy. To validate the effectiveness of the extraction methods, the mixed signal m(t) was divided into five segments, each covering 120 h. Segment 1 covers 1-120 h, segment 2  
Figure 2: The spectrum of the simulated clock bias calculated using windowed Fourier transform and Fourier basis pursuit (FBP). As the Hanning and Blackman windows introduce amplitude distortions, corresponding window function compensation factors are applied. For clearer analysis of periodic phenomena in the time series, the frequency axis is converted to the period axis. (**a**) the spectrum of FBP; (**b**) the spectrum based on rectangular window; (**c**) the spectrum based on Hanning window; (**d**) the spectrum based on Blackman window. covers 121-240 h, segment 3 covers 241-360 h, segment 4 covers 361-480 h, and segment 5 covers 481-600 h. The results of extracting periodic terms using the LSM, FBPBPF, FIR, and IIR are shown in Figures 3-5. Zero-phase digital filtering was applied to both the FIR and IIR bandpass filters to avoid phase delays. In order to suppress noise and spectral leakage, the design of the filter focuses on the preservation of frequency components close to the main lobe. The FIR filter was designed with a Kaiser window, and the IIR filter was configured as a Butterworth filter, each with zero-phase filtering to maintain signal alignment and improve extraction accuracy across boundary and full-length signal regions [19; 20; 36]. In these figures, each row represents a simulated data segment, and each column corresponds to a different extraction method. The blue line represents the simulated periodic term from Equation (16), the green line represents the reconstructed periodic term for each method, and the red line represents the extraction residuals, which is the difference between the simulated and reconstructed periodic terms. As shown in Figure 3, the extraction residuals of the LSM and FBPBPF remain relatively low at the data boundaries, while the FIR and IIR show optimal extraction performance only in the central regions, with residuals increasing significantly at the boundaries. This is due to insufficient data at the signal boundaries for the FIR and IIR filters to compute effectively, which causes distortion in these regions. The accuracy of the extraction at the boundaries is very important. Table 1 compares the relative error of 70 values at the left and right boundaries using these methods. Figure 3: Simulated 6 h periodic term (blue), reconstructed 6 h periodic term (green), and extraction residual (red). The simulated data are divided into five segments; each row in the figure represents the experimental results for one of these time series segments, and each column represents a different method for extracting the periodic terms.