# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 4 Experimental Analysis
### Simulation
This issue distorts the time-frequency characteristics of the signal, complicating frequency identification in the frequency domain and reducing the accuracy of the LSM fitting in the time domain. Similarly, as shown in Figure 5 for the FIR and IIR, the noise frequency components overlap with the FIR and IIR filter bands, resulting in noise contamination that distorts the reconstructed periodic terms. Compared to the LSM, FIR, and IIR, the FBPBPF demonstrates stronger resistance to noise, allowing it to extract more complete periodic terms even under low-frequency noise interference, as shown in Figure 5b,j. While the FBPBPF, FIR, and IIR all selectively process target frequency components in the frequency domain, the FBPBPF excels in avoiding boundary effects in reconstructed signal boundaries, as shown in Figure 3. The FBPBPF achieves significantly higher accuracy in extracting the 6 h periodic term with time-varying frequency, outperforming the FIR and IIR, with minimal boundary distortion  
Figure 5: Simulated 24 h periodic term (blue), reconstructed 24 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 term. (**a,e,i,m,q**) extracting periodic term with LSM; (**b,f,j,n,r**) extracting periodic term with FBPBPF; (**c,g,k,o,s**) extracting periodic term with FIR; (**d,h,l,p,t**) extracting periodic term with IIR. in the reconstructed signal. Additionally, as shown in Figure 4b,f,r, the FBPBPF effectively adapts to changes in amplitude over time for the 12 h periodic term. To evaluate the accuracy of these four methods in extracting the periodic term, the relative error was used as the evaluation metric:  
\[Relative\ error=\frac{RMSE}{RMS}=\frac{\sqrt{\sum_{i=1}^{N}{(x_{i}-\hat{x}_{i} )}^{2}/N}}{\sqrt{\sum_{i=1}^{N}{x_{i}^{2}/N}}} \tag{17}\]  
where \(x_{i}\) is the true value for the \(i\)-th data point, \(\hat{x}_{i}\) is the estimated value for the \(i\)-th data point, and \(N\) is the number of data points in each segment. The relative error statistics for the periodic term reconstruction are shown in Table 1. To account for boundary effects, the relative errors were calculated separately for the entire regions and for the boundary regions. LSM and FBPBPF show good performance in the boundary regions, with significantly lower errors in extracting shorter periodic terms (such as the 6 h term) than FIR and IIR. However, the entire accuracy of LSM was slightly lower than that of FIR when extracting the time-varying periodic terms (such as the 12 h term). FIR and IIR display higher extraction errors in the boundary regions, particularly for longer periodic terms like the 24 h term, likely due to limitations in handling boundary effects and reconstructing long-period components. Overall, the FBPBPF achieved the lowest error in both entire and boundary regions. Compared to the LSM, FIR, and IIR, the FBPBPF reduces the extraction error in the entire region by an average of 6.81%, 19.96%, and 26.55%, respectively, and reduces the extraction error in the boundary regions by an average of 11.47%, 63.67%, and 60.63%, respectively. FBPBPF showed a modest improvement over LSM for the 6 h term and significantly outperformed LSM, FIR, and IIR for the 12 h and 24 h terms, demonstrating robustness and accuracy in periodic term reconstruction.