# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 4 Experimental Analysis
### Simulation
(**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**,**r**) extracting periodic term with IIR. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Region**} & \multicolumn{4}{c}{**06 h (\%)**} & \multicolumn{4}{c}{**12 h (\%)**} & \multicolumn{4}{c}{**24 h (\%)**} \\ \cline{2-13}  & **LSM** & **FBP** & **FIR** & **IIR** & **LSM** & **FBP** & **FIR** & **IIR** & **LSM** & **FBP** & **FIR** & **IIR** \\ \hline \multirow{5}{*}{Entire} & 1 & 8.56 & 6.1 & 23.78 & 20.42 & 26.38 & 8.22 & 17.32 & 40.19 & 11.45 & 7.51 & 59.92 & 47.89 \\  & 2 & 9.95 & 6.07 & 22.31 & 21.52 & 33.42 & 20.63 & 28.43 & 48.55 & 22.75 & 18.25 & 59.94 & 52.57 \\  & 3 & 12.09 & 8.81 & 24.89 & 26.88 & 14.56 & 6.5 & 19.22 & 41.8 & 13.13 & 5.6 & 44.27 & 47.09 \\  & 4 & 8.52 & 4.94 & 18.42 & 24.02 & 25.03 & 18.37 & 24.05 & 34.27 & 31.78 & 21.17 & 45.21 & 52.21 \\  & 5 & 11.22 & 5.66 & 17.04 & 23.07 & 20.24 & 12.42 & 15.02 & 31.93 & 15.22 & 12.03 & 41.9 & 48.15 \\ \hline \multirow{5}{*}{Bdry} & 1 & 14.61 & 10.77 & 67.32 & 57.22 & 43.19 & 10.78 & 49.76 & 83.61 & 12.12 & 6.68 & 155.24 & 92.24 \\  & 2 & 17.56 & 10.53 & 62.48 & 59.22 & 42.13 & 28.47 & 50.56 & 58.26 & 32.54 & 24.56 & 153.53 & 95.38 \\ \cline{1-1}  & 3 & 21.45 & 16.71 & 67.78 & 66.35 & 25.25 & 9.27 & 50.53 & 78.73 & 17.27 & 5.57 & 109.46 & 93.77 \\ \cline{1-1}  & 4 & 16.62 & 9.52 & 50.63 & 60.57 & 28.61 & 23.43 & 45.05 & 57.32 & 46.12 & 20.91 & 106.68 & 91.98 \\ \cline{1-1}  & 5 & 18.41 & 13.36 & 36.87 & 57.7 & 29.46 & 5.59 & 45.18 & 75.64 & 17.06 & 13.98 & 114.16 & 91.69 \\ \hline \multirow{5}{*}{Entire Avg} & 10.06 & 6.31 & 21.28 & 23.18 & 23.92 & 13.22 & 20.80 & 39.34 & 18.86 & 12.91 & 50.24 & 49.58 \\ \cline{1-1}  & Bdry Avg & 17.73 & 12.17 & 57.01 & 60.21 & 33.72 & 15.50 & 48.21 & 70.71 & 25.02 & 14.34 & 127.81 & 93.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The relative errors are categorized by entire regions (1440 values) and boundary regions (140 values). Statistical results of relative errors for periodic terms (6 h, 12 h, and 24 h) by different methods (LSM, FBPBPF, FIR, and IIR) over time segments (segments 1 to 5). The entire average of all periodic terms for different methods is 17.62, 10.81, 30.78, and 37.37. The boundary average of all periodic terms for different methods is 25.49, 14.01, 77.68, and 74.64. Figure 4: Simulated 12 h periodic term (blue), reconstructed 12 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. As shown in Figure 4 for segment 1 and segment 5, compared to the LSM, the periodic terms reconstructed with the FBPBPF, FIR, and IIR exhibit time-varying amplitudes between \(t=60\) and \(t=90\), and these methods yield relatively smaller extraction residuals. Figure 4a,q show that the LSM struggles to suppress residuals effectively when extracting periodic terms, as the amplitude estimate of the LSM is constant. For quasi-periodic signals with time-varying amplitudes, using a fixed amplitude parameter in the periodic term model leads to significant extraction residuals. As shown in Figure 5 for the LSM, even when the frequency and amplitude of the 24 h periodic term remain constant, the LSM demonstrates limited extraction accuracy, primarily due to spectral leakage caused by low-frequency noise interference and insufficient sampling length.