# Extraction of Periodic Terms in Satellite Clock Bias Based on Fourier Basis Pursuit Bandpass Filter
## 4 Experimental Analysis  
### Simulation  
To validate the effectiveness of the FBP spectrum in suppressing spectral leakage and the accuracy of the FBPBPF in extracting periodic terms, simulations were performed on BDS satellite clock bias data. These data included the commonly observed 6 h, 12 h, and 24 h periodic signals, combined with various types of atomic clock noise [10]. By superimposing these signals, mixed data with a signal-to-noise ratio (SNR) of \(-20.44\) dB was generated, a level lower than typically observed in actual satellite clock bias data. Frequency domain comparisons were made between the WFT and the FBP to examine the differences. Based on the spectral analysis results, periodic terms were extracted from the mixed data using the LSM, FBPBPF, FIR, and IIR methods. The simulation model for the clock bias data is as follows:\[m(t)=A_{1}(t)cos\bigg{(}\frac{2\pi t(1+0.00005(t-300))}{6}+\frac{\pi}{5}\bigg{)}+A_ {2}(t)cos\bigg{(}\frac{2\pi t}{12}+\frac{\pi}{3}\bigg{)}+A_{3}(t)cos\bigg{(} \frac{2\pi t}{24}+\frac{\pi}{4}\bigg{)}+n(t) \tag{16}\]  
where the sampling interval is 1/12 h, and the total sampling duration is 600 h. The first three terms on the right side represent simulated 6 h, 12 h, and 24 h periodic signals, with their respective amplitudes \(A_{1}(t)\), \(A_{2}(t)\), and \(A_{3}(t)\) shown in Figure 1. The frequency of the 6 h periodic component gradually increases over time as defined in Equation (16), simulating real-world frequency fluctuations. In Figure 1a, \(A_{2}\) represents the amplitude of the 12 h periodic component, which varies over time to model realistic amplitude fluctuations. The amplitude and frequency of the 24 h periodic component remain constant. The fourth term on the right side of the equation represents the noise component in the clock bias model, whose time-domain plot is shown in Figure 1c. The left side of the equation represents the mixed data, comprising superimposed periodic and noise terms, with its time-domain plot also shown in Figure 1c. As shown in Figure 1d, the Allan deviation was calculated for the mixed signal, noise-only, and periodic-only components at different averaging times to assess frequency stability. Observations indicate that the Allan deviation curve of the noise component remains smooth, showing no distinct peaks or fluctuations, confirming that it is unaffected by the periodic terms. Figure 2b shows the spectrum generated by applying a rectangular window to the original signal. Due to noise interference and data truncation, both the 12 h and 24 h periodic terms exhibit substantial spectral leakage and peak frequency shifts. For example, as indicated by the arrows in Figure 2b, the peak frequencies of the 12 h and 24 h components appear at 1/11.85 h\({}^{-1}\) and 1/24.12 h\({}^{-1}\), respectively, differing from their expected frequencies in the simulation model. Additionally, the main lobe is insufficiently concentrated, and significant sidelobe leakage is evident, as highlighted in the red box. The right sidelobes of the 6 h and 12 h periodic terms are further impacted by leakage from the 12 h and 24 h components, resulting in significantly lower energy on the right side than on the left. Figure 1: Simulated satellite clock bias. (**a**) the amplitude of periodic term; (**b**) time domain of periodic term; (**c**) time domain of mixed signal and noise; (**d**) Allan deviation of the simulated signal. Figure 2c,d show the spectra produced using Hanning and Blackman windows, respectively. Both of these windows reduce spectral leakage from data truncation compared to the rectangular window. However, they also cause main lobe broadening. As shown in the green box in Figure 2c,d, the Blackman window suppresses sidelobes more effectively than the Hanning window, though it further broadens the main lobe. The arrows in Figure 2c,d indicate that the peak frequency of the 24 h periodic term shifts to 1/23.30 h\({}^{-1}\), showing a greater deviation from the true frequency than with the rectangular window.