# Analysis of BDS-3 Onboard Clocks Based on GFZ Precise Clock Products
## 5 Spectrum Analysis  
The fast Fourier transform (FFT) method is effective for computing discrete Fourier transforms (DFT). From this, we can judge whether there are periodic changes in time series on the spectrum. The spectral analysis series \(X(k)\) at epoch \(k\) is described as follows [27]:  
\[X(k)=\sum_{n=0}^{N-1}x(n)e^{-j\frac{2\pi}{M}kn} \tag{6}\]  
where \(x(n)\) is the clock offset fitting residuals fitted by a second-order polynomial; \(n\) is the number of an element in the residual series; \(e\) is Euler's number; \(j\) is an imaginary unit; and \(M\) is the number of the time series. However, there are missing samples in the precise clock product, and the elimination of gross errors could influence spectrum analysis. Thus, in this paper, the non-uniform fast Fourier transform (NUFFT) is used to analyze the clock periodical characteristics in the residual series with gaps [28; 29]. We use the fitting residuals of Equation (3) to analyze the periodic term, and the corresponding results are shown in Figure 13. The amplitude spectrum is the average value of the same types of satellite clocks, considering that their periodicities are similar.  
It can be seen from Figure 13 that the spectral analysis of BDS-2 and BDS-3 clocks show obvious periodicity. The main periods of BDS-2 MEO onboard clocks are about 12.88 (1 cycle per revolution, CPR), 6.44 (2 CPR), and 25.76 h (0.5 CPR), with amplitudes of 0.15, 0.08, and 0.04 ns, respectively. Being different from BDS-2 MEO satellites, the clocks of BDS-2 IGSO and GEO satellites show additional periods that are \(n\) times the length of 1 CPR (\(n>2\)).  
PHMs equipped on BDS-3 IGSO satellites show the same periods as BDS-2 IGSO onboard clocks, but their amplitudes are smaller than those for BDS-2s. The amplitude of 1 CPR is 0.15 ns, which is about two-thirds of the amplitude of that of BDS-2 IGSO onboard clocks. The PHMs equipped on BDS-3 GEO satellites also show the same period as BDS-2 GEO onboard clocks, but their amplitudes are greater than those of BDS-2 clocks. BDS-3 PHMs and Rb clocks aboard MEO satellites have slightly different period characteristics from BDS-2 MEO satellite clocks. The main periods of BDS-3 MEO Rb clocks are 12.88 (1 CPR), 6.44 (2 CPR), and 4.29 h (3 CPR). However, the significant periods of PHMs are about 6.44 (2 CPR) and 12.88 h (1 CPR). There are no \(n\) (\(n>2\)) CPR periods in both Rb clocks and PHMs. Besides, it can be found that the amplitudes of the period terms of PHMs are significantly smaller than those of Rb clocks.  
Figure 12: The 1000-second, 10,000-second and 1-day OADEVs of BDS-2 and BDS-3 onboard clocks.  
Since the satellite's clock offset estimated by ground tracking observation may absorb orbit errors, the periodic variation of a BDS-3 onboard clock may be related to the orbital period of the satellite [16; 30; 31]. According to the results of spectrum analysis, the amplitude of IGSO satellite clocks is much greater than that of the MEO satellite clocks, which is consistent with the poor orbit determination accuracy of the IGSO satellite. In addition, the comparison between BDS-3 PHMs and Rb clocks shows that the performance of PHMs is significantly better than Rb clocks. Furthermore, the performance of BDS-3 satellite clocks is better than that of BDS-2 onboard clocks.