# Analysis of BDS-3 Onboard Clocks Based on GFZ Precise Clock Products
## 4 Frequency Stability Analysis  
Clock frequency stability reflects the noise level of the frequency provided by an atomic clock. Allan deviation is a widely used indicator to describe frequency stability and random characteristics. Compared to the classic Allan deviation, the improved overlapping Allan deviation (OADEV) can utilize all possible combinations to calculate variance [23]. The OADEV of a clock at time interval \(\tau\) can be expressed as:  
\[\sigma^{2}(\tau)=\frac{1}{2(M-2m)\tau^{2}}\sum_{i=1}^{M-2m}(x_{i+2m}-2x_{i+m} +x_{i})^{2} \tag{5}\]  
where \(M\) is the number of the time series; \(m\) is the smoothing factor; and \(\tau=m\tau_{0}\) is the smoothing time. The 300 days of fitting residual data derived from Equation (3) are used to analyze frequency stability. Figure 10 shows a comparison of the results of frequency stability of different types clocks for BDS-3 over averaging intervals from 30 to 86,400 s. The upper two plots in Figure 10 depict the OADEVs for Rb clocks and PHMs on MEO satellites. Their OADEVs for averaging intervals of 1000 s and shorter decrease with \(\tau^{-\frac{1}{2}}\), which corresponds to random-walk phase noise. As the averaging intervals are more than 10,000 s, this trend turns into a power-law variation of \(\tau^{-1}\), which corresponds to flicker phase noise [24]. The large broad band of high deviations near 13,600 s and low deviations near 40,000 s may be the response of the OADEV to a 12.88-hour periodic signal. This bump is even more obvious in Rb clocks, which is similar to the bump found in GPS satellites [25; 26]. It is also  
Figure 9: RMS of clock fitting residuals of BDS-2 and BDS-3 GEO, IGSO, and MEO satellites. interesting to note that most Rb clocks and PHMs have very similar performance over the entire range of averaging intervals, except C44, C45, and C46. The spectral analysis of the periodic signal is analyzed in the next section. The bottom two plots in Figure 10 depict the OADEVs for PHMs on IGSO and GEO satellites. Compared to the PHMs on MEO satellites, the performance is significantly worse over an average interval of 30 to 20,000 s, which is consistent with the corresponding fitting residual data of the three types of satellites. The OADEVs for PHMs on IGSO and GEO follow approximately the same power-law trend within 1000 s as those for PHMs on MEO satellites. Unlike the clocks on MEO satellites, however, there is no large broad band of low or high deviations. Figure 11 compares the frequency stability of BDS-2 and BDS-3 satellite onboard clocks based on OADEV. It can be seen intuitively that the frequency stability of the BDS-3 satellite clocks is greatly improved over that of BDS-2 satellite clocks. Note that the frequency stability series of C11 fluctuates and oscillates, which differs from the behavior of the rest of the satellites. The same situation has also occurred in others' studies [17]. The reasons for this situation are not clear yet and need to be further studied. Figure 10: Clock offsets OADEV comparison of different BDS-3 clocks. OADEV at the time intervals of 1000 s, 10,000 s, and 1 day for all BDS clocks are compared in Figure 12. Inspection of the temporal frequency stability for each satellite reveals the following patterns by clock type:  
* BDS-3 MEO PHM: These clocks have the best frequency stability at 1000 s and 10,000 s among all BDS clocks. The OADEVs are \(4.03\times 10^{-14}\), \(2.24\times 10^{-14}\), and \(0.22\times 10^{-14}\) s/s at 1000 s, 10,000 s, and 1 day, respectively. * BDS-3 MEO Rb: It can be seen that the stability of MEO Rb clocks is slightly worse than that of BDS-3 MEO PHMs, with the OADEVs being \(4.3\times 10^{-14}\), \(2.56\times 10^{-14}\), and \(0.25\times 10^{-14}\) s/s at 1000 s, 10,000 s, and 1 day, respectively.