# Joint Timekeeping of Navigation Satellite Constellation with Inter-Satellite Links
## 4 Performance Evaluation of ISLT
### Frequency Stability Analysis  
The establishment of the ISLT requires periodic time comparison between the master satellite and slave satellites, a process which is not satisfied by the actual states of BeiDou ISLs. As the noise characters of ISL and MPOD relative clock deviations are consistent, the difference values of MPOD clock deviations were used as inter-satellite clock deviations in the analysis. By implementing the ISLT timescale algorithm, we obtained the ISLT expressed as the clock deviation relative to the reference clock. By using Equation (1), the relative deviation was translated to the absolute clock deviation relative to the same reference timescale with the satellite clocks. The weight allocation according to Equation (39) of the clock ensemble is shown in Figure 5. The frequency of C19 was more stable in the short term (averaging time smaller than 7000 s) than that of C26 and was allocated with a larger weight, while the frequency of C27 was more stable in the long term (averaging time larger than 7000 s) than that of C30 and was allocated with a larger weight. Figure 4: The noise character of relative clock deviations. Figure 3: The precision of inter-satellite time comparison. This contrast demonstrates that the weights of the ISLT timescale had comprehensively considered both the long-term and short-term frequency stability. The Allan deviations of the satellite clocks are drawn again in Figure 6 and compared with the ISLT jointly kept by the nine satellite clocks. The ISLT\({}_{\text{KFW}}\) was generated by the KPW algorithm in [14], while the ISLT was generated by the algorithm in this research. The short-term stability of the ISLT and ISLT\({}_{\text{KFW}}\) was almost the same, but the long-term frequency of the ISLT was more stable, demonstrating that our weights were better (Figure 5(b)). Among the Rb clocks, the C19 clock was the most stable, and the C37 clock was the worst. The frequency stability after 10,000 s of averaging time of all the Rb clocks decreased due to the influence of RWFM. The frequencies of PHMs are generally more stable than Rb clocks. Expect for those of C26, the PHMs of other three satellites showed a similar frequency stability, with their daily stability reaching \(8\times 10^{-15}\), so they were allocated with the three largest weights. For an averaging time of less than 7000 s, the frequency of the ISLT was more stable than that of all the spaceborne clocks. However, the Allan deviation slightly increased around 10,000 s of averaging time, possibly because of the mismodeling non-power-law behaviors of PHMs or the flicker frequency noises (FFM) of Rb clocks. Even so, the daily stability of the ISLT reached \(6\times 10^{-15}\), which was better than the best PHM of the four satellites. Figure 5: Weights of the clock ensemble (1 March–31 March 2019). Figure 6: _Cont._  
Figure 7 demonstrates the ISLTs kept by different ensembles of satellite clocks, as illustrated in Table 3 to evaluate the influence of the clock ensemble. Since the long-term stability of PHMs was obviously superior to Rb clocks, the frequency of the ISLT2 was more stable than the ISLT1 in the long term and approaches to the ISLT after 30,000 s of averaging time. Meanwhile, the short-term stability of the ISLT1 was better than the ISLT2 because the weights of the ISLT1 were more dependent on the short-term stability of the Rb clock ensemble. The ADEVs of the ISLT3 and ISLT4 indicate that the malfunction of a timekeeping clock has little influence on the stability of the ISLT, especially for the one with a small weight. \begin{table}
\begin{tabular}{c c} \hline
**ISLT Symbol** & **Clock Ensemble** \\ \hline ISLT & All the nine satellite clocks \\ ISLT1 & All the five Rb clocks \\ ISLT2 & All the four PHMs \\ ISLT3 & C19 C20 C21 C22 C26 C27 C29 C30 \\ ISLT4 & C19 C20 C21 C22 C26 C29 C30 C37 \\ \hline \end{tabular}
\end{table} 
Table 3: The ensemble of satellite clocks that kept the ISLT. Figure 6: Frequency stability comparison of different timescales.