# Clock Ensemble Algorithm Test in the Establishment of Space-Based Time Reference
## 3 Experimental Analysis
### Frequency Stability of the Timescales Generated by the Three Algorithms  
Unlike the simulation test, we cannot find the true value of TA to compare. Thus, the timescales of group A are subtracted from the other three groups to eliminate the offsets and analyze their frequency stability. For the NKT algorithm, the timescale is generated by three sets of time deviations in each group according to different weights. The weight is proportional to the inverse of the Allan deviation with a sampling interval of 86,400 s. The four timescales and their differences are shown in Figure 2. In the figure, the dotted line represents the four timescales, and the solid line represents the difference in the timescale. Allan deviation of 86,400 s sampling interval for each timescale and timescale difference is given in Table 2.  
Groups A and B are composed of BDS onboard clocks, group C is composed of GPS onboard clocks, and group D is composed of Galileo system onboard clocks. In Figure 2, TA_A-TA_B represents the frequency stability of the BDS timescale, which is 1.47 \(\times\) 10\({}^{-13}\) at 300 s intervals and 1.09 \(\times\) 10\({}^{-14}\) at 86,400 s intervals; TA_A-TA_C indicates the frequency stability of the GPS timescale, which is 3.94 \(\times\) 10\({}^{-13}\) at 300 s intervals and 1.24 \(\times\) 10\({}^{-14}\) at 86,400 s intervals; TA_A-TA_D indicates the frequency stability of the Galileo timescale, which is 8.72 \(\times\) 10\({}^{-14}\) at 300 s intervals and 8.40 \(\times\) 10\({}^{-15}\) at 86,400 s intervals. In the NKT algorithm, the short-term frequency stability of the Galileo system's timescale is better than that of BDS, and BDS is slightly better than GPS. In addition, in the graph of Allan \begin{table}
\begin{tabular}{c c c} \hline \hline
**Time Series** & \
\begin{tabular}{c} **Allan Deviation** \\ **(\(\tau\) = 300 s)** \\ \end{tabular} &
\begin{tabular}{c} **Allan Deviation** \\ **(\(\tau\) = 86,400 s)** \\ \end{tabular} \\ \hline TA\_A & 7.88 \(\times\) 10\({}^{-14}\) & 5.93 \(\times\) 10\({}^{-15}\) \\ TA\_B & 1.21 \(\times\) 10\({}^{-13}\) & 9.10 \(\times\) 10\({}^{-15}\) \\ TA\_C & 3.87 \(\times\) 10\({}^{-13}\) & 1.13 \(\times\) 10\({}^{-14}\) \\ TA\_D & 3.57 \(\times\) 10\({}^{-14}\) & 5.31 \(\times\) 10\({}^{-15}\) \\ TA\_A-TA\_B & 1.47 \(\times\) 10\({}^{-13}\) & 1.09 \(\times\) 10\({}^{-14}\) \\ TA\_A-TA\_C & 3.94 \(\times\) 10\({}^{-13}\) & 1.24 \(\times\) 10\({}^{-14}\) \\ TA\_A-TA\_D & 8.72 \(\times\) 10\({}^{-14}\) & 8.40 \(\times\) 10\({}^{-15}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Allan deviation of each group (NKT). The average interval is 300 s and 86,400 s.  
Figure 2: Allan deviations of NKT. The dotted line shows the timescale generated by the NKT algorithm, and the dotted line shows the difference in timescales. deviation, the part of the curve with slope \(-1\) represents WFM, and the part with slope \(1\) represents RWFM. In Figure 2, we can see the obvious slopes of \(1\) and \(-1\), indicating that the generated time scale still has WFM and RWFM. For the RKT algorithm, x-reduction is performed in Kalman filtering and then obtains reduced timescale by calculating the weight of each group of clock deviation. The weights are also proportional to the inverse of the Allan deviation with a sampling interval of 86,400 s. The four timescales and their differences are shown in Figure 3, and Allan deviation of 86400s sampling interval for each timescale and timescale difference is given in Table 3.  
In Figure 3, the reduced Kalman timescale contrasts little with the natural Kalman timescale. As shown in Table 3, for short-term (\(\tau\) = 300 s) frequency stability, RKT has a small edge over NKT, with an increased range of about 0.01-0.1 orders of magnitude. For medium and long-term (\(\tau\) = 86,400 s) frequency stability, the results of the two algorithms are nearly identical.