## 4 Performance Evaluation and Discussion
### Example 2: Abrupt Dynamics and Signal Fading Scenario  
The second example is used to further verify the proposed algorithm's validity. In Example 1, the dynamics contain acceleration, while Example 2 not only contains acceleration but also contains higher-order dynamics, which are abrupt turns. This example implements the signal fading and higher-order dynamics simultaneously. Figure 7 demonstrates the signal strength and dynamics for Example 2. For the signal power, the \(C/N_{0}\) started with 37 dB-Hz, and then was descended by 2 dB per 30 s. The maximum attenuation was 22 dB at 180 s and was kept at 25 dB-Hz for 25 s. The dynamics of the signal are illustrated in Table 2, which has been divided into ten intervals. Figure 7c labels the ten intervals of dynamics to reflect the Doppler variation of PRN 14. Figure 5: The \(C/N_{0}\) and Doppler estimated results of the four algorithms for Example 1. (**a,c**) are the results for PRN 14 satellite signal; (**b,d**) are the results for PRN 19 satellite signal. (PRN: pseudo-random number). Figure 6: The results of the adaptive factor \(\lambda_{k}\) in Example 1. The height of the receiver remains constant during the whole simulation process. Figure 8 shows the trajectory of the receiver. The receiver motion coexisted with acceleration, deceleration, and three abrupt turns. The first two turns turned with 90 degrees and the third one turned with 180 degrees. This scenario is used to simulate the aircraft flying in the sky and passing through obstructions, like forests and so on. For the parameters set in this test, apart from \(\mathbf{R}_{k}\), the other initial parameters in the four methods are identical to those in Example 1. \(\mathbf{R}_{k}\) is initialized using (4) with \(C/N_{0}=37\) dB-Hz and \((c/n_{0})_{0}=10^{3.7}\). The \(C/N_{0}\) and Doppler estimated results after 60 s for PRN 14 and PRN 20 are shown in Figure 9. Figure 9a,c show the estimations for PRN 14. Figure 9b,d show the estimations for PRN 20. In Figure 9a,b, all four tracking algorithms can obtain accurate estimated values when the \(C/N_{0}\) is relatively higher and the receiver motion includes acceleration and abrupt turns. As the \(C/N_{0}\) decreases below 27 dB-Hz, only the proposed method can follow the signal fading and dynamic signals during the whole process. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Movement State** & **Time (s)** \\ \hline
1 & Stationary & 0–60 \\
2 & Accelerated (\(a=40\) m/s\({}^{2}\)) & 61–85 \\
3 & Constant velocity (\(v=1000\) m/s) & 86–90 \\
4 & Decelerated (\(a=-27.5\) m/s\({}^{2}\)) & 91–122 \\
5 & 1 st clockwise turn with 90\({}^{\circ}\) (radius \(r=763.94\) m) & 123–132 \\
6 & Constant velocity (\(v=120\) m/s) & 133–150 \\
7 & 2 nd clockwise turn with 90\({}^{\circ}\) (radius \(r=763.94\) m) & 151–160 \\
8 & Constant velocity (\(v=120\) m/s) & 161–170 \\
9 & 3 rd clockwise turn with 180\({}^{\circ}\) (radius \(r=763.94\) m) & 171–190 \\
10 & Decelerated (\(a=-8\) m/s\({}^{2}\)) & 191–205 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of vehicle motion. Figure 7: The signal \(C/N_{0}\), velocity and the Doppler of the signals for Example 2: (**a**) the \(C/N_{0}\) variation of the signals; (**b**) the velocity variation of the signals; (**c**) the Doppler frequency variation for PRN 14 of the signals (the numbers with circles corresponds to the numbers in Table 2). When the \(C/N_{0}\) decreases to 25 dB-Hz and the receiver experiences the third clockwise turn, the PLL method and the KF tracking both cannot work well and lose lock. Because the STKF tracking does not have system stability detection and \(\mathbf{Q}_{k}\) adaptation, the STKF may introduce inaccurate fading factors and perform worse than KF. As shown in Figure 9, the STKF is worse than KF in PRN 20. The adaptive factor of the proposed AKF algorithms for Example 2 is shown in Figure 10. When the system encounters sudden dynamics, the adaptive factor \(\lambda_{k}\) will be larger, otherwise, the adaptive factor will be set to 1.