## 4 Performance Evaluation and Discussion
### Example 1: Signal Fading Rapidly and Acceleration Scenario  
Figure 4 demonstrates the simulated fading signal with dynamics. In Figure 4a, the \(C/N_{0}\) began with 45 dB-Hz for 20 s. Then the \(C/N_{0}\) was descended by 1 dB per 5 s until it decreased to 25 dB-Hz. Then the signal kept the 25 dB-Hz for 60 s. After 180 s, the \(C/N_{0}\) was increased by 1 dB per 5 s until reaching the initial signal power. In Figure 4b, during the first 20 s, the receiver halted. After 20 s, the receiver accelerated with 32 m/s\({}^{2}\) and the velocity reached 3200 m/s at \(t\) = 120 s. After 120 s, the receiver remained at 3200 m/s until \(t\) = 180 s. Then, the receiver decelerated with \(-\)32 m/s\({}^{2}\) for 100 s and then stopped at 280 s. The velocity set in the scenario often occurs in aircraft, which is a high dynamic scenario. In four tracking methods, the parameters settings are as follows:  
Method 1:  
The initial parameter settings are listed in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Description** & **Values** \\ \hline \(\mathbf{Q}_{k}\) & Process noise covariance matrix & Using Equation (3) with \(q_{4}\)= 0.3 \(\left(\mathrm{m}^{2}/\mathrm{s}^{6}\right)/\mathrm{Hz}\) \\ \(\mathbf{R}_{k}\) & Measurement noise covariance matrix & Using Equation (4) with \(\left(C/N_{0}\right)_{0}\) = 45 dB-Hz \\ \(T\) & Update period of the tracking loop & 4 ms \\ \(N\) & Window size for estimated innovation covariances & 20 \\ \(\chi_{2}^{2}(1)\) & Threshold value to detect system stability & 6.6349 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The initial parameters settings for Method 1. Figure 4: The signal \(C/N_{0}\) and the velocity of the signals for Example 1: (**a**) the \(C/N_{0}\) variations of the signals; (**b**) the velocity variations of the signals. 1. Initialize the process noise covariance matrix \(\mathbf{Q}_{k}\) using (3) with \(q_{a}\)= 0.3 \(\left(\mathrm{m}^{2}/\mathrm{s}^{6}\right)/\mathrm{Hz}\). Initialize the measurement noise covariance \(\mathbf{R}_{k}\) using (4) with \(\left(C/N_{0}\right)_{0}\) = 45 dB-Hz and \(\left(c/n_{0}\right)_{0}=10^{4.5}\). The update period of the tracking loop is 4 ms. For the weak and dynamic signals, the update period is a compromise parameter, which should be long enough to work under weak signal conditions and should be as short as possible to work under dynamic conditions simultaneously. Then we set the update period as 4 ms.
2. In the proposed method, the window size for estimated innovation covariances is 20. To make a compromise between the unbiased estimation and the short-term statistics, reference [29] has stated that good window size is between 10 and 30. 3. Set the significance level \(\alpha\) to 0.01, and the threshold value \(\chi^{2}_{0.01}(1)\) is 6.6349. It means that the probability of \(\left\{\beta_{k}\geq 6.6349\right\}\) is 0.01. If the event \(\left\{\beta_{k}\geq 6.6349\right\}\) occurs, the hypothesis \(\gamma_{1}\) will hold and the proposed AKF tracking will be implemented. Method 2: in the PLL tracking method, the PLL's bandwidth is 15 Hz. The update period of the tracking loop is the same as that in Method 1. Method 3: in the standard KF tracking, the settings of \(\mathbf{Q}_{k}\), \(\mathbf{R}_{k}\), and loop update interval are the same as those in Method 1. Method 4: in STKF tracking, the settings of \(\mathbf{Q}_{k}\), \(\mathbf{R}_{k}\), and loop update interval are the same as those in Method 1. The \(C/N_{0}\) and Doppler estimated results of the four algorithms for PRN 14 and PRN 19 are shown in Figure 5. Figure 5a,c are the estimated results for PRN 14. Meanwhile, Figure 5b,d are the estimated results for PRN 19. Figure 5 demonstrates that the proposed AKF tracking approach performs best, which can follow the variable signal strength. Other tracking methods cannot follow the weak and dynamic power and deviate from the actual value when the signal is weak. In Figure 5c,d, the Doppler rates for PRN 14 and PRN 19 are 39 Hz/s and 50 Hz/s from 20 s to 120 s, respectively.