## 5 Convergence Analysis  
The convergence of iteration mainly depends on three aspects: convergence standard, maximum number of iterations and initial value.  
(1) Convergence standard: The convergence criteria are defined as  
\[\left|L\left(\hat{\theta}^{i+1}\middle|r_{N}\right)-L\left(\hat{\theta}^{i} \middle|r_{N}\right)\right|<0.01 \tag{25}\]  
That is, if the difference between the logarithmic likelihood function at time I + 1 and the logarithmic likelihood function at time i of the maximum likelihood estimation is less than the set value, it is considered unnecessary to iterate further, that is, the algorithm has converged.  
(2) Maximum number of iterations: The maximum number of times that the iteration process terminates, given in advance, automatically terminates if it does not converge. This is set to 10 times for this article.  
(3) Initial value: set with predictor.  
In order to evaluate the convergence speed and performance of the algorithm, the following two kinds of square root errors are introduced in this paper [29],  
\[\text{RMSE}_{time}(k) =\sqrt{\frac{1}{M}\sum_{j=1}^{M}\left[\hat{\theta}_{k}^{j}-\theta _{k}^{j}\right]^{2}} \tag{26}\] \[\text{RMSE}_{total}(j) =\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left[\hat{\theta}_{k}^{j}-\theta _{k}^{j}\right]^{2}} \tag{27}\]  
where, \(\hat{\theta}_{j}(k)\) represents the estimated value of \(\theta\) in the \(j\)th Monte Carlo simulation at time \(k\), \(N\) is the length of discrete time, and \(M\) is the number of Monte Carlo simulations. \(RMSE_{time}(k)\) represents the average convergence speed of algorithms in simulations, \(RMSE_{total}\) indicates the performance of the algorithm in a simulation, the smaller the value, the better the performance of the algorithm.  
Figure 9: Comparison of the mean values of the V FAPLL (MLE-based), VFAPLL (EKF-based) and FAPLL (MLE-based) RMS Doppler frequency errors under different jerk scenarios when CNR = 23 dB/Hz.  
The Figure 10 compares the convergence performance of VFAPLL (MLE) and VFAPLL (EKF) when CNR = 23 dB/Hz. It is known from the figure that the average convergence time of VFAPLL (MLE) loop is shorter than that of VFAPLL (EKF).  
Tables 1 and 2 compare the performance of the VFAPLL (MLE) and VFAPLL (EKF) using RMSE\({}_{total}\) and average consumption time as the index, as follows:  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Carrier-to-Noise Ratio**} & **Average** & **Average Value of RMSE\({}_{total}\)** \\  & **Consumption Time of** & **of VFAPLL (MLE)** \\  & **VAPLL (MLE)** & **(s)** \\ \hline
25 & 0.023 & 0.02 \\
23 & 0.026 & 0.029 \\
21 & 0.036 & 0.035 \\
20 & 0.042 & 0.046 \\
19 & 0.056 & 0.05 \\
15 & 0.065 & 0.053 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The average consumption time and Average value of RMSE\({}_{total}\) of VFAPLL (MLE) when CNR is 15, 19, 20, 21, 23, and 25 dB/Hz.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Carrier-to-Noise Ratio**} & **Average Consumption Time** & **Average Value of RMSE\({}_{total}\)** **of** \\  & **of VFAPLL (EKF)** & **VAPLL (EKF)** \\  & **(s)** & **(s)** \\ \hline
25 & 0.026 & 0.029 \\
23 & 0.035 & 0.039 \\
21 & 0.039 & 0.042 \\
20 & 0.053 & 0.05 \\
19 & 0.06 & 0.055 \\
15 & 0.065 & 0.06 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The average consumption time and Average value of RMSE\({}_{total}\) of VFAPLL (EKF) when CNR is 15, 19, 20, 21, 23, and 25 dB/Hz.  
Figure 10: The relationship between RMSE\({}_{time}\) and moment \(k\) when CNR = 23 dB/Hz.  
It can be seen from the table that the average time of running VFAPLL (MLE) was less than that of running VFAPLL (EKF), which proves that the overall complexity of VFAPLL (MLE) is lower than that of VFAPLL (EKF) (Including frequency discriminator). The comparison of RMSE\({}_{total}\) shows that the loop tracking performance of VFAPLL (MLE) is better than that of VFAPLL (EKF).