## 4 Simulation and Verification
The definition of \(RMSE\) was as follows [29]:  
\[RMSE=\sqrt{\frac{1}{M}\sum_{i=1}^{M}\left(\ X_{n}(k)-\hat{X}_{n}^{i}(k|k) \right)^{2}} \tag{24}\]  
where \(X_{n}(k)\) and \(\hat{X}_{n}^{i}(k|k)\) represent the real value and filtered estimated value at the same time, respectively. Figure 6 shows the relationship between the RMS frequency error and the carrier-to-noise ratio of the three loops of the VFAPLL (MLE), VFAPLL (EKF) and FAPLL (MLE). Figure 7 shows the loss-of-lock probability of these three loops to the signal Doppler frequency under different carrier-to-noise ratios. Figure 6: Doppler frequency root mean square error of the VFAPLL (MLE), VFAPLL (EKF) and FAPLL (MLE) at different carrier-to-noise ratios. Figure 7: Loss-of-lock probability of the VFAPLL (MLE), VFAPLL (EKF) and FAPLL (MLE) at different CNRs. Figure 6 shows that the VFAPLL (MLE) loop has significantly improved frequency tracking accuracy compared with the VFAPLL (EKF) and FAPLL (MLE), whether in the low CNR range of 20-25 dB/Hz or in the extremely low CNR range of 15-20 dB/Hz. When the carrier -to-noise ratio of the input signal is at a very low value of 15 dB/Hz, the tracking accuracy of VFAPLL (MLE) was improved by 8 and 18 dB compared with the other two loops respectively. When C/N0 = 21.5, the tracking accuracy of the VFAPLL (MLE) was 6 dB higher than that of the VFAPLL (EKF) and 15 dB higher than that of the FAPLL (MLE). When C/N0 = 22.5, the mean square error of the frequency estimation of the VFAPLL (MLE) was 28 Hz, which was 6 dB higher than that of the VFAPLL (EKF) and 15 dB higher than that of the FAPLL (MLE). However, with the increase in the carrier noise ratio C/N0, this improvement gradually decreases. Figure 7 shows that the tracking threshold of the VFAPLL (MLE) was 22.5 dB/Hz, and the tracking thresholds of the VFAPLL (EKF) and FAPLL(MLE) were 24.5 and 25.5 dB/Hz, respectively, which indicates that compared with the other two loops, the VFAPLL (MLE) achieved 2 and 3 dB/Hz improvements, respectively, on the loss-of-lock threshold. In addition, it can be seen from the figure that when CNR \(\leq\) 18.5 dB/Hz, the probability of FAPLL (MLE) losing lock was approximately 1, while VFAPLL (MLE) still had the possibility of locking signal under extremely low CNR of 15 dB/Hz, and when CNR was 24.5 dB/Hz or more, the probability of VFAPLL (MLE) losing lock became very low. In summary, the VFAPLL (MLE) has higher tracking accuracy than the VFAPLL (EKF) and FAPLL (MLE) in both low-SNR and very low-SNR conditions. The following is an analysis of the robustness of the simulated loops. Based on the highly dynamic scenarios given in Figure 3a,b, we conducted a statistical comparison of the measurement accuracy of the Doppler frequency for the VFAPLL (MLE), VFAPLL (EKF), and FAPLL (MLE), as shown in Figures 8 and 9. Figures 8 and 9 show a comparison of the average values of the root mean square (RMS) Doppler frequency errors of VFAPLL (MLE), VFAPLL (EKF), and FAPLL (MLE) under different acceleration and jerk scenarios, respectively. The results show that the VFAPLL (MLE) had a smaller average RMS Doppler frequency error than the other two loops in the abovementioned highly dynamic scenarios, indicating that the VFAPLL (MLE) had the highest robustness of the three loops. This was mainly due to the efficient use of iterations by the LM algorithm and the advantages of the vector tracking loop. Figure 8: Comparison of the mean values of the VFAPLL (MLE), VFAPLL (EKF) and FAPLL (MLE). RMS Doppler frequency errors under different acceleration scenarios when CNR = 23 dB/Hz.