## 4 Simulation and Verification  
The receiver adopts a combination of vector and scalar tracking loops. To compare the VFAPLL (MLE), VFAPLL (EKF) and FAPLL (MLE) under the same conditions, parameters such as their loop filter bandwidth and the predetection integral were selected to be consistent; that is, the predetection integration time was 1 ms, \(B_{PLL}=15\) Hz, and \(B_{PLL}=25\) Hz. The MATLAB software was used to simulate and generate the GPS digital intermediate frequency signal using a signal frequency \(f_{IF}=4.092\) MHz, sampling frequency \(f_{s}=16.368\) MHz, ratio between carrier and noise CNR =35 dB/Hz, and a 1 ms C/A code period. During 0-25s, the carrier moved in a uniform straight line. Then, the number of channels with signals in the system was 12. During 26-59s, the carrier continued to move in a uniform straight line, but during this period, the number of signal channels decreased to three or less, and this weak signal state lasted until 120s. The carrier periodically performed the high dynamic motion as shown in Figure 2 from 60s to 85s, during which the carrier was in an extremely harsh high dynamic weak signal environment. After 85s, the carrier returned to uniform linear motion, and after 120s, the number of channels with signal returned to 12 [28]. In this paper, the highly dynamic model of the Jet Propulsion Laboratory (JPL) in the United States was used to simulate the highly dynamic part in the motion state of the carrier, as shown in Figure 3a,b. During 0-3 s, the receiver performed uniform acceleration movement at \(-25\) g, and the jerk was zero. At 3 s, the jerk increased rapidly to 100 g/s and lasted for 0.5s, meanwhile, the acceleration increased from \(-25\) to 25 g, and the acceleration motion was then made uniform at 25 g. When it reached 5.5 s, the jerk sharply changed from 0 to \(-100\) g/s, and lasted for 0.5 s, whereas the acceleration decreased from 25 to \(-25\) g, and the acceleration motion was then made uniform at \(-25\) g [28]. The simulation tests the tracking performance of the VFAPLL and FAPLL based on the maximum likelihood estimation algorithm and VFAPLL based on the extended Kalman filter. Figures 4 and 5 show the speed and position errors generated by the three loops in the movement process described above. Figures 4 and 5 show that the tracking performance of the three loops was good during the initial period of 0-24 s, but during the period of 25-120 s, the loops were in a weak signal environment, and the position and speed errors of the three loops significantly increased:  
Figure 3: Trajectory of the receiver relative to the GPS satellite. (**a**) Acceleration model for highly dynamic motion; (**b**) jerk model for highly dynamic motion [28]. at the same time, there was a gap in tracking performance. The VFAPLL (MLE) had the best performance both in a weak signal environment and in a harsh, highly dynamic weak signal environment during 60-85 s; however, the FAPLL (MLE) could not adapt to the high dynamic stress due to the limited frequency discrimination range of the FLL and began to lose lock after 63 s. After 120 s, the number of channels with signals was restored to 12; accordingly, the tracking errors of the VFAPLL (MLE) and VFAPLL (EKF) were greatly reduced and gradually returned to the state during 0-24 s. During the operation of the above loops, the maximum speed error difference between the VFAPLL (MLE) and VFAPLL (EKF) was 2.69 m/s, and the maximum position error difference was 57 m.  
Figure 4: Velocity error. Figure 5: Position error. The frequency tracking accuracy of the loop was analyzed through the relationship between the root mean square (\(RMS\)) frequency error and the carrier-to-noise ratio (CNR) of the loop, and the frequency tracking accuracy of VFAPLL (MLE), VFAPLL (EKF) and FAPLL (MLE) was compared.