## 4 Simulation and Analysis
### Comparison of Dynamic Adaptability  
To compare the dynamic adaptability of the standard VLL, the VLL variant described in [11], and the HD-VLL, all three of these systems are simulated and analyzed.  
Firstly, we selected the parameters used in reference [11]: a frequency hopping rate of 10 khops/s, dynamics of 10 g/s, and a CNR of 33 dB-Hz. The three VLLs were simulated with a bandwidth of 5 Hz. The velocity estimate curves and velocity error curves of three tracking loop structures are shown in Figure 10.  
As shown in Figure 10, All three VLLs are capable of steady-state tracking under the parameters used in reference [11].  
Next, we increase the dynamics by assuming the same parameters as those used in Section 4.1, and a bandwidth of 5 Hz is adopted for the three VLLs. The velocity estimate curves and velocity error curves of the three tracking loop structures are shown in Figure 11.  
In Figure 11, it can be seen that the HD-VLL is able to lock and enter steady-state tracking successfully, while the VLL and the VLL in [11] will lose lock at the moment of maximum jerk in the sinusoidal motion and cannot maintain steady-state tracking.  
Then, to further observe the performance of VLL, the VLL in [11], and the HD-VLL under high dynamic conditions, we set up a simulation scenario where the frequency of sinusoidal motion increases with time: the velocity is \([20 sin(2t^{2})+30]\) m/s, the acceleration is \(80 tcos(2t^{2})\) m/s\({}^{2}\), and the jerk is \([80 tcos(2t^{2})-320t^{2}\)sin(2t^{2})]\) m/s\({}^{3}\). It can be seen that the  
Figure 10: The tracking results of the VLL, the VLL in [11], and the HD-VLL with a jerk of 10 g/s. (**a**) The velocity estimate curve. (**b**) The velocity error curves.  
maximum values of acceleration and jerk will increase with time. The velocity estimate curves and velocity error curves of the three tracking loop structures are shown in Figure 12.  
Figure 11: The tracking results of the VLL, the VLL in [11], and the HD-VLL with a velocity of \([20 sin(10t)+30]\). **(a)** The velocity estimate curve. **(b)** The velocity error curves.  
Figure 12: The tracking results of the VLL, the VLL in [11], and the HD-VLL with a velocity of \([20 sin(2t^{2})+30]\). **(a)** The velocity estimate curve. **(b)** The velocity error curve.  
As shown in Figure 12, at the beginning of tracking, the acceleration and jerk values are small, and the three VLLs are able to track, but as the acceleration and jerk increase, the HD-VLL can still maintain stable tracking, while the VLL and VLL in [11] cannot adapt to high dynamics and thus lose lock. Therefore, the HD-VLL proposed in this paper has better dynamic adaptability than the VLL and VLL in [11].  
Finally, in order to make the VLL adapt to the given dynamic condition, the bandwidth of the VLL is increased to 8 Hz and 11 Hz, as shown in Figure 13.  
It can be seen in Figure 13 that the VLL is able to lock and enter steady-state tracking for bandwidths of 8 Hz and 11 Hz. The larger the bandwidth, the higher the dynamic the VLL can adapt to. However, as the bandwidth increases, the steady-state tracking error also becomes larger, which worsens the PLL's tracking performance in VLL-SA-PLL structures. Thus, we adopt 8 Hz for the VLL in a subsequent simulation.