## 1 Introduction
Since the VLL used for hybrid DS/FH spread spectrum signals shares structural similarities with the frequency locked loop (FLL) employed for DSS signals, we can draw inspiration from the tracking algorithms developed for DSS signals under high dynamic conditions and design tracking algorithms suitable for VLL. It was found that the filter parameters (i.e., the loop bandwidth) are essentially governed by the noise characteristic and dynamics. To this end, the authors of [13] derive the optimal loop bandwidth. Furthermore, the authors of [14; 15; 16] propose tracking algorithms that can adaptively adjust the loop bandwidth. However, both the dynamic adaptability and real-time performance of these adaptive algorithms are limited to a certain extent. The superiority of an FLL mainly comes from its wide pull-in range. Therefore, an FLL is often used to assist a phase locked loop (PLL) in a coupled structure to maintain the lock of the carrier tracking loop when there is a possible large frequency error under the high dynamic environment. The authors of [17] utilize a tracking algorithm switching between an FLL and PLL; the authors of [18] employ a loop structure of the second-order FLL and serially assisted third-order PLL; the authors of [19] weigh the discriminating results of a second-order FLL and third-order PLL, respectively, and adjust the weights according to different dynamic conditions; and the authors of [20] further analyze the steady-state \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**References** & **Hopping Rates (hops/s)** & **Dynamics** & **SNR (dB)** & **Loop Structures** \\ \hline
[8] & \(\leq\)1.2 k & Acceleration 30 g & \(\geq\)13 & Second-order PLL \\
[9] & 10 k & Unspecified & Unspecified & FLL and PLL Switching \\
[10] & 100 k & Unspecified & \(\geq\)0 & PLL \\
[11] & 10 k & Jerk 10 g/s & \(\geq\)\(-\)30 & Second-order VLL assisted third-order PLL \\
[12] & 9 k & Jerk 10 g/s & \(-\)25 & Second-order VLL assisted third-order PLL \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of hopping rates, dynamics, SNR, and loop structures. tracking performance of a second-order FLL and parallelly assisted third-order PLL. However, these algorithms only discuss the improvement of loop structure without considering the improvement in the loop filter. An iterative filter design for an FLL and parallelly assisted PLL is presented in [21], but the iterative process may increase the computational complexity; the authors of [22] analyze the filter structure of a conventional third-order PLL and design a high dynamic PLL, but by simply considering the PLL-only case. The authors of [23] propose a high dynamic loop filter based on a second-order FLL and serially assisted third-order PLL, which achieves stable tracking in a high dynamic condition with an acceleration of 120 g and a jerk of 30 g/s. Based on our previous thorough discussions, we recognized the issue of frequency agility in hybrid DS/FH spread spectrum signals. This characteristic of frequency agility poses significant challenges to the traditional structure of an FLL-assisted PLL. However, it is noteworthy that although the carrier frequency of the hybrid DS/FH spread spectrum signal undergoes hops, the relative motion speed remains continuously variable. Consequently, we utilize the VLL instead of FLL to track hybrid DS/FH spread spectrum signals. Nevertheless, when facing high dynamic scenarios characterized by significant acceleration and jerk, the rapid variation in signals makes it difficult for a traditional VLL to effectively cope and maintain stable tracking. To achieve stable tracking of a hybrid DS/FH spread spectrum signal under high dynamic conditions, a high dynamic VLL (HD-VLL) based on the high dynamic loop filter is proposed in this paper. We initially delve into the accumulated phase tracking error caused by acceleration and jerk.