## 2 VFL-Assisted PLL Carrier Tracking Loop Design
Based on the theory of parameter estimation, the maximum likelihood estimation method is applied to VFAPLL (MLE) loop to improve the robustness of the loop in high dynamic environment. The loop is modeled using the Jerk model of high-speed and highly maneuverable targets. In the model, the state vector is composed of the Doppler frequency \(f_{0}\), the first-order rate of change of the Doppler frequency \(f_{1}\), and the second-order rate of change of the Doppler frequency \(f_{2}\). The state equation is as follows [26]:  
\[\mathbf{X}_{k}= \begin{bmatrix}1&T&\frac{1}{2}T^{2}\\ 0&1&T\\ 0&0&1\end{bmatrix} \mathbf{X}_{k-1}+ \begin{bmatrix}\frac{1}{2}T^{2}\\ T\\ 1\end{bmatrix} \mathbf{W}_{k-1} \tag{1}\]  
where \(\mathbf{X}_{k}=[ \begin{smallmatrix}f_{0}&f_{1}&f_{2}\end{smallmatrix} ]^{T}\), \(\mathbf{W}_{k-1}\) is the state disturbance noise, and its covariance matrix \(\mathbf{Q}(k)=E[\mathbf{W}_{k}\mathbf{W}_{k}^{T}]\). The expression of \(\mathbf{Q}(k)\) is shown in Formula (2). The jerk model adds another dimension to the acceleration model; that is, it can estimate the jerk, so a more accurate estimation of the acceleration parameters can be obtained. Consequently, the jerk model is very suitable for modeling highly dynamic carriers. \[\mathbf{Q}(k)= \begin{bmatrix}\frac{T^{5}}{2\mathbf{0}}&\frac{T^{4}}{8}&\frac{T^{3}}{6 }\\ \frac{T^{4}}{8}&\frac{T^{2}}{3}&\frac{T^{2}}{2}\\ \frac{T^{3}}{6}&\frac{T^{2}}{2}&T\end{bmatrix} \mathbf{N}_{y}T \tag{2}\]  
where \(\mathbf{N}_{y}\) represents the random jitter of the third derivative of the carrier frequency. After coherent integration processing by the integral clearer, the output integral signal with phase and frequency deviation is [17]:  
\[g(k)= \begin{bmatrix}g_{l}(k)\\ g_{\mathcal{Q}}(k)\end{bmatrix} = \begin{bmatrix}AD(k)R(\Delta\tau)\frac{\sin(f_{ l}T_{coh})}{f_{e}T_{coh}}cos\theta(k)\\ AD(k)R(\Delta\tau)\frac{\sin(f_{e}T_{coh})}{f_{e}T_{coh}}sin\theta(k)\end{bmatrix} + \begin{bmatrix}n_{l}(k)\\ n_{\mathcal{Q}}(k)\end{bmatrix}  \tag{3}\]  
where \(A\) is the signal amplitude, \(D(k)\) represents the data code, and \(R(\Delta\tau)\) represents the C/A code autocorrelation function with a maximum value of 1. As the code delay of the instant branch is within one-half chip when the tracking is initialized, \(R(\Delta\tau)\) can be normalized to 1. Variable \(\Delta\tau\) is the phase difference between the copied C/A code and the received C/A code, \(n_{1}(k)\), \(n_{Q}(k)\) are zero-mean white Gaussian noise, and the \(\sin\text{c}(f_{e}T_{coh})\) term is considered to be approximately equal to 1. Therefore, the observation equation of the loop is:  
\[Z(k)= \begin{bmatrix}Acos(\theta_{me})\\ Asin(\theta_{me})\end{bmatrix} +B(k) \tag{4}\]  
where \(\theta_{me}(k)\) is the average residual phase over time \(T\). \[ \begin{array}{ll}\theta_{me}(k)&=\frac{1}{T}\int_{(k-1)T}^{kT}\!\!\!\left[ \theta_{c}(k)+2\pi f_{c}(k)T+2\pi f_{c}{}^{\prime}(k)\frac{T^{2}}{2}+2\pi f_{c }{}^{\prime\prime}(k)\frac{T^{3}}{6}\right]d\tau\\ &\qquad\qquad-\frac{1}{T}\int_{(k-1)T}^{kT}\!\!\!\left[\theta_{nco}(k)+2\pi f_ {nco}(k)T+\Delta\theta_{nco}\right]d\tau\\ &=\theta_{e}(k)+\pi f_{c}(k)T+\pi f_{c}{}^{\prime}(k)\frac{T^{2}}{3}+\pi f_{c }{}^{\prime\prime}(k)\frac{T^{3}}{12}-\pi f_{nco}(k)T\end{array}  \tag{5}\]  
The covariance of the observation noise matrix \(V(k)\) is \(R_{k}\), where \(R_{k}=E\!\left[B_{k}\;B_{k}^{T}\right]\), such that  
\[R_{k}\!\!=\!\!\!\left[ \begin{array}{ll}\sigma_{n}^{2}&\theta_{\gamma}\\ \theta&\sigma_{n}^{2}\end{array} \right] \tag{6}\]