# Design and Validation of a Cascading Vector Tracking Loop in High Dynamic Environments
## 2 Structure and Methods
### Cascading Tracking Methods
For the feedback, the carrier phase difference must be provided by the pre-filter, and the rest of the frequency feedback values can be provided by the relevant velocity results of the main filter rather than the pre-filter frequency results. Therefore, in this tracking structure, the main filter and its feedback ensure that the channel with a strong signal can assist the channel tracking with a weak signal. Furthermore, the pre-filter can be a rampart for blocking extremely bad signals from contaminating other channels. Further, if the vector receiver works in the low dynamic environment, with the cascading filtering, the pre-filter and the main filter can work in different rates. The pre-filter frequency is the same as that of the coherent integration, but the main filter can decelerate the frequency of computation to reduce the calculation because the carrier signal changes caused by low-speed motion are not very large. In Figure 2 and Equation (6), considering the feedback, the states of the pre-filter are \(\Delta\theta\), \(\omega_{k}\), \(\omega^{\prime}{}_{k}\), and \(\omega^{\prime\prime}{}_{k}\). \(\Delta\theta\) is the difference between the true carrier phase and the receiver NCO's carrier phase at the start of an integral interval. Therefore, the carrier phase relationship can be expressed as Equation (6), where \(T\) is the length of the accumulation interval, and \(f_{\rm IF}\) is the intermediate frequency (IF). \(\left(\omega_{nco}\right)_{k}\) is the feedback to the carrier's NCO at time \(k\);  
\[ \begin{array}{l}\Delta\theta(k+1)\\ =\theta_{1}(k)+(\omega(k)+2\pi f_{IF})T+\frac{1}{2}\omega^{\prime}(k)T^{2}+ \frac{1}{6}\omega^{\prime\prime}(k)T^{3}-(\theta_{2}(k)+(\omega_{nco}+2\pi f _{IF})T)\\ =\Delta\theta(k)+\omega(k)T+\frac{1}{2}\omega^{\prime}(k)T^{2}+\frac{1}{6} \omega^{\prime\prime}(k)T^{3}-\omega_{nco}T\\ \end{array}  \tag{6}\]  
Combined with the Doppler angular frequency, the carrier phase kinetic model is expressed as Equations (7) and (8). At moment \(k\), \(\omega_{k}\), \(\omega^{\prime}{}_{k}\), and \(\omega^{\prime\prime}{}_{k}\) express the estimated Doppler angular frequency and the first and second order change rates, respectively. \(w_{k}\) is the state noise matrix of the kinetic model. The detailed noise covariance matrixes of the carrier pre-filter formula are provided in [21];  
\[\left[ \begin{array}{c}\Delta\theta\\ \omega\\ \omega^{\prime\prime}\\ \omega^{\prime\prime}\end{array} \right]_{k+1}=\left[ \begin{array}{cccc}1&T& \frac{T^{2}}{2}&\frac{T^{3}}{6}\\ 0&1&T&\frac{T}{2}\\ 0&0&1&T\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}\Delta\theta\\ \omega\\ \omega^{\prime}\\ \omega^{\prime\prime}\end{array} \right]_{k}-(\omega_{nco})_{k}\left[ \begin{array} []{c}T\\ 0\\ 0\\ 0\end{array} \right]+w_{k}, \tag{7}\]  
Figure 3: Cascading vector tracking loop structure chart. code frequency, and \(CN_{i}\) is the CNR of channel \(i\). \(g_{i}\) is the slope between the code phase difference and the corresponding pseudocode distance difference [22]. \(d\) is the spacing of correlators. When the narrow correlation technique is not considered, the spacing is generally considered as 0.5 code slices. Through Equations (11)-(13), the nonlinear observation equation can be integrated as follows (16). Each tracked channel provides three observation values to the main filter through the corresponding pre-filter:  
\[\left[ \begin{array}{c}\frac{Z_{\text{req},1}}{S_{1}}\cdot c\\ Z_{freq,1}\\ Z_{freq,1}\\ \vdots\\ \frac{Z_{\text{req},n}}{S_{n}}\cdot c\\ Z_{freq,n}\\ Z_{freq,n}\end{array} \right]=\left[ \begin{array}{c}h_{1}\left(\overset{ \rightarrow}{X_{k}}\right)\\ h_{2}\left(\overset{\rightarrow}{X_{k}}\right)\\ h_{3}\left(\overset{\rightarrow}{X_{k}}\right)\\ \vdots\\ h_{1}\left(\overset{\rightarrow}{X_{k}}\right)\\ h_{2}\left(\overset{\rightarrow}{X_{k}}\right)\\ h_{3}\left(\overset{\rightarrow}{X_{k}}\right)\end{array}