# Performance and Evaluation of GNSS Receiver Vector Tracking Loop Based on Adaptive Cascade Filter
## 2 Design of Cascaded Adaptive Vector Tracking Loop
### Signal Tracking Loop Based on KF  
Won [10,11] has proved that the KF method can provide a C/N\({}_{0}\) gain of 2-3 dB in terms of carrier phase and Doppler tracking. Therefore, in the signal tracking process, the filter equation is established with the code phase error \(\Delta\tau^{i}\) (chips), carrier phase error \(\Delta\theta^{i}\) (rad), and carrier frequency error \(\Delta f^{i}\) (Hz), carrier frequency rate error \(\Delta\alpha^{i}\) (Hz/s) as the state variables. After discretization, it can be expressed as [22]:  
\[\left[ \begin{array}{c}\Delta\tau^{i}\\ \Delta\theta^{i}\\ \Delta f^{i}\\ \Delta\alpha^{i}\end{array} \right]_{k+1}=\left[ \begin{array}{cccc}1&0&\beta T &\beta\frac{T^{2}}{\Delta\sigma}\\ 0&1&2\pi T&\pi T^{2}\\ 0&0&1&T\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}\Delta\tau^{i}\\ \Delta\theta^{i}\\ \Delta f^{i}\\ \Delta\alpha^{i}\end{array} \right]_{k}+W_{k} \tag{1}\]where the coefficient \(\beta\)= \(f_{code}/f_{carrier}\)= \(1/1540\) is used to convert the units of cycles to units of chips. \(f_{code},f_{carrier}\) represent code and carrier frequency. \(W_{k}\) represents the process noise vector. \(k\) represents time interval...represents the coherent integration time. \(i\) represents the channel number. The output of the discriminator is the average value of the carrier phase error \(\delta\overline{q}^{i}\) and the code phase error \(\delta\overline{\tau}^{i}\) (the output is obtained through time integration, called the \"average effect\") [1]:  
\[\left[ \begin{array}{c}\delta\overline{q}^{i}\\ \delta\overline{\tau}^{i}\end{array} \right]_{k+1}=\left[ \begin{array}{ccc}0& 1&-\pi T&\frac{\pi T^{2}}{3}\\ 1&0&\frac{-\beta T}{2}&\frac{\beta T^{2}}{6}\end{array} \right]\left[ \begin{array} []{c}\Delta\tau^{i}\\ \Delta\theta^{i}\\ \Delta f^{i}\\ \Delta\alpha^{i}\end{array} \right]_{k+1}+V_{k} \tag{2}\]  
where \(V_{k}\) represents the measurement noise vector. The code phase error and carrier error are calculated by using the early minus later phase discriminator and the two-quadrant arctangent phase discriminator, respectively:  
\[\delta\overline{\tau}^{i}=\tan^{-1}\left(\frac{\left(I_{E}^{2}+Q_{E}^{2} \right)-\left(I_{L}^{2}+Q_{L}^{2}\right)}{\left(I_{E}^{2}+Q_{E}^{2}\right)+ \left(I_{L}^{2}+Q_{L}^{2}\right)}\right) \tag{3}\]  
\[\delta\overline{q}^{i}=\tan^{-1}(Q_{P}/I_{P}) \tag{4}\]  
where \(I_{P},Q_{P},I_{E},Q_{E},I_{L},Q_{L}\) are the coherent integral output of the prompt, early, and later forward and quadrature directions of the correlator, respectively. When the C/N\({}_{0}\) ratio is low, the discriminator output is noisy. The adaptive strategy can be used to improve the tracking accuracy of the carrier phase in high dynamic scenarios. Measurement noise depends on the discriminator output and the coherent integration time. Figure 1: Design block diagram of adaptive cascaded vector tracking loop. where the diagonal element is proportional to the noise variance of the code phase error \(\Delta\tau^{i}\) and the carrier phase error \(\Delta\theta^{i}\)[5]. \[\left\{ \begin{array}{l}R_{\Delta\tau^{i}}=\sigma_{\Delta\tau}^{2}T=\frac{d_{0} T}{4T\sqrt{N_{0}}}\Big{(}1+\frac{2}{(2-d_{0})T\sqrt{N_{0}}}\Big{)}\\ R_{\Delta\theta^{i}}=\sigma_{\Delta\theta^{i}}^{2}T=\frac{T}{2T\sqrt{N_{0}}} \Big{(}1+\frac{2}{2T\sqrt{N_{0}}}\Big{)}\end{array} \right. \tag{6}\]  
where \(\sigma_{\Delta\tau^{i}}^{2},\sigma_{\Delta\theta^{i}}^{2}\) are the error variance values, \(T\) is the coherent integration time, and \(d_{0}\) is the code correlator interval. The C/N\({}_{0}\) ratio estimation method adopts the Variance Sum Method (VSM) for each coherent integration output result I, Q (coherent integration time is 20 ms) [6].