## 2 Construction of Baseband Signal Preprocessing Model Based on Dual-filter Structure
\begin{array}{ccc}1&T_{coh}&\frac{1}{2}T_{coh}^{2} \\ 0&1&T_{coh}\\ 0&0&1\end{array} \right] \tag{20}\]  
\[M_{k-1}^{code}=\left[ \begin{array}{ccc}-T_{coh}u_{k-1}^{code}&0&0\end{array}  \right]^{T} \tag{21}\]  
\[H_{k}^{code}=\left[ \begin{array}{ccc}1&-\frac{1}{2}T_{coh}&\frac{1}{6}T_{coh }^{2}\\ 0&\frac{f_{L1}}{f_{CA}}&0\end{array} \right] \tag{22}\]  
\[N_{k}^{code}=\left[ \begin{array}{ccc}\frac{1}{2}T_{coh}u_{k-1}^{code}&0\end{array}  \right]^{T} \tag{23}\]  
\[Z_{k}^{code}=\left[ \begin{array}{ccc}\Delta\theta_{k,k-1}^{code}\\ f_{k}^{curr}-f_{IF}+f_{L1}\end{array} \right] \tag{24}\]  
\(W_{k-1}^{code}\), \(V_{k}^{code}\) are respectively the system and observation noises of pre-filter 2, while the observation \(\Delta\theta_{k,k-1}^{code}\) obtained from code phase discriminator can be denoted by:  
\[\Delta\theta_{k,k-1}^{code}=\frac{1}{2}\frac{\sqrt{I_{k}^{2}+Q_{E}^{2}}}{\sqrt {I_{k}^{2}+Q_{E}^{2}}+\sqrt{I_{L}^{2}+Q_{L}^{2}}} \tag{25}\]  
\(u_{k-1}^{code}\) is the local code frequency at epoch \(k-\ 1\). Moreover, at epoch \(k\), the local code frequency, namely the feedback for code tracking, can be expressed as:  
\[u_{k}^{code}=\frac{\Delta\hat{\theta}_{k}^{code}}{T_{coh}}+\hat{f}_{k}^{code}+ \frac{1}{2}\hat{f}_{k}^{code}T_{coh} \tag{26}\]  
Since code estimation is processed after carrier stripping, its errors have no effect on carrier tracking. At the same time, once these two pre-filters are skillfully tuned, they can accurately track GPS signals in high dynamic situations. Taking pre-filter 1 for example, its equivalent bandwidth can be calculated from the elements of the gain matrix \(K_{k}^{curr}\) and is proportional to \(Q_{k}^{curr}\left(1,1\right)\), \(Q_{k}^{curr}\left(2,2\right)\) and \(Q_{k}^{curr}\left(3,3\right)\)[17]. Herein, \(Q_{k}^{curr}\left(1,1\right)\), \(Q_{k}^{curr}\left(2,2\right)\) and \(Q_{k}^{curr}\left(3,3\right)\) are the diagonal elements of \(Q_{k}^{curr}\). \(Q^{curr}\left(1,1\right)\) and \(Q^{curr}\left(2,2\right)\) (the epoch \(k\) is omitted to reveal these two values are constant) can be obtained from the receiver clock [11], while \(Q_{k}^{curr}\left(3,3\right)\) is related to the vehicle dynamics. If \(Q_{k}^{curr}\left(3,3\right)\) is estimated in real-time by any efficient adaptive algorithm, the loop bandwidth will vary in a time-varying optimal manner. This paper uses another more practical method: \(Q^{curr}\left(3,3\right)\) is set to a large constant value based on the general knowledge of vehicle dynamics. Since \(Q^{carr}\left(1,1\right)\) and \(Q^{carr}\left(2,2\right)\) are unchanged, the low variances of \(\Delta\hat{\theta}_{k}^{carr}\) and \(\hat{f}_{k}^{carr}\) can be maintained [3]. In other words, although the variance of \(\hat{f}^{carr}\) increases, phase and frequency estimations which normally deserve concern can hardly be affected. Additionally, \(R^{carr}\) is set according to the knowledge of the phase noise variance, while \(P_{0}^{carr}\) is usually larger than \(Q^{carr}\) to leave space for state corrections. In this case, pre-filter 1 gradually settles down to a wide bandwidth, and hence is robust to high dynamics. Benefiting from the above mentioned advantages, this baseband signal preprocessing model can be appropriate for highly dynamic situations and make GPS have better tracking and navigation performance.