## 2 Construction of Baseband Signal Preprocessing Model Based on Dual-filter Structure  
The classical GPS tracking channel, which typically adopts second-order or third-order phase control technology, has fixed loop bandwidth. Although a narrower bandwidth is helpful to reduce the thermal noise, in highly dynamic situations, the tracking channel may not retain useful high frequency information and suffers from significant dynamic stress error. These usually lead to signal distortion and tracking loss. Thus, a trade-off between noise resistance and response to dynamics is required while setting the loop bandwidth. In view of the abovementioned fact, the classical control approach cannot cope well with high dynamics, and hence KF, which belongs to the modern control category, is taken into account. KF is a kind of optimal estimator, which allows precise modeling of the tracking loop. With its help, baseband signal preprocessing can achieve accurate control of the local signal generation, even in highly dynamic situations [15]. The baseband signal preprocessing model constructed in this paper contains two pre-filters. In this model, the estimation of carrier and code tracking errors are relatively separate, the same as the manner adopted by classical GPS receiver. Figure 1 just takes carrier loop for example to show the details of this model. It is observed that the loop filter in the classical tracking channel has been replaced by a pre-filter equipped with state feedback. Obviously, designing a suitable KF is the key to accurate carrier tracking. This paper sets \(X_{k}^{carrier}=\left[\Delta\theta_{k}^{carrier}\ f_{k}^{carrier}\ f_{k}^{carrier} \right]^{T}\) as the state vector of pre-filter 1, where \(k\) is the epoch, \(\Delta\theta_{k}^{carrier}\) is the carrier phase error (unit: rad), \(f_{k}^{carrier}\) is the input carrier frequency, and \(\hat{f}_{k}^{carrier}\) is the changing rate of the input carrier frequency. These state variables have the following recurrence relations:  
\[ \begin{array}{rl}\Delta\theta_{k,k-1}^{carrier}&=\frac{1}{T_{coh}}\int_{0}^{ T_{coh}}\left(\Delta\theta_{k-1}^{carrier}+2\pi f_{k-1}^{carrier}t+\pi\hat{f}_{k-1}^{carrier}t^{2} \right)dt-\frac{1}{T_{coh}}\int_{0}^{T_{coh}}2\pi u_{k-1}^{carrier}tdt\\ &=\Delta\theta_{k-1}^{carrier}+\pi f_{k-1}^{carrier}T_{coh}+\frac{1}{3}\pi\hat{f }_{k-1}^{carrier}T_{coh}^{2}-\pi u_{k-1}^{carrier}T_{coh}\end{array}  \tag{1}\]  
where \(T_{coh}\) is the coherent integration time, \(u_{k-1}^{carrier}\) is the local carrier frequency at epoch \(k-1\). The output of carrier phase discriminator at epoch \(k\), notated as \(\Delta\theta_{k,k-1}^{carrier}\), is the average carrier phase error over an integration interval, due to the presence of Doppler shift. Thus, this output can be expanded as:  
\[ \begin{array}{rl}\Delta\theta_{k,k-1}^{carrier}&=\frac{1}{T_{coh}}\int_{0}^ {T_{coh}}\left(\Delta\theta_{k-1}^{carrier}+2\pi f_{k-1}^{carrier}t+\pi\hat{f}_ {k-1}^{carrier}t^{2}\right)dt-\frac{1}{T_{coh}}\int_{0}^{T_{coh}}2\pi u_{k-1}^ {carrier}tdt\\ &=\Delta\theta_{k-1}^{carrier}+\pi f_{k-1}^{carrier}T_{coh}+\frac{1}{3}\pi\hat{ f}_{k-1}^{carrier}\tau_{coh}^{2}-\pi u_{k-1}^{carrier}T_{coh}\end{array}  \tag{2}\]  
Unfortunately, if Equation (2) is directly used to generate the observation model for pre-filter 1, there will be a time lag between the state vector and observation [16]. Hence, by moving \(\Delta\theta_{k-1}^{carrier}\), \(f_{k-1}^{carrier}\) and \(\hat{f}_{k-1}^{carrier}\) to the left side of the equal sign, Equation (1) is rewritten as follows (more details are available in Appendix A):  
\[\left\{ \begin{array}{l}\Delta\theta_{k-1}^{carrier}=\Delta\theta_{k}^{carrier }-2\pi f_{k}^{carrier}T_{coh}+\pi\hat{f}_{k}^{carrier}T_{coh}^{2}+2\pi u_{k-1} ^{carrier}T_{coh}\\ f_{k-1}^{carrier}=f_{k}^{carrier}-\hat{f}_{k}^{carrier}T_{coh}\\ \hat{f}_{k-1}^{carrier}=\hat{f}_{k}^{carrier}\end{array} \right.