## 2 Construction of Baseband Signal Preprocessing Model Based on Dual-filter Structure
\tag{3}\]  
Substitution of Equation (3) into Equation (2) results in:  
\[\Delta\theta_{k,k-1}^{carrier}=\Delta\theta_{k}^{carrier}-\pi f_{k}^{carrier}T _{coh}+\frac{1}{3}\pi\hat{f}_{k}^{carrier}T_{coh}^{2}+\pi u_{k-1}^{carrier}T_{ coh} \tag{4}\]  
Figure 1: The carrier loop using pre-filter. Then, in terms of Equations (1) and (4), the state equation and observation equation of pre-filter 1 can be expressed as:  
\[\left\{ \begin{array}{l}X_{k}^{carr}=\Phi_{k,k-1}^{carr}X_{k-1}^{carr}+M_{k-1}^{ carr}+W_{k-1}^{carr}\\ Z_{k}^{carr}=H_{k}^{carr}X_{k}^{carr}+N_{k}^{carr}+V_{k}^{carr}\end{array} \right. \tag{5}\]  
where:  
\[\Phi_{k,k-1}^{carr}=\left[ \begin{array}{ccc}1&2\pi T_{coh}&\pi T_{coh}^{2}\\ 0&1&T_{coh}\\ 0&0&1\end{array} \right] \tag{6}\]  
\[M_{k-1}^{carr}=\left[ \begin{array}{ccc}-2\pi T_{coh}u_{k-1}^{carr}&0&0\\ \end{array} \right]^{T} \tag{7}\]  
\[H_{k}^{carr}=\left[ \begin{array}{ccc}1&-\pi T_{coh}&\frac{1}{3}\pi T_{coh}^{ 2}\\ \end{array} \right] \tag{8}\]  
\[N_{k}^{carr}=\pi T_{coh}u_{k-1}^{carr} \tag{9}\]  
\(Z_{k}^{carr}\) is the observation, and can be directly obtained from carrier phase discriminator, namely:  
\[Z_{k}^{carr}=\Delta\theta_{k,k-1}^{carr}=\arctan\frac{Q_{p}}{I_{p}} \tag{10}\]  
\(W_{k-1}^{carr},\;V_{k}^{carr}\) represent the system noise and observation noise, respectively. Assume these two noises have the following characteristics:  
\[\left\{ \begin{array}{l}E\left[W_{k}\right]=0,E\left[W_{k}W_{j}^{T}\right]=Q _{k}^{carr}\delta_{kj}\\ E\left[V_{k}\right]=0,E\left[V_{k}V_{j}^{T}\right]=R_{k}^{carr}\delta_{kj}\\ E\left[W_{k}V_{j}^{T}\right]=0\end{array} \right. \tag{11}\]  
where \(\delta_{kj}\) is the Kronecker delta. Then, the corresponding iterative algorithm for pre-filter 1 is given below:  
\[\hat{X}_{k,k-1}^{carr}=\Phi_{k,k-1}^{carr}\hat{X}_{k-1}^{carr}+\hat{M}_{k-1}^{ carr} \tag{12}\]  
\[P_{k,k-1}^{carr}=\Phi_{k,k-1}^{carr}P_{k-1}^{carr}\Phi_{k,k-1}^{carr,T}+Q_{k-1} ^{carr} \tag{13}\]  
\[K_{k}^{carr}=P_{k,k-1}^{carr}H_{k}^{carr,T}\left(H_{k}^{carr}P_{k,k-1}^{carr}H_ {k}^{carr,T}+R_{k}^{carr}\right)^{-1} \tag{14}\]  
\[P_{k}^{carr}=\left(I-K_{k}^{carr}H_{k}^{carr}\right)P_{k,k-1}^{carr} \tag{15}\]  
\[\hat{X}_{k}^{carr}=\hat{X}_{k,k-1}^{carr}+K_{k}^{carr}\left(Z_{k}^{carr}-N_{k} ^{carr}-H_{k}^{carr}\hat{X}_{k,k-1}^{carr}\right) \tag{16}\]  
The carrier loop is in essence a Phase Lock Loop (PLL), and aims to achieve the elimination of the carrier phase error. Since \(\Delta\theta_{k+1}^{carr}=\Delta\theta_{k}^{carr}+2\pi\left(f_{k}^{carr}-u_{ k}^{carr}\right)T_{coh}+\pi\hat{f}_{k}^{carr}T_{coh}^{2}\), by assuming \(\Delta\theta_{k+1}^{carr}=\ 0\), the feedback at epoch \(k\) can be obtained as:  
\[u_{k}^{carr}=\frac{\Delta\hat{\theta}_{k}^{carr}}{2\pi T_{coh}}+\hat{f}_{k}^{ carr}+\frac{1}{2}\hat{f}_{k}^{carr}T_{coh} \tag{17}\]  
It can be seen that \(u_{k}^{carr}\) is not exactly equivalent to the estimated input carrier frequency \(\hat{f}_{k}^{carr}\). And further discussions about \(u_{k}^{carr}\) and \(\hat{f}_{k}^{carr}\) can be found in Sections 3 and 4.1. Similarly, the state vector of pre-filter 2 is set to \(X_{k}^{\text{code}}=\left[\Delta\theta_{k}^{code}\ f_{k}^{code}\ \dot{f}_{k}^{code}\right]^{T}\), where \(\Delta\theta_{k}^{code}\) is the code phase error (unit: chip), \(f_{k}^{code}\) is the input code frequency, and \(\dot{f}_{k}^{code}\) is the changing rate of the input code frequency. Meanwhile, in consideration of the following formula:  
\[f_{d}=f^{curr}-f_{IF}=\frac{f_{L1}}{f_{CA}}(f^{code}-f_{CA}) \tag{18}\]  
where \(f_{d}\) is the Doppler shift, \(f_{IF}\) is the Intermediate Frequency (IF), \(f_{L1}\) (1575.42 MHz) is the L1 carrier frequency, \(f_{CA}\) (1.023 MHz) is the nominal code frequency, the state equation and observation equation of pre-filter 2 can be written as:  
\[\left\{ \begin{array}{l}X_{k}^{code}=\Phi_{kk-1}^{code}X_{k-1}^{code}+M_{k-1}^{ code}+W_{k-1}^{code}\\ Z_{k}^{code}=H_{k}^{code}X_{k}^{code}+N_{k}^{code}+V_{k}^{code}\end{array} \right. \tag{19}\]  
where:  
\[\Phi_{k,k-1}^{code}=\left[