## Appendix B  
The filter, whose state vector is \(\left[\Delta\theta_{k}^{carr}\;\Delta f_{k}^{carr}\;\Delta\dot{f}_{k}^{carr} \;\Delta\theta_{k}^{code}\right]^{T}\), is a popular baseband signal preprocessing model. Herein, \(\Delta f_{k}^{carr}=f_{k}^{carr}-u_{k}^{carr}\), \(\Delta\dot{f}_{k}^{carr}=\dot{f}_{k}^{carr}-0\). To focus on analyzingthe benefits of decoupling between carrier tracking and code tracking, this filer should be further improved. First, consider the equations derived from Equation (18):  
\[f^{code}=\beta f^{carr}+\beta\left(f_{L1}-f_{IF}\right) \tag{19}\]  
\[\dot{f}^{code}=\beta\dot{f}^{carr} \tag{20}\]  
where \(\beta=\frac{f_{CA}}{f_{L1}}\). The new state variables \(\Delta\theta_{k}^{carr}\), \(f_{k}^{carr}\), \(\dot{f}_{k}^{carr}\) and \(\Delta\theta_{k}^{code}\) have the following recurrence relations:  
\[\left\{ \begin{array}{l}\Delta\theta_{k}^{carr}=\Delta\theta_{k-1}^{carr}+2 \pi f_{k-1}^{carr}T_{coh}+\pi\dot{f}_{k-1}^{carr}T_{coh}^{2}-2\pi u_{k-1}^{carr}T _{coh}\\ f_{k}^{carr}=f_{k-1}^{carr}+\dot{f}_{k-1}^{carr}T_{coh}\\ \dot{f}_{k}^{carr}=\dot{f}_{k-1}^{carr}\\ \Delta\theta_{k}^{code}=\Delta\theta_{k-1}^{code}+\beta f_{k-1}^{carr}T_{coh}+ \frac{1}{2}\beta\dot{f}_{k-1}^{carr}T_{coh}^{2}-u_{k-1}^{code}T_{coh}+\beta \left(f_{L1}-f_{IF}\right)T_{coh}\end{array} \right. \tag{21}\]  
The relation between the observation \(\Delta\theta_{k,k-1}^{carr}\) and state variables can still be represented by Equation (4), while \(\Delta\theta_{k,k-1}^{code}\) is written as:  
\[\Delta\theta_{k,k-1}^{code}=\Delta\theta_{k}^{code}-\frac{1}{2}\beta f_{k}^{ carr}T_{coh}+\frac{1}{6}\beta\dot{f}_{k}^{carr}T_{coh}^{2}+\frac{1}{2}u_{k-1}^{ code}T_{coh}-\frac{1}{2}\beta\left(f_{L1}-f_{IF}\right)T_{coh} \tag{22}\]  
Then, the state equation and observation equation of pre-filter-S can be expressed as:  
\[\left\{ \begin{array}{l}X_{k}^{caco}=\Phi_{k,k-1}^{caco}X_{k-1}^{caco}+M_{k- 1}^{caco}+W_{k-1}^{caco}\\ Z_{k}^{caco}=H_{k}^{caco}X_{k}^{caco}+N_{k}^{caco}+V_{k}^{caco}\end{array} \right. \tag{23}\]  
where:  
\[\Phi_{k,k-1}^{caco}=\left[ \begin{array}{cccc}1&2\pi T_{coh}&\pi T_{coh}^{2}& 0\\ 0&1&T_{coh}&0\\ 0&0&1&0\\ 0&\beta T_{coh}&\frac{1}{2}\beta T_{coh}^{2}&1\end{array} \right] \tag{24}\]  
\[M_{k-1}^{caco}=\left[ \begin{array}{cccc}-2\pi T_{coh}u_{k-1}^{carr}&0&0&\beta \left(f_{L1}-f_{IF}\right)T_{coh}-T_{coh}u_{k-1}^{code}\end{array} \right]^{T} \tag{25}\]  
\[H_{k}^{caco}=\left[ \begin{array}{cccc}1&-\pi T_{coh}&\frac{1}{3}\pi T_{coh}^ {2}&0\\ 0&-\frac{1}{2}\beta T_{coh}&\frac{1}{6}\beta T_{coh}^{2}&1\end{array} \right] \tag{26}\]  
\[N_{k}^{caco}=\left[ \begin{array}{cccc}\pi T_{coh}u_{k-1}^{carr}&-\frac{1}{2} \beta\left(f_{L1}-f_{IF}\right)T_{coh}+\frac{1}{2}T_{coh}u_{k-1}^{code}\end{array}  \right]^{T} \tag{27}\]  
\[Z_{k}^{caco}=\left[ \begin{array}{cccc}\Delta\theta_{k,k-1}^{carr}\\ \Delta\theta_{k,k-1}^{code}\end{array} \right] \tag{28}\]  
while the feedback for carrier tracking is still represented by Equation (17), the feedback for code tracking is expressed as:  
\[u_{k}^{code}=\frac{1}{T_{coh}}\Delta\hat{\theta}_{k}^{code}+\beta\hat{f}_{k}^{ carr}+\frac{1}{2}\hat{\beta\dot{f}_{k}^{carr}T_{coh}+\beta\left(f_{L1}-f_{IF}\right) \tag{29}\]Furthermore, it is noted that \(\Delta\theta_{k,k-1}^{code}\) is uncorrelated with \(\Delta\theta_{k,k-1}^{corr}\)[15]. In fact, based on statistical analysis, it can be drawn that the correlation coefficient between \(\Delta\theta_{k,k-1}^{corr}\) and \(\Delta\theta_{k,k-1}^{code}\) is usually smaller than 0.01. The measurement covariance matrix \(R_{k}^{caco}\), therefore, is set as:  
\[R_{k}^{caco}=\left[ \begin{array}{cc}\sigma_{1}^{2}&0\\ 0&\sigma_{2}^{2}\end{array} \right] \tag{10}\]  
where \(\sigma_{1}^{2}\) and \(\sigma_{2}^{2}\) are the variances at the outputs of the carrier and code discriminators respectively.