# Weak and Dynamic GNSS Signal Tracking Strategies for Flight Missions in the Space Service Volume
## 3 Adaptive Bandwidth Tracking Strategy
### Adaptive Four-State KF-Based Algorithm  
A three-state (phase error, Doppler frequency, and Doppler frequency rate) adaptive KF-based method was adopted in [15]. To improve the loop performance in weak-signal and high-dynamic environments, we introduce the state of code phase error and extend the three-state to four-state. Replacing DLL (Delay Lock Loop), FLL (Frequency Lock Loop), and PLL filters of conventional tracking structure with a combined adaptive Kalman filter, the block diagram of the four-state KF-based strategy is depicted in Figure 3. The error-state vector consists of code phase error \(\delta d_{k}\), carrier phase error \(\delta\phi_{k}\), Doppler frequency error \(\delta f_{d,k}\), and its rate error \(\delta\dot{f}_{d,k}\) at time \(k\):  
\[\mathbf{X}_{k}=\left[\delta d_{k},\delta\phi_{k},\delta f_{d,k},\delta\dot{f}_ {d,k}\right]^{T} \tag{15}\]  
\(\delta f_{d,k}\) equals the difference between the actual Doppler frequency and the frequency feedback to numerically controlled oscillator(NCO). For L1 C/A signal, the ratio of code chip rate \(f_{code}=1.023\) Mcps and nominal radio frequency \(f_{RF}=1575.42\) MHz is constant, so the state forward prediction is expressed as:  
\[\mathbf{\hat{x}}_{k,k-1}=\mathbf{\Phi}_{k,k-1}\mathbf{\hat{x}}_{k-1}=\left[ \begin{array}{cccc}1&0&\dfrac{f_{code}T_{coh}}{f_{RF}}&\dfrac{f_{code}T_{coh} ^{2}}{2f_{RF}}\\ &1&2\pi T_{coh}&\dfrac{f_{code}T_{coh}^{2}}{2f_{RF}}\\ &&2\pi&2\pi T_{coh}\\ &&&2\pi\end{array} \right]\mathbf{\hat{x}}_{k-1}=\left[ \begin{array}{cccc}1&0& \dfrac{T_{coh}}{1540}&\dfrac{T_{coh}^{2}}{2\cdot 1540}\\ &1&2\pi T_{coh}&\dfrac{T_{coh}^{2}}{2\cdot 1540}\\ &&2\pi&2\pi T_{coh}\\ &&&2\pi\end{array} \right]\mathbf{\hat{x}}_{k-1} \tag{16}\]  
Since coherent integration is used to process weak signals, the measurements obtained from discriminators are the average phase differences during CIT, rather than instant phase errors. Thus, we have:  
\[\delta\phi_{k}=\dfrac{1}{T_{coh}}\int_{Coh}^{T_{coh}}\delta\phi_{k-1}+2\pi \delta f_{d,k-1}\tau+\dfrac{1}{2}2\pi\delta\dot{f}_{d,k-1}\tau^{2}d\tau=\delta \phi_{k-1}+\pi\delta f_{d,k-1}T_{coh}+\dfrac{1}{3}\pi\delta\dot{f}_{d,k-1}T_{ coh}^{2} \tag{17}\]  
Likely, the measurement equation of code phase error is:  
\[\delta d_{k}=\left[ \begin{array}{cccc}1&0&\dfrac{\pi T_{coh}}{1540}&\dfrac{ \pi T_{coh}^{2}}{3\cdot 1540}\end{array} \right]\mathbf{X}_{k-1} \tag{18}\]Rewriting Equations (17) and (18) in the form of matrix and get the measurement matrix:  
\[\mathbf{H}_{k}=\left[ \begin{array}{ccc}1&0&\dfrac{\pi T_{coh}}{1540}&\dfrac{ \pi T_{coh}^{2}}{3\cdot 1540}\\ 0&1&\pi T_{coh}&\dfrac{1}{3}\pi T_{coh}^{2}\end{array} \right] \tag{19}\]  
The formal measurement equation is expressed as:  
\[\mathbf{Z}_{k}=\mathbf{H}_{k}\mathbf{\Phi}_{k,k-1}\mathbf{X}_{k-1}+v_{k} \tag{20}\]  
where \(v_{k}\) stands for the difference between actual measurement obtained from discriminator and the theoretical forward prediction. The process noise matrix is computed as follows:  
\[\mathbf{Q}_{k}=E[(\mathbf{\hat{X}}_{k}-\mathbf{\hat{X}}_{k-1})(\mathbf{\hat{X }}_{k}-\mathbf{\hat{X}}_{k-1})^{T}]\]  
\[=S_{f}f_{L1}^{2}\left[ \begin{array}{cccc}\dfrac{T_{coh}}{1540^{2}}&\dfrac{T _{coh}}{1540}&0&0\\ \dfrac{T_{coh}}{1540}&T_{coh}&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right]+S_{g}f_{L1}^{2}\left[ \begin{array}{cccc}\dfrac{T _{coh}^{3}}{3\cdot 1540^{2}}&\dfrac{T_{coh}^{3}}{3\cdot 1540}&\dfrac{T_{coh}^{2}}{2 \cdot 1540}&0\\ \dfrac{T_{coh}^{3}}{3\cdot 1540}&\dfrac{T_{coh}^{3}}{3}&\dfrac{T_{coh}^{2}}{2 \cdot 0}&0\\ \dfrac{T_{coh}^{2}}{2\cdot 1540}&\dfrac{T_{coh}^{2}}{2}&T_{coh}&0\\ 0&0&0&0\end{array} \right] \tag{21}\]  
\[+q_{a}\left(\dfrac{f_{L1}}{c}\right)^{2}\left[