# 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
\begin{array}{cccc}\dfrac{T _{coh}^{5}}{20\cdot 1540^{2}}&\dfrac{T_{coh}^{5}}{20\cdot 1540}&\dfrac{T_{coh}^{4}}{ 8\cdot 1540}&\dfrac{T_{coh}^{3}}{6\cdot 1540}\\ \dfrac{T_{coh}^{5}}{20\cdot 1540}&\dfrac{T_{coh}^{5}}{20}&\dfrac{T_{coh}^{4}}{8 }&\dfrac{T_{coh}^{3}}{6}\\ \dfrac{T_{coh}^{4}}{8\cdot 1540}&\dfrac{T_{coh}^{4}}{8}&\dfrac{T_{coh}^{3}}{3}& \dfrac{T_{coh}^{2}}{2}\\ \dfrac{T_{coh}^{3}}{6\cdot 1540}&\dfrac{T_{coh}^{3}}{6}&\dfrac{T_{coh}^{2}}{2}&T_{coh} \end{array} \right]+q_{d}\left[ \begin{array}{cccc}T_{coh}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array} \right]\]  
where \(q_{a}\) is the spectral density of LOS acceleration random walk model, \(q_{d}\) is the code density of LOS acceleration random walk model, \(S_{f}\) and \(S_{g}\) are relevant to the phase and frequency of clock oscillator. Generally, \(S_{f}=2h_{0}\) and \(S_{g}=8\pi^{2}h_{-2}\). The coefficients \(h_{0}\) and \(h_{-2}\) are determined by the type of clock oscillator [16; 17], which can be referred in Table 1. \begin{table}
\begin{tabular}{c c c} \hline
**Oscillator Type** & \(\mathbf{h_{0}\) **(s\({}^{2}\)/Hz)** & \(\mathbf{h_{-2}}\) **(1/Hz)** \\ \hline OCXO & \(1.0\times 10^{-21}\) & \(1.0\times 10^{-20}\) \\ TCXO & \(2.51\times 10^{-26}\) & \(1.0\times 10^{-22}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Coefficients for oven-controlled crystal oscillator (OCXO) and temperature compensated crystal oscillator (TCXO). Additionally, the measurement noise covariance matrix is related to the output of the discriminator, and it is a function of real-time signal power:  
\[\mathbf{R}_{k}=\left[ \begin{array}{cc}\sigma_{d}^{2}&\\ &\sigma_{\Phi}^{2}\end{array} \right]=\left[ \begin{array}{cc}d(2-d)[\frac{1}{2} +\frac{1}{T_{coh}C/N_{0}(2-d)}]^{2}&\\ &\frac{1}{2T_{coh}C/N_{0}}(1+\frac{1}{2T_{coh}C/N_{0}})\mathbf{I}\end{array} \right] \tag{22}\]  
where \(d\) is the code phase, \(\sigma_{d}^{2}\) is noise variance of the DLL discriminator, \(\sigma_{\Phi}^{2}\) is noise variance of the PLL discriminator, and \(\mathbf{I}\) represents the unit diagonal matrix. The iterative process of KF contains the computation of forward prediction covariance matrix \(\mathbf{P}_{k,k-1}\), adaptive gain matrix \(\mathbf{K}_{k}\), new state covariance matrix \(\mathbf{P}_{k}\) and new state vector \(\mathbf{X}_{k}\):  
\[\left\{ \begin{array}{cc}\mathbf{P}_{k,k-1}=\mathbf{\Phi}_{k,k-1}\mathbf{P}_ {k-1}\mathbf{\Phi}_{k,k-1}^{T}+\mathbf{Q}_{k}\\ \mathbf{K}_{k}=\mathbf{P}_{k,k-1}\mathbf{H}_{k}^{T}[\mathbf{R}_{k}+\mathbf{H }_{k}\mathbf{P}_{k,k-1}\mathbf{H}_{k}^{T}]^{-1}\\ \mathbf{P}_{k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k,k-1}\\ \mathbf{\hat{X}}_{k}=\mathbf{\hat{X}}_{k,k-1}+\mathbf{K}_{k}v_{k}\end{array} \right. \tag{23}\]  
Compared to conventional tracking strategy, the advantage of an adaptive KF is that the equivalent LNBW can be easily adjusted by tuning the coefficients in the noise matrix. In-depth analysis was given by [18] about how the adaptive KF varies the LNBW in a time-varying optimal manner as the signal power and dynamics change. The equivalent LNBW in steady state is:  
\[\left(B_{L,\alpha}\right)_{k}\cong\frac{\mathbf{K}_{k}(\mathbf{Q}_{k},\mathbf{ R}_{k})}{c_{3}T_{coh}}=\frac{(\mathbf{\Phi}_{k,k-1}\mathbf{P}_{k-1}\mathbf{ \Phi}_{k,k-1}^{T}+\mathbf{Q}_{k})\mathbf{H}_{k}^{T}[\mathbf{R}_{k}+\mathbf{H }_{k}(\mathbf{\Phi}_{k,k-1}\mathbf{P}_{k-1}\mathbf{\Phi}_{k,k-1}^{T}+ \mathbf{Q}_{k})\mathbf{H}_{k}^{T}]^{-1}}{c_{3}T_{coh}} \tag{24}\]  
where the filter coefficient \(c_{3}\) for the third-order PLL equals 3.048 and the gain matrix \(\mathbf{K}_{k}\) is a function of \(\mathbf{Q}_{k}\) and \(\mathbf{R}_{k}\). It can be proved than \(\mathbf{K}_{k}\) is positively correlated with \(\mathbf{Q}_{k}\) and negatively correlated  
Figure 3: Block diagrams of adaptive four-state KF-based tracking strategy. with \(\mathbf{R}_{k}\), thus \(\mathbf{K}_{k}\propto\frac{\mathbf{Q}_{k}}{\mathbf{R}_{k}}\). This indicates the realization of adaptive bandwidth in weak-signal and high dynamic environment:  
1.