# A Review of High-Sensitivity Tracking Techniques for Satellite Navigation Signals
## 2 Methods
### Vector Tracking Loops  
The core principle of VT lies in the fact that the pseudo-range and pseudo-range rates observed by the user are influenced not only by the user's position and velocity but also by the geometry of the satellite constellation. The inherent relationships between the multiple tracking channels of the receiver can be leveraged to enhance tracking performance. The key to the VT algorithm is to fully exploit this inter-channel information and share it across the channels. A flowchart illustrating this process is shown in Figure 5.  
Unlike scalar tracking, which utilizes separate loop filters for each channel, VT eliminates these filters and employs a unified navigation filter that links all channels together [30]. The VT algorithm uses the receiver's position, velocity, acceleration, clock bias, clock drift, and other parameters as state variables. To reduce computational complexity, the EKF estimates the errors of these state variables rather than the variables themselves, and adjusts the navigation output based on the estimated state errors. The state transition equation for VT is [97]:  
\[X_{\text{e}}=\left[ \begin{array}{cccccccc}\delta x&\delta y&\delta z&\delta \hat{x}&\delta\hat{y}&\delta\hat{z}&\delta\hat{x}&\delta\hat{y}&\delta\hat{z}&b &b\end{array} \right]^{T} \tag{17}\]  
The first nine terms represent the errors in the receiver's position, velocity, and acceleration in the ECEF coordinate system, while the last two terms represent the receiver's clock bias and clock drift errors. \begin{table}
\begin{tabular}{p{85.4 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Filter Type** & **Advantages** & **Challenges** & **Representative References** \\ \hline Wiener Filter & Effective at suppressing thermal and oscillator noise; low implementation complexity. & Fixed gain; less adaptive to environmental changes. & [PERSON] (2012) [23] \\ \hline Kalman Filter & Provides optimal tracking performance with white Gaussian noise; flexible model design. & Sensitive to model inaccuracies; assumes known noise statistics. & [PERSON] (2009) [33], [PERSON] (2009) [31], [PERSON] (2012) [35], [PERSON] (2025) [86] \\ \hline Extended Kalman Filter & Can handle nonlinear measurement equations; effective in dynamic weak signal scenarios. & Requires Jacobian calculation; highly linear or model. & [PERSON] (2004) [10], [PERSON] (2018) [41], [PERSON] (2013) [24], [PERSON] (2002) [94], [PERSON] (2010) [96] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of loop filter optimization techniques in GNSS tracking. In practical applications, the state vector can be adapted based on the receiver's dynamic requirements. In low-dynamic scenarios, the effects of the receiver's acceleration and higher-order terms can be neglected to simplify the state vector and filter design. Based on [PERSON]'s law of kinematics, the system's state transition equation is:  
\[X_{k+1} =\left[ \begin{array}{cccccc}\partial x_{k+1}&\partial y_{k+1}& \partial z_{k+1}&\partial\hat{x}_{k+1}&\partial\hat{y}_{k+1}&\partial\hat{z}_{ k+1}&\partial\hat{x}_{k+1}&\partial\hat{y}_{k+1}&\partial\hat{z}_{k+1}&b_{k+1}&b_{k+1} \end{array} \right]^{T}\] \[=\left[ \begin{array}{cccccc}I_{3\times 3}&TI_{3\times 3}&\frac{T^{2}}{2}I_{3 \times 3}&0&0\\ 0_{3\times 3}&I_{3\times 3}&TI_{3\times 3}&0&0\\ 0_{3\times 3}&0_{3\times 3}&I_{3\times 3}&0&0\\ 0_{3\times 1}&0_{3\times 1}&0_{3\times 1}&1&T\\ 0_{3\times 1}&0_{3\times 1}&0_{3\times 1}&0&1\end{array} \right]_{1\times 11}\] \[\times\left[ \begin{array}{cccccc}\delta x_{k}\\ \delta y_{k}\\ \delta z_{k}\\ \delta\hat{y}_{k}\\ \delta\hat{x}_{k}\\ \delta\hat{y}_{k}\\ \delta\hat{y}_{k}\\ \delta\hat{z}_{k}\\ \delta\hat{y}_{k}\\ \delta\hat{z}_{k}\\ b_{k}\\ \hat{b}_{k}\end{array} \right]+\mathbf{W}_{k}\] \[\times\left[ \begin{array}{cccccc}\delta x_{k}\\ \delta y_{k}\\ \delta z_{k}\\ \delta y_{k}\\ \delta z_{k}\\ \delta y_{k}\\ \delta z_{k}\\ b_{k}\\ \end{array}