# A Review of High-Sensitivity Tracking Techniques for Satellite Navigation Signals
## 2 Methods
### Scalar Tracking
#### 2.2.3 Optimizing the Loop Filter
\right],\quad Q_{c}=\left[ \begin{array}{ccc}Q_{\theta}&0&0 \\ 0&Q_{d}&0\\ 0&0&Q_{a}\end{array} \right] \tag{10}\]  
where \(Q_{\theta}\) represents the receiver clock phase bias, \(Q_{d}\) corresponds to the receiver clock frequency drift, and \(Q_{a}\) is caused by the acceleration along the LOS direction [21]. \[Q =\int_{0}^{\Delta t}\Phi_{k+1,k}Q_{c}\Phi_{k+1,k}^{T}\,dt \tag{11}\] \[=Q_{\theta}\left[ \begin{array}{ccc}\Delta t&0&0\\ 0&0&0\\ 0&0&0\end{array} \right]+Q_{d}\left[ \begin{array}{ccc}\frac{\Delta t^{3}}{3 }&\frac{\Delta t^{2}}{2}&0\\ \frac{\Delta t^{2}}{2}&\Delta t&0\\ 0&0&0\end{array} \right]+Q_{a}\left[ \begin{array}{ccc}\frac{\Delta t^{5}}{ 20}&\frac{\Delta t^{4}}{3}&\frac{\Delta t^{3}}{6}\\ \frac{\Delta t^{3}}{8}&\frac{\Delta t^{3}}{3}&\frac{\Delta t^{2}}{2}\\ \frac{\Delta t^{6}}{6}&\frac{\Delta t^{2}}{2}&\Delta t\end{array} \right]\]  
Figure 4: Methods of KF and EKF optimization. The measurement equation is [21]:  
\[\theta_{e,k}^{\text{mea}}=\left[ \begin{array}{cc}1&\frac{\Delta t}{2}&\frac{ \Delta t^{2}}{6}\end{array} \right]\left[ \begin{array}{c}\theta_{e,k-1}\\ f_{d,k-1}\\ f_{a,k-1}\end{array} \right]-\frac{\Delta t}{2}f_{d,k-1}^{\text{NCO}}+V_{k} \tag{12}\]  
where \(\theta_{e,k-1}^{\text{mea}}\) is the measured value of the carrier phase error, and \(V_{k-1}\) is a Gaussian white noise sequence. [PERSON] and [PERSON] et al. used the KF algorithm as an alternative to the conventional loop filter and optimized the KF algorithm's parameters using a control method. They mathematically demonstrated the equivalence of the KF-based tracking loop and the conventional third-order PLL [31; 33; 35]. Building on this, [PERSON] compared the KF algorithm with the EKF and showed that both algorithms offer similar performance when only the loop filter is replaced, with the KF being optimal due to its lower complexity [86]. [PERSON] and [PERSON]. further optimized the KF algorithm by proposing the Adaptive Kalman Filter (AKF), which improves tracking performance by exploiting the adaptive adjustment capability of the AKF [87; 88]. Following this, [PERSON] combined the Carrier Noise Ratio (CNR) estimation algorithm with AKF. By using the estimated CNR to adjust the KF algorithm, this method further enhances tracking performance [89]. Finally, wang adapted the AKF algorithm to dynamically changing environmental conditions by incorporating factors such as platform dynamics, vibration-induced oscillator noise, and tropospheric phase scintillation, thus achieving stronger tracking performance [90]. The second approach, shown in the right panel of Figure 4, estimates signal parameter errors by using the baseband I and Q values as measurements for the KF algorithm. This approach benefits from the fact that the noise characteristics of the I and Q values nearly satisfy the White Gaussian assumption, which is a key assumption in the KF algorithm. The state equation for this method is [24]:  
\[\left[ \begin{array}{c}x_{p,k+1}\\ x_{w,k+1}\\ x_{a,k+1}\\ x_{j,k+1}\end{array} \right] =\left[ \begin{array}{ccc}1&\Delta T_{k}&\frac{\Delta T_{k}^{2}} {2}&\frac{\Delta T_{k}^{2}}{6}\\ 0&1&\Delta T_{k}&\frac{\Delta T_{k}^{2}}{2}\\ 0&0&1&\Delta T_{k}\\ 0&0&0&1\end{array} \right]\left[ \begin{array}{c}x_{p,k}\\ x_{w,k}\\ x_{a,k}\\ x_{j,k}\end{array} \right]\] \[\quad\quad\quad\quad\quad-\left[ \begin{array}{c}\Delta T_{k}\\ 0\\ 0\\ 0\end{array} \right]w_{\text{NCO}_{k}}+\mathbf{w}_{\phi k} \tag{13}\] \[=\Phi\mathbf{x}_{k}-\left[ \begin{array}{c}\Delta T_{k}\\ 0\\ 0\\ 0\end{array} \right]w_{\text{NCO}_{k}}+\mathbf{w}_{\phi k}\]  
where \(x_{p}\) is the phase difference between the satellite signal carrier and the local carrier in the GNSS receiver, \(x_{w}\) is the frequency Doppler shift of the satellite signal carrier, \(x_{a}\) is the first-order rate of change of the carrier Doppler shift, and \(x_{j}\) is the second-order rate of change of the carrier Doppler shift. \(w_{\text{NCO}_{k}}\) is the angular frequency Doppler shift of the carrier NCO in the GNSS receiver, and \(\Phi\) is the state transition matrix.