# A Review of High-Sensitivity Tracking Techniques for Satellite Navigation Signals
## 2 Methods
### Scalar Tracking
#### 2.2.3 Optimizing the Loop Filter  
Optimizing the loop filter is a crucial research area for enhancing the tracking sensitivity of GNSS receivers. This improvement can be categorized into two main approaches: the Wiener filter and the Kalman Filter (KF). Wiener filters are widely used due to their effectiveness in mitigating thermal and oscillator noise within the loop. As demonstrated in [PERSON]'s work, designs incorporating Wiener filters offer superior tracking performance compared to conventional PLL designs [23]. KF optimization can be further divided into two approaches: the linear KF, which treats discriminator outputs as observations, and the Extended Kalman Filter (EKF), which processes outputs from the I and Q correlators with non-linear observations. The block diagrams illustrating the flow in both approaches are shown in Figure 4.  
The first approach, shown in the left panel of Figure 4, combines the baseband I and Q values to generate signal parameter error values via a nonlinear discriminator function, which are then processed using a KF. The advantage of this approach lies in the simplicity of its design matrix, as the discriminator output is directly used as the measurement residual. However, a major drawback is that the covariance of the measurement noise is no longer white Gaussian, primarily because the nonlinear discriminator function causes a loss of the white Gaussian noise properties [85]. Nonetheless, by assuming a quasi-Gaussian distribution, a robust implementation can still be achieved. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Discriminator Type** & **Advantages** & **Limitations** & **Representative References** \\ \hline \multirow{3}{*}{Maximum Likelihood} & High estimation accuracy using DTFT; effective in low SNR conditions. & Computational complexity; sensitive to bit flips. & [PERSON] (2008) [78] \\ \cline{2-4}  & Uses multiple NCOs; straightforward structure. & Performance degrades in dynamic environments. & [PERSON] (2014) [25], [PERSON] (2018) [27] \\ \hline \multirow{3}{*}{FFT-Based Estimation} & Applies FFT on I/Q coherent integration results in closed-loop tracking; enables real-time processing. & Implementation complexity; less explored in the literature. & [PERSON] (2016) [11], [PERSON] (2017) [83], [PERSON] (2015) [84], [PERSON] (2009) [15], [PERSON] (2008) [78], [PERSON] (2009) [82] \\ \hline \multirow{3}{*}{Multiband/Early Delayed} & Allows higher frequency resolution; applicable to dynamic environments. & Pseudorange accuracy lower than carrier phase; estimator design more complex. & [PERSON] (2003) [80], [PERSON] (2015) [84] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of discriminator optimization techniques in GNSS receivers. The carrier tracking loop based on the KF algorithm is modeled as follows [21]:  
\[\left[ \begin{array}{c}\theta_{e,k+1}\\ f_{d,k+1}\\ f_{a,k+1}\end{array} \right]=\left[ \begin{array}{ccc}1&\Delta t&\frac{\Delta t^{2} }{2}\\ 0&1&\Delta t\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}\theta_{e,k}\\ f_{d,k}\\ f_{a,k}\end{array} \right]-\left[ \begin{array}{c}\Delta t\\ 0\\ 0\end{array} \right]f_{d,k}^{NCO}+W_{n} \tag{9}\]  
where \(\theta_{e,k}\) represents the carrier phase error between the input signal and the NCO output, \(f_{d,k}\) is the Doppler frequency in the input signal, and \(f_{a,k}\) is the rate of change of the Doppler frequency due to acceleration along the Line-of-Sight (LOS) direction between the satellite and the receiver. \(f_{d,k-1}^{NCO}\) is the Doppler frequency replicated by the NCO, and \(\Delta t\) is the updating period (or pre-integration time). \(W_{n}=\left[ \begin{array}{cc}W_{\theta}&W_{d}&W_{a}\end{array} \right]^{T}\) is the noise vector, consisting of a sequence of discrete Gaussian white noise components, with covariance matrix \(Q\). The system matrix and spectral intensity in continuous time are given by [21]:  
\[\Phi_{k+1,k}=\left[ \begin{array}{ccc}1&\Delta t&\frac{\Delta t^{2}}{2}\\ 0&1&\Delta t\\ 0&0&1\end{array}