# A Review of High-Sensitivity Tracking Techniques for Satellite Navigation Signals
## 2 Methods
### Scalar Tracking
#### 2.2.3 Optimizing the Loop Filter
The noise vector \(W_{n}=\left[ \begin{array}{cccc}W_{p}&W_{w}&W_{a}&W_{j}\end{array} \right]^{T}\) is modeled as discrete Gaussian white noise with a mean of 0 and covariance matrix \(Q_{k}\)[24]. \[\mathbf{Q}_{\phi}=\left[ \begin{array}{cccc}Q_{p}&0&0&0\\ 0&Q_{w}&0&0\\ 0&0&Q_{a}&0\\ 0&0&0&Q_{j}\end{array} \right] \tag{14}\]  
where \(Q_{p}\) is caused by the receiver clock phase bias, \(Q_{w}\) is due to the receiver clock frequency drift, \(Q_{a}\) is caused by acceleration along the LOS, and \(Q_{j}\) is caused by receiver jitter along the line between the satellite and the receiver [24]. \[\mathbf{Q}_{k+1}=\int_{0}^{\Delta T_{k}}\mathbf{\Phi Q}_{\phi}\mathbf{\Phi}^{ T}\,dt=Q_{j}\left[ \begin{array}{cccc}\frac{\Delta T_{k}^{T}}{25}&\frac{ \Delta T_{k}^{6}}{72}&\frac{\Delta T_{k}^{5}}{30}&\frac{\Delta T_{k}^{4}}{24} \\ \frac{\Delta T_{k}^{4}}{72}&\frac{\Delta T_{k}^{5}}{20}&\frac{\Delta T_{k}^{4} }{8}&\frac{\Delta T_{k}^{4}}{6}\\ \frac{\Delta T_{k}^{4}}{30}&\frac{\Delta T_{k}^{4}}{8}&\frac{\Delta T_{k}^{4} }{3}&\frac{\Delta T_{k}^{4}}{2}\\ \frac{\Delta T_{k}^{4}}{24}&\frac{\Delta T_{k}^{4}}{6}&\frac{\Delta T_{k}^{4} }{2}&\Delta T_{k}\end{array} \right] \tag{15}\]  
The measurement equation is [24]:  
\[\left[ \begin{array}{c}I_{k+1}\\ Q_{k+1}\end{array} \right]=\left[ \begin{array}{c}\cos(\Delta\phi_{k})R( \Delta\tau_{k})\\ \sin(\Delta\phi_{k})R(\Delta\tau_{k})\end{array} \right]+\left[ \begin{array}[] {c}n_{k}^{I}\\ n_{k}^{Q}\end{array} \right] \tag{16}\]  
[PERSON] was the first to propose the use of the Extended Kalman Filter (EKF) to directly process the I and Q correlation values, resulting in a tracking loop with enhanced dynamic stress resistance and improved sensitivity under weak signal conditions [10]. Building on this foundational work, [PERSON] evaluated the performance of EKF-based tracking loops and investigated methods for updating the Numerically Controlled Oscillator (NCO) frequency and phase. His design, which incorporated a four-state control input, significantly improved tracking robustness [91]. [PERSON] further advanced the approach by introducing a multidimensional NCO update scheme, leading to superior tracking performance in complex environments [41]. In addition to these efforts, [PERSON] et al. conducted a theoretical comparison between traditional Phase-Locked Loop (PLL)-based structures and Kalman filter-based tracking algorithms, demonstrating that Kalman filters offer lower dynamic stress error and improved steady-state behavior [24]. To address the challenges of weak signal reception, [PERSON] et al. proposed a novel adaptive EKF framework that dynamically updates measurement noise statistics based on innovation sequences, thereby enhancing loop stability and adaptability [92]. [PERSON] et al. developed an EKF architecture capable of real-time estimation of carrier phase, Doppler shift, Doppler rate, and code phase under high-dynamic conditions, achieving reliable performance at acceleration levels up to 100g [93]. [PERSON] and [PERSON] introduced an EKF-based tracking algorithm specifically designed for weak GPS signals, combining code and carrier tracking in a unified estimation framework. Their implementation was shown to maintain lock with signal strengths as low as 15 dB-Hz, making it suitable for high-altitude and scintillation-prone environments [94; 95]. [PERSON] et al. emphasized the practical aspects of implementing EKF-based tracking in software-defined receivers and highlighted the importance of appropriate NCO update strategies for achieving high-accuracy PVT (position/velocity/time) solutions [91]. Furthermore, [PERSON] et al. proposed an EKF-based Vector Delay Lock Loop (VDLL) that leverages cross-satellite coupling to maintain lock even when individual channels momentarily lose tracking, providing an effective solution for robust multi-satellite navigation [96]. Despite the theoretical advantages of Kalman Filter structures, their practical performance remains highly dependent on the accuracy of system models and the characterization of noise statistics.