# A Review of High-Sensitivity Tracking Techniques for Satellite Navigation Signals
## 2 Methods
### Conventional Approach  
Traditional open-loop tracking does not distinguish between the acquisition and tracking phases, treating them as a single process. This method is akin to batch processing, where a complete view of the signal is constructed by analyzing each data batch. During this process, the input signal is correlated with a replica signal generated based on the code and carrier search space. This correlation operation is batch-based, and parallel computation is  
\begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt} p{142.3 pt} p{142.3 pt}} \hline \hline
**Technique** & **Advantages** & **Challenges** & **Representative References** \\ \hline Centralized Ultra-Tight & Single filter processes multi-channel tracking data; robust against multipath and signal interruptions. & High computational complexity; sensitive to error propagation from model mismatches. & [PERSON] (1996) [13], [PERSON] (2009) [120], [PERSON] (2020) [121], [PERSON] (2021) [122], [PERSON] (2022) [123], [PERSON] (2023) [124], [PERSON] (2024) [125] \\ \hline Cascaded Ultra-Tight & Two-stage structure reduces processing burden and allows adaptive updates; better dynamic adaptability. & Design of pre-filters is critical; potential risk of accumulated estimation errors. & [PERSON] (2003) [126], [PERSON] (2010) [32], [PERSON] (2006) [128], [PERSON] (2010) [129], [PERSON] (2013) [130], [PERSON] (2012) [131], [PERSON] (2016) [132] \\ \hline \hline \end{tabular}
\end{table}
Table 7: Summary of ultra-tight combining techniques.  
achieved through joint time-frequency domain techniques, such as parallel processing of search space transfer correlation coefficients via Fast Fourier Transform (FFT) [137].  
The output of the open-loop tracking framework, as shown in Figure 9, is a three-dimensional signal image of the current data batch, with dimensions including code frequency shift, Doppler shift, and signal energy. Utilizing this principle, the CA-code autocorrelation function is computed in parallel for all possible values of the code shift [15]  
\[\left[R(\tau_{n})\right]_{n=1}^{N_{\text{code}}}=\text{Re}\Big{\{}\text{IFFT} \Big{(}\text{FFT}\Big{(}\big{[}C(A(t_{n}))\big{]}_{n=1}^{N_{\text{code}}} \Big{)}\Big{)}\cdot\text{FFT}\Big{(}\big{[}C(A(t_{n}))\big{]}_{n=1}^{N_{\text {code}}}\Big{)}\Big{\}} \tag{21}\]  
where Re is the real part, \(\tau_{n}=\frac{T_{\text{code}}}{N_{\text{code}}}\) is the CA code period (1 ms), and \(N_{\text{code}}\) is the number of code samples within the period boundaries. Equation (21) is used to compute in-phase and quadrature correlation values (i and q, respectively) over a 1 ms interval for all possible code shifts and a specific Doppler frequency within the frequency search space.  
\[\left[I_{n,f_{h}}\right]_{n=1}^{N_{\text{code}}}=\text{Re}\Big{\{}\text{IDFT} \Big{(}\text{DFT}\Big{(}\big{[}S(t_{n})\cdot C(t_{n})\big{]}_{n=1}^{N_{\text {code}}}\Big{)}\Big{)}\cdot\text{DFT}\Big{(}\big{[}C(A(t_{n}))\big{]}_{n=1}^{N_ {\text{code}}}\Big{)}\Big{\}} \tag{22}\]  
\[\left[Q_{n,f_{h}}\right]_{n=1}^{N_{\text{code}}}=\text{Im}\Big{\{}\text{IDFT} \Big{(}\text{DFT}\Big{(}\big{[}S(t_{n})\cdot C(t_{n})\big{]}_{n=1}^{N_{\text {code}}}\Big{)}\Big{)}\cdot\text{DFT}\Big{(}\big{[}C(A(t_{n}))\big{]}_{n=1}^{N _{\text{code}}}\Big{)}\Big{\}} \tag{23}\]  
where Re and Im represent the real and imaginary parts, and \(C(t_{n})=\sin(2\pi(f_{IF}+f_{e})t_{n})+j\cos(2\pi(f_{IF}+f_{e})t_{n})\) is the complex carrier. The i and q values are computed for all discrete frequencies of the search space, yielding \(i_{n}\) and \(q_{n}\), where \(n=1,\ldots,N_{\text{code}}\) and \(k=-K,\ldots,K\), thus utilizing the above expression to construct a three-dimensional view, providing the Doppler frequency and pseudo-code phase.  
Although traditional open-loop tracking has received less research attention, current innovation trends focus on improvements in discriminators and bridging the gap between open-loop and closed-loop tracking.