# A Two-Stage Kalman Filter-Based Carrier Tracking Loop for Weak GNSS Signals
## 5 Performance Evaluation
### Static Weak Signal Environment Simulation
#### 5.1.1 Sensitivity Test and Analysis  
For the nine channels, the C/\(N_{0}\) values of the input signals are all set to 45 dB-Hz for the first minute. Then, the signal power is decreased by 2 dB every minute until the C/\(N_{0}\) reaches 15 dB-Hz. Thus, the total simulation time of this signal is 960 s. Moreover, the frequency of the simulated IF signal is 1.42 MHz, the sampling rate is 10 MHz, and the signal resolution is 4-bit. In the simulation, C/\(N_{0}\) is estimated via the power ratio method, as presented in [19], using wideband and narrowband powers from the sums of the I and Q samples. Hence, bit synchronization is indispensable for C/\(N_{0}\) estimation. If a navigation bit transition appears during the computation of the sums, incorrect power estimation could occur. For the two tracking methods discussed above, bit synchronization was achieved before the fine tracking. Therefore, the power ratio method can be employed in the fine tracking stage. In this experiment, C/\(N_{0}\) is estimated every 5 s.  
For comparison, four algorithms are evaluated for static weak signals. The parameters of the four tracking algorithms are listed in Table 2. The algorithms, 2-stage conv1 and 2-stage conv2, have conventional two-stage carrier tracking architectures with different fixed bandwidths and CITs in both coarse tracking and fine tracking. The algorithms, 2-stage KF1 and 2-stage KF2, are the proposed KF-based two-stage carrier tracking architectures with different PLL bandwidths and CITs. For 2-stage KF1 and 2-stage KF2, the initial parameters in the fine tracking should be set first. The initialization of \(\mathbf{Q}\) and \(\mathbf{R}\) depends on (13) and (16), where \(q_{a}\)= \(0\)(m\({}^{2}\)/s\({}^{6}\))/Hz and \(c/n_{0}=10^{\frac{6}{10}}\). The state error covariance, \(\mathbf{P}_{0}^{-}\), is initialized by the coarse tracking results. \(\sqrt{\mathbf{P}_{0}^{-}(2,2)}\) denotes the initial Doppler frequency uncertainty, which was illustrated in [13]. Because the uncertainty of the coarse tracking Doppler results is approximately 100 Hz, \(\sqrt{\mathbf{P}_{0}^{-}(2,2)}\) is set to \(2\pi\):500 with a sufficient margin. Because the phase error result of the coarse tracking is less than 1 cycle, \(\sqrt{\mathbf{P}_{0}^{-}(1,1)}\) is set to 2\(\pi\):1. Due to the static signal, \(\sqrt{\mathbf{P}_{0}^{-}(3,3)}\) is set to 0. Accordingly:  
\[\mathbf{P}_{0}^{-}=\text{diag}[(2\pi\ )^{2}(2\pi\cdot 500)^{2}0] \tag{19}\]  
The initial state vector, \(\hat{\mathbf{x}}_{0}=\left[ \begin{array}{cc}\Delta\phi_{0}&\hat{\omega}_{0 }&\hat{\omega}_{0}\end{array} \right]^{T}\), represents the estimated values from the coarse tracking results. Its initialization depends on Equation (18). For simplicity, only the performance of PRN 14 is illustrated here. Figure 5 shows the sensitivity of the four tracking algorithms. According to the top panel, the C/\(N_{0}\) estimates of the four tracking algorithms follow the actual values when C/\(N_{0}\) is high. As C/\(N_{0}\) decreases, the mismatch between the estimated and actual C/\(N_{0}\) becomes increasingly large. The middle panel shows the Doppler frequency that is obtained via the four tracking algorithms. For the static receiver, the Doppler rate is \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Algorithms**} & \multicolumn{4}{c}{**Tracking Stage**} \\ \cline{2-6}  & \multicolumn{4}{c}{**Coarse Tracking**} & \multicolumn{2}{c}{**Fine Tracking**} \\ \cline{2-6}  & \(B_{nPLL}\)**(Hz)** & \(B_{nPLL}\)**(Hz)** & **CIT (ms)** & \(B_{nPLL}\)**(Hz)** & **CIT (ms)** \\ \hline
2-stage conv1 & 15 & 10 & 4 & 15 & 4 \\
2-stage conv2 & 5 & 10 & 10 & 5 & 20 \\
2-stage KF1 & 15 & 10 & 4 & Not applicable & 4 \\
2-stage KF2 & 5 & 10 & 10 & Not applicable & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tracking algorithms that are considered in the assessment. relatively low, which is due to the clock drift of the receiver and the satellite motion. In the bottom panel, the tracking thresholds of the four tracking algorithms are presented.