# A Two-Stage Kalman Filter-Based Carrier Tracking Loop for Weak GNSS Signals
## 5 Performance Evaluation
### Static Weak Signal Environment Simulation
#### 5.1.2 Convergence Speed Test and Analysis  
Figure 6 shows the carrier phase tracking error when C/\(N_{0}\) = 39 dB-Hz and the simulation time is 120 s with the algorithms, 2-stage conv1 and 2-stage KF1, that are described in Table 2. The top panel of the figure is the carrier phase tracking error for the whole tracking stage. The magenta dotted line divides the whole tracking stage into the coarse tracking stage and the fine tracking stage. The bottom panel presents the carrier phase error in detail in the coarse tracking stage. In this simulation, the coarse tracking stage lasts for approximately 2 s, and the remaining time is the fine tracking stage.  
In the top panel of Figure 6, the carrier phase errors of the two algorithms are both close to zero, which proves that the two algorithms can track the carrier phase parameters properly. For the 2-stage KF1 algorithm, the seamless connection of the phase error results between the coarse tracking and the fine tracking demonstrates the compatibility between the two tracking stages in the proposed method. In the bottom panel of Figure 6, the carrier phase error results of the two algorithms are the same. This is because the same algorithm is used in the coarse tracking stage.  
According to Figure 6, in the first-stage tracking, the carrier phase errors of the two methods both have converged into the steady state. Therefore, there is no need for comparison between the proposed method and the conventional 2-stage method in terms of the convergence speed.  
Figure 6: Carrier phase error of the algorithms, 2-stage conv1 and 2-stage KF1, when C/\(N_{0}\) = 39 dB-Hz.  
To evaluate the convergence speed, the proposed method is compared with the standard KF-based PLL tracking approach, which contains only one stage. The C/N0 value of the simulated signal is set to 39 dB-Hz, and the simulation time is 120 s. For comparison, the standard KF-based PLL with a CIT of 4 ms is compared with the algorithm 2-stage KF1, which is described in Table 2. For the two algorithms, the settings of **Q** and **R** are the same according to (13) and (16), where \(q_{a}\)= \(0\left(\text{m}^{2}/\text{s}^{6}\right)/\text{Hz}\) and \(c/n_{0}=10^{\frac{29}{10}}\). The state error covariance, \(\textbf{P}_{0}^{-}\), in the standard KF method is set according to the acquisition results.  
Figure 7 shows the carrier phase errors and Doppler frequency estimations of the two different KF-based PLL tracking methods. The top and bottom panels of the figure show the carrier phase errors and the Doppler frequency estimated by the two KF-based tracking approaches, respectively. The small boxes in the top and bottom panels are magnified views of the first 6 s. Evidently, the proposed method converges faster than the standard KF tracking approach.  
The noise of the estimated Doppler frequency in the bottom panel of Figure 7 is larger than that in the middle panel of Figure 5. This is because the Doppler frequency extracted from the filter output contains more noise than that extracted from the velocity accumulator and this was illustrated in detail in [4].  
The faster convergence of the proposed method is attributable to the coarse tracking stage that refines the acquisition results. The initial state vectors and the state error covariance of the KF-based fine tracking rely on the coarse tracking results. However, the standard KF-based tracking method uses the acquisition results to initialize the state vector, but the acquisition results are not as accurate as the coarse tracking results of the proposed method. Accordingly, the proposed two-stage KF-based tracking method can achieve faster convergence than the standard KF-based tracking loop.  
Figure 7: Carrier phase error and Doppler frequency estimations for PRN 9 satellite signals of two KF-based PLL tracking methods under a static signal of C/N0 = 39 dB-Hz. (KF: Kalman filter).