# A Two-Stage Kalman Filter-Based Carrier Tracking Loop for Weak GNSS Signals
## 5 Performance Evaluation
### Dynamic Weak Signal Environment Simulation  
To test the performance of the proposed algorithm under low dynamic weak signal conditions, a low dynamic situation is designed with attenuation for the signal simulation. Figure 8 shows the simulated weak dynamic signal. Regarding the signal strength, the signal started at 40 dB-Hz and descended by 2 dB every minute. The minimum power reached 24 dB-Hz at 540 s and was maintained at this level for 60 s. Finally, a recovery period with 2 dB per minute was sustained until it was resumed to the original value. For the dynamic scene simulation, the middle and bottom panels of the figure show the acceleration and velocity, respectively, of the receiver. The figure shows that four dynamic periods are simulated with different accelerations of 12 m/s\({}^{2}\), 5 m/s\({}^{2}\), 1 m/s\({}^{2}\), and 10 m/s\({}^{2}\). In each period, the receiver first accelerated with a constant acceleration, after which it moved with constant speed for tens of seconds, and finally, it slowed down with a negative acceleration until it stopped. The IF of the simulated signal is 1.42 MHz, the sampling rate is 10 MHz, and the signal resolution is 4-bit. For comparison, two different algorithms are designed to evaluate the dynamic weak signals. The settings of CIT, \(B_{nPLL}\) and \(B_{nFLL}\), are the same as those of the algorithms, 2-stage conv1 and 2-stage KF1, described in Table 2.  
For two-stage KF1, the initial parameters in the fine tracking are set as follows:  
\[\mathbf{P}_{0}^{-}=\mathrm{diag}[(2\pi\,)^{2}(2\pi\times 500)^{2}10] \tag{20}\]  
The initialization of \(\mathbf{Q}\) and \(\mathbf{R}\) depends on (13) and (16), where \(q_{a}\) is \(0.05\big{(}\mathrm{m}^{2}/\mathrm{s}^{6}\big{)}/\mathrm{Hz}\) and \(c/n_{0}=10^{\frac{40}{10}}\). The top panel of Figure 9 shows the estimated \(\mathrm{C}/N_{0}\) with the two tracking algorithms. The algorithm, 2-stage conv1, can follow the actual signal strength only when the signal power is high. When \(\mathrm{C}/N_{0}\) reaches 26 dB-Hz, the \(\mathrm{C}/N_{0}\) estimation exhibits a large error, and the receiver loses the lock at about \(t=460\) s. In contrast, 2-stage KF1 can still lock the signal even under the harsher environment, such as the dynamic period \(\mathrm{C}\), and the \(\mathrm{C}/N_{0}\) of the signal can be accurately estimated when the signal power is recovered to the high level. Figure 8: Dynamic weak signals used for the simulation. The middle panel of Figure 9 shows the velocity estimated by the receiver. In general, the velocity of the receiver should be computed by more than four satellites. During the dynamic period, A, both algorithms can estimate the velocity correctly when the acceleration is 12 m/s\({}^{2}\) and when the C/\(N_{0}\) is between 38 dB-Hz and 36 dB-Hz. In the dynamic period, B, the acceleration is 5 m/s\({}^{2}\) and the C/\(N_{0}\) is between 30 dB-Hz and 26 dB-Hz. The algorithm 2-stage KF1 can estimate the speed correctly during the whole period, B. In contrast, the algorithm 2-stage conv1 loses the lock at about 424 s when C/\(N_{0}\) is 26 dB-Hz. During the dynamic period, C, the C/\(N_{0}\) is as low as 24 dB-Hz, and the acceleration is 1 m/s\({}^{2}\). The algorithm 2-stage conv1 cannot track the weak and dynamic signal after the period, B, while 2-stage KF1 can reliably maintain the lock with more than four satellites and compute the receiver velocity correctly. This observation indicates that the algorithm, 2-stage KF1, can achieve a better performance under dynamic weak signal conditions due to its variable gain in the fine tracking stage. In the dynamic period, D, C/\(N_{0}\) recovers to 34 dB-Hz, and the acceleration is 10 m/s\({}^{2}\). The receiver employed with 2-stage conv1 is unable to recover to track the signal even though the signal power is recovered to a higher level. With the algorithm 2-stage KF1, the receiver can endure this variable environment and estimate the speed reliably.