# A Carrier Estimation Method Based on MLE and KF for Weak GNSS Signals
## 5 Simulation Results
### Monte Carlo Simulations for Sensitivity and Accuracy  
To provide intuitive quantification analysis, MC simulations are performed in terms of the sensitivity, accuracy and bit error rate of the proposed loop. A classic two-order FLL-assisted three-order PLL (FPLL) is also simulated to make a comparison [19]. This loop integrates both the FLL and the PLL characteristics. It is designed to satisfy the need for both the dynamic robustness of FLL plus the accuracy performance of the PLL. The discriminator algorithms of two-order FLL and three-order PLL are shown in Table 3. In order to keep consistent with the proposed loops, the integration time of PLL is set to 20 ms. Because the discriminator of FLL uses two integration results in an update, its integration time is set to 10 ms. The bandwidth of FLL and PLL is 4 Hz and 18 Hz, respectively, which is the optimal choice for low and high dynamics [19]. The simulation time is 10 s and 200 MC simulations are made. The other parameters are the same as those in Table 1.  
The tracking performance of the proposed loop and FPLL in pedestrian-level dynamic circumstances is shown in Figure 9. The tracking probability is the ratio of successful tracking to Monte Carlo simulations. A successful tracking means that 1-sigma frequency error is less than 5 Hz and the maximum frequency error is less than 20 Hz during 10 s. The sensitivity is defined as the C/N0 where the tracking probability reaches 50%. It can be seen from Figure 9a that the sensitivity of the ML-KF and ML loop is approximately 19.5 dB-Hz, which is 3 dB-Hz lower than FPLL (22.5 dB-Hz). Therefore, the proposed MLE-based loops have lower \(C/N_{0}\) tracking threshold than the conventional FPLL. The frequency errors are shown in Figure 9b. The ML-KF loop shows the highest accuracy when \(C/N_{0}\) is below 32 dB-Hz. In contrast, the accuracy of the ML loop is much lower due to the lack of filtering. FPLL shows high accuracy when the \(C/N_{0}\) is above 30 dB-Hz. However, its performance decreases rapidly with \(C/N_{0}\) decreases. Figure 9c shows the bit error rate. It can be seen that the ML-KF loop has an approximately 8% error rate reduction compared with conventional FPLL. In summary, the ML-KF loop shows the optimal performance in sensitivity, precision and bit error rate in the weak signal and pedestrian-level dynamic circumstance. \begin{table}
\begin{tabular}{c c} \hline \hline  & \multicolumn{2}{c}{**Discriminator Algorithm**} \\ \hline \multirow{3}{*}{**Two-order FLL**} & \(\frac{ATAN2(cross\_dot)}{2\pi T}\) \\  & where \(ATAN2(*)\) denotes four-quadrant arctangent, \\  & \(\dot{dot}=R(r_{n-1})R(r_{n})+I(r_{n-1})I(r_{n})\), \\  & \(cross=R(r_{n-1})I(r_{n})-R(r_{n})I(r_{n-1})\) \\ \hline \multirow{3}{*}{**Three-order PLL**} & \(ATAN\left(\frac{I(r_{n})}{Q(r_{n})}\right)\) \\  & where \(ATAN(*)\) denotes the two-quadrant arctangent. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Discriminator algorithms of two-order frequency-locked loop (FLL) and three-order phase-locked loop (PLL). Figure 10 is similar to Figure 9 and shows the tracking performance comparison in the weak signal and vehicle-level dynamic circumstance. It can be seen from Figure 10a that the sensitivity of the ML-KF and ML loop reduces by 2.5 dB and 1 dB, respectively. The ML loop reaches to the lowest \(C/N_{0}\) tracking threshold. In contrast, the sensitivity of the conventional FPLL seems almost unchanged because two-order FLL can provide dynamic assistant for PLL. The frequency errors and bit error rate of all the three loops have a growing trend compared with Figure 9. However, the ML-KF loop still can provide the highest tracking accuracy and the lowest bit error rate. Therefore, the ML-KF loop still has better performance than the conventional FPLL in terms of sensitivity, accuracy and bit error rate. In summary, the proposed ML-KF loop can reach the better performance than the conventional FPLL in terms of sensitivity, accuracy and bit error rate in weak signal and low dynamic circumstances.