# An Enhanced Non-Coherent Pre-Filter Design for Tracking Error Estimation in GNSS Receivers
## 4 Performance Evaluation
### Varying Carrier Frequency Error Scene  
In test scene C, the carrier frequency error was varying as Equation (33), where \(K=20\) Hz; \(\alpha=-1\); \(\omega=-2\pi\) rad/s. \(T_{\text{coh}}\) was set to 1 ms while \(C/N_{0}\) was set to 45 dB-Hz, and \(d\) was set to 1 chip. This configuration is closer to what really happened in the signal tracking loop when the GNSS receiver was faced with high dynamic motion. The convergence with oscillation of \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tracking Error** & \multicolumn{3}{c}{**Root Mean Square of Estimation Error for Tracking Error**} \\ \cline{2-4}
**Estimation Method** & **Carrier Phase Error (cycle)** & **Carrier Frequency Error (Hz)** & **Code Phase Error (chip)** \\ \hline Discriminator & 0.023002 & 32.4045 & 0.1715 \\ Coherent pre-filter & 0.007829 & 0.7464 & 0.0237 \\ Non-coherent pre-filter & 0.009757 & 0.8930 & 0.0322 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance comparison of tracking error estimation methods in test scene A.  
Figure 12: Carrier phase error estimation in test scene B. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Tracking Error** & \multicolumn{3}{c}{**Root Mean Square of Estimation Error for Tracking Error**} \\ \cline{2-4}
**Estimation Method** & **Carrier Phase Error (cycle)** & **Carrier Frequency Error (Hz)** & **Code Phase Error (chip)** \\ \hline Discriminator & 0.35206 & 28.2365 & 0.08783 \\ Coherent pre-filter & 0.05035 & 0.40724 & 0.00723 \\ Non-coherent pre-filter & 0.05475 & 0.43523 & 0.00813 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison of tracking error estimation methods in test scene B.  
tracking error often occurs in a highly dynamic scene. This test is designed to evaluate the estimation accuracies of discriminator and pre-filter algorithms in a highly dynamic scenario. \[\delta f(t)=K\cdot e^{\alpha t}\cdot\sin(\omega t) \tag{33}\]  
Figure 13 shows the carrier phase/frequency tracking error estimation. Similarly, there are still estimation outliers for the ATAN discriminator when the real carrier phase error exceeds the ATAN discriminator's effective working region. There are also some estimation outliers for the pre-filter. After 2250 ms, the coherent pre-filter cannot estimate the carrier phase error correctly. The estimation outliers of the coherent pre-filter appear because the navigation bits cannot be estimated correctly according to the sign of \(I_{\text{E}}\) in test scene C. Thus, the state variables of the coherent pre-filter converge to the wrong value. Even though are there estimation outliers in the non-coherent pre-filter for carrier phase tracking error estimation, it can estimate the tracking error correctly and accurately most of the time. The outliers in the non-coherent pre-filter appear because of the constant use of \(R_{k2}\), as discussed in Section 3.2. By the time real carrier phase error approaches \(-0.5\) cycle or \(0.5\) cycle, the non-coherent pre-filter has worked with \(R_{k2}\) for a very long time. Thus, the estimation error is relatively large. The FDE structure cannot switch the \(R_{k}\) from \(R_{k2}\) to \(R_{k1}\) quickly in this condition. The number of estimation outliers of the non-coherent pre-filter is obviously less than that of the coherent pre-filter. Thus, the robustness of the non-coherent pre-filter is stronger than that of the coherent pre-filter in a high tracking error dynamic scene. The RMS values of the estimation error of the three algorithms in scene C are shown in Table 4. It can be concluded that the estimation accuracy of the enhanced non-coherent pre-filter outperforms the ATAN/ATAN2/NC-EMLE discriminator and coherent pre-filter in high tracking error dynamic change condition, which often occurs when the tracking loop suffers from high dynamic movement of GNSS receivers.