# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 5 Simulation Results
The RMSE was calculated by averaging different satellites, treating each satellite as a varying one to eliminate the impact of geometry. Following the configuration of the curves in Figure 1, we considered another estimator, the ideal least square (ILS), an unrealistic estimator in which the true FIM is used as the weighted matrix; it is shown to account for the 2 SP with the cyan curves connected by crosses. It is worth noting that the bound of the ILS, which is denoted as CRB\({}_{\text{d}}^{b}\) in [23], is the same bound as that of CRB\({}_{\text{d}}\) in the DPE. The bounds on the estimators and their corresponding RMSEs are presented in Figure 2. In terms of the RMSE, in the \(C/N_{0}\) region between 25 and 55 dB-Hz, it is observed that the DPE and ILS exhibited similar performance, outperforming the WLS. The WLS achieved comparable performance to that of the other methods only when the \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**PRN** & **2** & **3** & **4** & **6** & **9** & **14** & **16** & **23** & **25** & **26** & **29** & **31** \\ \hline Azimuth & 62.4 & 190.1 & 254.6 & 108.7 & 128.1 & 276.2 & 230.7 & 158.5 & 10.8 & 244.7 & 338.7 & 297.5 \\ Elevation & 29.9 & 17.8 & 46.4 & 20.3 & 15.7 & 39.7 & 11.2 & 12.7 & 32.1 & 38.3 & 41.4 & 53.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The degrees of the azimuth and elevation angles of satellites. varying satellite had a \(C/N_{0}\) value of 35 dB-Hz. For this specific \(C/N_{0}\), all satellites had the same energy; the appropriately weighted matrix was the identity matrix, and in this case, the WLS and ILS were equivalent, and all estimators could reach their respective bounds. In the low \(C/N_{0}\) region, between 15 to 20 dB-Hz, both the WLS and ILS experienced a slight performance degradation. This degradation was attributed to the interference from other satellites affecting the signals of the varying satellite. Similarly, when the \(C/N_{0}\) of the varying satellite exceeded 50 dB-Hz, the varying satellite introduced interference to the rest, leading to significant performance degradation in both the WLS and ILS. The variance of cross-correlation errors, which was related to the energies of the received signals, had a more pronounced impact in the region with a large \(C/N_{0}\) gap. Both of these performance degradations were reflected in \(\text{CRB}_{w}^{a}\), but not in \(\text{CRB}_{w}^{b}\) and \(\text{CRB}_{1}^{b}\). On the contrary, the DPE provides an optimal approach where the effects of cross-correlation errors have already been taken into account. As a result, optimal performance is achieved, approaching the CRB in all ranges of \(C/N_{0}\). Although the signal energies of the satellites are not the same as those in Figure 2, this scenario is still considered ideal, and various approaches can be employed to eliminate individual anomalous observations, ensuring that the estimation accuracy is not significantly degraded. Comparing Figure 2 with the results in [23], the degradation of one satellite does not significantly affect the overall results. To further validate this concept, a more challenging scenario was tested. The received energy was described by a log-normal distribution, which is typically applied to land mobile channels [44]; the \(C/N_{0}\) value of the satellites was configured to follow a normal distribution with a variable mean and a variance of 3. The bounds on the estimators and the corresponding RMSEs are presented in Figure 3. In terms of RMSEs, the WLS exhibited the most significant performance degradation, while the ILS, despite experiencing a smaller degradation, failed to match the \(\text{CRB}_{1}^{b}\), as presented in Figures 1 and 2. The RMSE of the ILS suggests that the estimation error cannot be reduced even with an optimal weighted matrix provided for the 2 SP. On the other hand, the DPE demonstrated the best performance, maintaining a flat trajectory close to the CRB in all \(C/N_{0}\) ranges.