# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 5 Simulation Results  
In this section, we present the numerical and simulation results. The variances of the position estimators obtained with the 2 SP and DPE are compared. The former was computed by first computing estimates of synchronization parameters and transforming them with the WLS procedure, which is the common choice in 2 SP-based GNSS receivers. The diagonal entries in the weighted matrix \(\mathbf{W}\) were set to the carrier-to-noise density ratios (\(C/N_{0}\)) of the corresponding satellites. The latter was obtained by solving the ML estimator of the position. This was performed by using the grid search method, with the search step size set to be less than half of the sampling distance. We focused on a civilian GPS L1 signal, a spread-spectrum signal transmitted at a chip rate of 1.023 MHz on a carrier frequency of \(f_{c}=1575.42\) MHz. The received signals were filtered with a 1 MHz bandwidth filter and sampled at an intermediate frequency of \(f_{s}=8.184\) MHz. The reconstructed scenario from real ephemeris data corresponded to a realistic constellation geometry involving \(M=12\) satellites while considering an elevation mask of \(5^{\circ}\). The corresponding PRN code numbers, azimuth angles, and elevation angles of the satellites are summarized in Table 1.  
To simplify the plotting, we computed the CRB of the position vector as follows:  
\[\varepsilon_{\mathbf{p}}=\sqrt{\mathbb{E}(\|\mathbf{p}-\hat{\mathbf{p}}\|^{2})}=\sqrt{ \varepsilon_{x}^{2}+\varepsilon_{y}^{2}+\varepsilon_{z}^{2}},\]  
where \(\varepsilon_{x}^{2},\varepsilon_{y}^{2},\varepsilon_{z}^{2}\) are the CRBs for each coordinate and were obtained with the corresponding CRB at the true position. With this setup, we compared the RMSEs of both position estimators against their respective theoretical lower bounds provided by the CRBs. Figure 1 shows the curves plotted against the \(C/N_{0}\) of the satellites, and all were assumed to be equal in this simulation. The blue curve, which is connected by stars, represents the RMSE of the DPE, and its corresponding bound is the black curve denoted as CRB\({}_{\text{d}}\). The red curve, which is connected by circles, represents the RMSE of the 2 SP with WLS, and its corresponding bound from Equation (28) is the green curve denoted as CRB\({}_{\text{w}}^{a}\). As a contrast, the yellow curve represents the CRB from [23] and is shown to account for the 2 SP; it is denoted as CRB\({}_{\text{w}}^{b}\). Note that in this scenario, all satellites had the same \(C/N_{0}\), and the optimal weighted matrix was the identity matrix, which implies that CRB\({}_{\text{w}}^{b}\) and CRB\({}_{\text{d}}\) are the same in Figure 1. Regarding the RMSEs, both estimators exhibited similar performance for \(C/N_{0}\) values larger than 20 dB-Hz, and both approximations approached their corresponding CRBs. In the low \(C/N_{0}\) region, between 15 to 20 dB-Hz, WLS yielded a higher RMSE. Notably, the DPE estimator demonstrated greater robustness to high-energy noise, with a break-point that was approximately 6 dB-Hz lower than that of WLS. Concerning the derived CRBs, CRB\({}_{\text{w}}^{a}\) and CRB\({}_{\text{w}}^{b}\) mostly overlapped. In this case, the transmitted noise was the main source of error, and the cross-correlation errors were low when all satellite signals had the same \(C/N_{0}\). Although the difference between their bounds was small, a closer look at the enlarged portion of Figure 1 reveals that CRB\({}_{\text{w}}^{a}\) was slightly higher than CRB\({}_{\text{w}}^{b}\) in the high \(C/N_{0}\) region, between 50 to 55 dB-Hz, where the power of cross-correlation errors could reach a magnitude comparable to that of the noise. In comparison, the CRB\({}_{\text{w}}^{a}\) proposed in this study aligned more consistently with the RMSE of WLS. To assess the performance with varying received signal energies, we considered a scenario where all satellites, except one, had the same \(C/N_{0}\) at 30 dB-Hz, while one varying satellite altered its \(C/N_{0}\) value.