# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 6 Discussion  
Previous studies on the asymptotic performance of the 2 SP can be seen as simplifying the joint time delay and Doppler shift estimation problem in GNSSs through the EXIP, which relaxes the constraints on the structure of the synchronization parameter vector; then, by using a diagonal weighted matrix to determine the confidence of these intermediate estimates, the performance of the approximate initial model is achieved by refining them through WLS minimization. As shown by the curves of the WLS in Figure 1, this allows the approximation of the CRB in certain \(C/N_{0}\) intervals. The improvement can be illustrated from the point of view of information theory; the 2 SP discards all but the most likely code phases and associated signal-to-noise ratios after the acquisition stage and then performs positioning [45]. In contrast, the DPE uses all of the information of the signal for positioning. This allows the DPE to find the best solution only from the space of possible states, so the erroneous peaks in the correlation function will most likely never be aligned, which improves  
Figure 3: RMSE and CRB with respect to the \(C/N_{0}\) values (dB-Hz); the \(C/N_{0}\) values of the satellites are normally distributed with a mean \(C/N_{0}\) variance of 3. Figure 2: RMSE and CRB with respect to the \(C/N_{0}\) values (dB-Hz) of the varying satellite. the noise resistance [46]. Specifically, the 2 SP can achieve the same asymptotic performance as that of the DPE when the weighted matrix is equal to the FIM of the time delay and Doppler shift [22], as shown by the curves of the ILS in Figure 2. Consequently, these studies converge on the same conclusion: the 2 SP can achieve equivalent asymptotic performance to that of the DPE when an appropriately weighted matrix is employed [23; 25]. However, these conclusions are derived under the condition that the estimation of the synchronization parameter in the first step is unbiased and asymptotically effective. In a GNSS, the cross-correlation error is non-negligible and is correlated with the synchronization parameters of other satellites, which leads to a bias in the synchronization parameters obtained from the first step of the individual estimation with the 2 SP. This study focuses on analyzing the asymptotic performance of the 2 SP compared to that of the DPE while taking the effect of cross-correlation errors into account. Specifically, it fills the analysis of the first step for the 2 SP, which has been previously overlooked. Theoretically, the difference in asymptotic performance can be obtained by comparing different optimization problems, as shown in Equations (8) and (27). It is evident that the 2 SP solves the synchronization parameters by maximizing the norms of \(\mathbf{y}\) and \(\mathbf{D}(\boldsymbol{\gamma})\). This represents a special case of the DPE where the multiplication \(\mathbf{D}(\boldsymbol{\gamma})\mathbf{D}^{\top}(\boldsymbol{\gamma})\) is approximated as a diagonal matrix. However, the pseudo-orthogonality of PRN codes causes the off-diagonal elements of the multiplication \(\mathbf{D}(\boldsymbol{\gamma})\mathbf{D}^{\top}(\boldsymbol{\gamma})\) to represent non-zero cross-correlations, making it a non-diagonal matrix. In this case, a simple scaled diagonal weighing matrix does not suffice [47]. In contrast, the DPE performs an ML estimation of the PTV state, which is calculated from all visible satellites' signals, with the aim of maximizing the norm of the projection of signal \(\mathbf{y}\) onto the column space of matrix \(\mathbf{D}(\boldsymbol{\gamma})\). The projection matrix plays the role of correcting for cross-correlation interference; then, the corrected received signals have the maximum effective projection in the signal structure matrix, thus eliminating the cross-correlation errors.