# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 4 Asymptotic Performance Analysis for the 2 SP
Removing the effects of measurement length, the variance of the equivalent AWGN, including the cross-correlation error, for a single snapshot signal of the \(i\)-th satellite is  
\[\lim_{K\rightarrow+\infty}\mathbf{C}(\hat{\gamma})=\left[\dfrac{\partial^{2} \ell}{\partial\gamma^{2}}\right]^{-1}\dfrac{\partial\mathbf{\omega}}{\partial \gamma^{\top}}\mathbf{WC}(\hat{\mathbf{\omega}})\mathbf{W}\dfrac{\partial\mathbf{ \omega}^{\top}}{\partial\mathbf{\gamma}}\left[\dfrac{\partial^{2}\ell}{\partial \gamma^{2}}\right]^{-1}, \tag{24}\]2  
**Remark 1**. : _Notice that although the PRN codes carry certain properties of a random sequence, they are entirely deterministic, leading to predictable cross-correlation values. Consequently, for each \(R_{i,j}\), the occurrence probability does not align with the estimation based on the cross-correlation of random sequences [39]. For given \(i\) and \(j\), their cross-correlation function \(R_{i,j}\), relative time delay \(\Delta\tau_{i,j}\), and relative Doppler shift \(\Delta f_{i,j}^{d}\) are computable, meaning that the cross-correlation error can be calculated and eliminated through the cross-correlation function between satellites. This offset renders the 2 SP estimate suboptimal, and \(\tilde{\omega}\) is strictly biased. Furthermore, these errors do not follow a standard Gaussian distribution [40; 41]. For the purpose of comparing the estimation performance between the 2 SP and the DPE, only the magnitude of the variance is considered. Therefore, the cross-correlation errors are considered as AWGN and represent a reduction in the quality of the desired satellite signal [42]._  
Using \(\mathbf{y}\) in the correlation step of the \(i\)-th channel is equivalent to considering \(\hat{\mathbf{x}}_{i}+\tilde{\zeta}_{i}\) as the observation. This equivalence results in an unavoidable performance degradation for the 2 SP. This results in a decrease in \(\mathbf{I}(\tilde{\omega})\):  
\[\lim_{K\rightarrow+\infty}\mathbf{C}(\hat{\gamma})=\left[\dfrac{\partial^{2} \ell}{\partial\gamma^{2}}\right]^{-1}\dfrac{\partial\mathbf{\omega}}{\partial \gamma^{\top}}\mathbf{WC}(\hat{\mathbf{\omega}})\mathbf{W}\dfrac{\partial\mathbf{ \omega}^{\top}}{\partial\mathbf{\gamma}}\left[\dfrac{\partial^{2}\ell}{\partial \gamma^{2}}\right]^{-1}, \tag{24}\]3  
where \(\mathbf{Q}=\text{diag}(\tilde{\delta})\) is a \(2M\times 2M\) diagonal matrix whose diagonal elements are given by the \(1\times 2M\) vector  
\[\lim_{K\rightarrow+\infty}\mathbf{C}(\hat{\gamma})=\left[\dfrac{\partial^{2} \ell}{\partial\gamma^{2}}\right]^{-1}\dfrac{\partial\mathbf{\omega}}{\partial \gamma^{\top}}\mathbf{WC}(\hat{\mathbf{\omega}})\mathbf{W}\dfrac{\partial\mathbf{ \omega}^{\top}}{\partial\mathbf{\gamma}}\left[\dfrac{\partial^{2}\ell}{\partial \gamma^{2}}\right]^{-1}, \tag{24}\]4  
By substituting Equation (29) into Equation (28) and comparing it with Equation (16), it is obvious that \(\sigma_{i}^{2}>\sigma^{2}\), thus proving that \(\mathbf{C}_{2}>\mathbf{C}_{1}\). The conclusion holds even if \(\partial\gamma/\partial\omega^{\top}\) is invertible, if the weighted matrix \(\mathbf{W}\) is an identity matrix, and in any other cases mentioned in Appendix A. **Remark 2**. : _In the above results, these increases in variance cannot be theoretically eliminated by a 2 SP-based receiver, although the loops similar to the delay lock loop can mitigate them below some thresholds. This study focuses on discussing the magnitude of the errors. More extended research on the bounds of the code-tracking error is described in [43]._