# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 4 Asymptotic Performance Analysis for the 2 SP
This transformation leads to a covariance matrix of \(\hat{\gamma}\) that is lower-bounded as follows:  
\[\lim_{K\rightarrow+\infty}\mathbf{C}(\hat{\gamma})=\left[\dfrac{\partial^{2} \ell}{\partial\gamma^{2}}\right]^{-1}\dfrac{\partial\mathbf{\omega}}{\partial \gamma^{\top}}\mathbf{WC}(\hat{\mathbf{\omega}})\mathbf{W}\dfrac{\partial\mathbf{ \omega}^{\top}}{\partial\mathbf{\gamma}}\left[\dfrac{\partial^{2}\ell}{\partial \gamma^{2}}\right]^{-1}, \tag{24}\]  
where \(\mathbf{C}(\hat{\gamma})\) is the covariance matrix of \(\hat{\gamma}\) for the 2 SP with a generic weighted matrix, and \(\mathbf{C}(\hat{\mathbf{\omega}})\) is the covariance matrix of \(\hat{\mathbf{\omega}}\) obtained in Equation (19). Recall that \(\hat{\omega}_{i}\) is estimated under the ML principle with \(\mathbf{x}_{i}\), and it is known that for a sufficiently large sample of data and certain regularity conditions, \(\hat{\omega}_{i}\) is normally distributed with mean \(\omega_{i}\) and the following covariance matrix:  
\[\lim_{K\rightarrow+\infty}\mathbf{C}(\hat{\omega}_{i})=\mathbf{I}^{-1}(\hat{ \omega}_{i})=\left[\dfrac{a_{i}^{2}}{\sigma^{2}}\dfrac{\partial\mathbf{d}( \omega_{i})}{\partial\omega_{i}^{\top}}\dfrac{\partial\mathbf{d}^{\top}( \omega_{i})}{\partial\omega_{i}}\right]^{-1}. \tag{25}\]  
Thus, we find that the covariance matrix of the estimated vector \(\hat{\mathbf{\omega}}\) consisting of \(\hat{\omega}_{i}\) converges to the inverse of \(\mathbf{I}(\mathbf{\omega})\). In many cases of interest (see Appendix A), \(\mathbf{C}(\hat{\gamma})\) has the same performance as that of the DPE, and we have  
\[\lim_{K\rightarrow\infty}\mathbf{C}(\hat{\gamma})=\left[\dfrac{\partial\mathbf{ \omega}}{\partial\gamma^{\top}}\mathbf{I}(\mathbf{\omega})\dfrac{\partial\mathbf{ \omega}^{\top}}{\partial\mathbf{\gamma}}\right]^{-1}. \tag{26}\]  
Equation (26) shows that \(\hat{\gamma}\) is an efficient estimator in the sense that each step of the estimation is ML and is called an unbiased indirect estimation. This result is similar to that of the EXIP [26], which indicates that the 2 SP can not outperform the DPE, but the two methods are approximately equal when choosing an appropriately weighted matrix for the second step of WLS in the 2 SP. However, the estimation process above does not precisely align with the actual procedure followed by the 2 SP. The problem of estimating the parameter \(\omega_{i}\) in Equation (17) involves the signal component \(\mathbf{x}_{i}\). However, the determinism of the pdf \(p\left(\mathbf{y}|\mathbf{x}_{i}\right)\) contradicts the randomness of the PRN codes, thereby preventing the correctness of processing each satellite individually. To address this problem, the 2 SP-based receivers exploit the near orthogonality in \(\mathbf{x}_{i}\), using \(\mathbf{y}\) instead of \(\mathbf{x}_{i}\) in the correlation step in Equation (18). This leads to the following optimization:\[\tilde{\omega}_{i}=\arg\max\frac{1}{K}\left\|\mathbf{yd}^{\top}(\omega_{i})\mathbf{ d}(\omega_{i})\mathbf{y}^{\top}\right\|,\]  
which is the actual correlation step for 2 SP-based GNSS receiver processing. It is straightforward to obtain  
\[\tilde{\omega}=\arg\max\left\|\mathbf{yD}^{\top}(\boldsymbol{\omega})\mathbf{ D}(\boldsymbol{\omega})\mathbf{y}^{\top}\right\|. \tag{27}\]  
where \(\tilde{\omega}\) denotes the synchronization parameter of the \(i\)-th satellite obtained through the 2 SP. Following the same derivation as that in Equation (17) to Equation (26), we have that the variance of \(\tilde{\gamma}\) is  
\[\mathbf{C}_{2}=\Bigg{[}\frac{\partial\boldsymbol{\omega}}{\partial\gamma^{ \top}}\mathbf{I}(\tilde{\boldsymbol{\omega}})\frac{\partial\boldsymbol{\omega }^{\top}}{\partial\gamma}\Bigg{]}^{-1}. \tag{28}\]  
Comparing Equation (28) with Equation (26), the distinction between the CRBs of unbiased indirect estimation and those of the 2 SP lies in the two distinct optimization problems corresponding to different FIMs, as represented in Equation (19) and Equation (27).