# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 3 Asymptotic Performance Analysis for the DPE
\begin{array}{cc}\mathbf{I}(\mathbf{\mathrm{a}})&\mathbf{I}^{\top}(\mathbf{ \gamma},\mathbf{\mathrm{a}})\\ \mathbf{I}(\mathbf{\gamma},\mathbf{\mathrm{a}})&\mathbf{I}(\mathbf{\gamma})\end{array}  \right], \tag{9}\]  
where \(\mathbf{I}(\mathbf{\mathrm{a}})\) is the FIM of \(\mathbf{\mathrm{a}}\), \(\mathbf{I}(\mathbf{\gamma})\) is the FIM of \(\mathbf{\gamma}\), and \(\mathbf{I}(\mathbf{\gamma},\mathbf{\mathrm{a}})\) is the cross-FIM of \(\mathbf{\gamma}\) and \(\mathbf{\mathrm{a}}\). The elements of each submatrix can be computed with the Slepian-Bang formula [34]. For \(u,v\in\{1,\cdots,M\}\) and \(p,q\in\{1,\cdots,7\}\), we have  
\[\left[\mathbf{I}(\mathbf{a})\right]_{u,v}= \frac{1}{\delta^{2}}\frac{\partial\mathbf{a}}{\partial[\mathbf{a}]_{u}} \mathbf{D}(\mathbf{\gamma})\mathbf{D}^{\top}(\mathbf{\gamma})\frac{\partial\mathbf{a}^{ \top}}{\partial[\mathbf{a}]_{v}},\] \[\left[\mathbf{I}(\mathbf{\gamma},\mathbf{a})\right]_{p,v}= \frac{1}{\delta^{2}}\frac{\partial\mathbf{D}(\mathbf{\gamma})}{ \partial[\mathbf{\gamma}]_{p}}\mathbf{D}^{\top}(\mathbf{\gamma})\frac{\partial\mathbf{a}^ {\top}}{\partial[\mathbf{a}]_{v}},\] \[\left[\mathbf{I}(\mathbf{\gamma})\right]_{p,q}= \frac{1}{\delta^{2}}\frac{a\partial\mathbf{D}(\mathbf{\gamma})}{ \partial[\mathbf{\gamma}]_{p}}\frac{\partial\mathbf{D}^{\top}(\mathbf{\gamma})}{ \partial[\mathbf{\gamma}]_{q}}\mathbf{a}^{\top}, \tag{10}\]  
where the symbol \([\cdot]_{u}\) denotes the \(u\)-th element of the vector, \(\partial\mathbf{a}/\partial[\mathbf{a}]_{u}\) is an all-zero \(1\times M\) vector, except for a 1 in the \(u\) position. Applying basic linear algebra, the derivative \(\partial\mathbf{D}^{\top}(\mathbf{\gamma})/\partial[\mathbf{\gamma}]_{p}\) is found as follows:  
\[\frac{\partial\mathbf{D}^{\top}(\mathbf{\gamma})}{\partial[\mathbf{\gamma}]_{p}}= \left[ \begin{array}{cc}\partial\mathbf{d}^{\top}(\omega_{1})&\ldots&\frac{ \partial\mathbf{d}^{\top}(\omega_{M})}{\partial\omega_{M}}\end{array} \right] \frac{\partial\mathbf{\omega}^{\top}}{\partial[\mathbf{\gamma}]_{p}}, \tag{11}\]  
where \(\partial\mathbf{d}^{\top}(\omega_{i})/\partial\omega_{i}\) stands for the derivative with respect to the element of \(\omega_{i}\) as follows:  
\[\left[ \begin{array}{cc}-c_{i}(t_{1})\dot{g}_{i}(t_{1}-\tau_{i})\cos(2\pi(f_{ c}+f_{d_{i}})t_{1}+\phi_{i})&2\pi t_{1}c_{i}(t_{1})g_{i}(t_{1}-\tau_{i})\sin(2\pi(f_{ c}+f_{d_{i}})+\phi_{i})\\ \vdots&\vdots\\ -c_{i}(t_{k})\dot{g}_{i}(t_{k}-\tau_{i})\cos(2\pi(f_{c}+f_{d_{i}})t_{K}+\phi_{ i})&2\pi t_{K}c_{i}(t_{k})g_{i}(t_{K}-\tau_{i})\sin(2\pi(f_{c}+f_{d_{i}})+\phi_{i}) \end{array} \right]. \tag{12}\]  
In the first column in Equation (12), \(\dot{g}_{i}(t_{k})\) is the derivative of time of the waveform \(g_{i}(t_{k})\). Finally, the derivative \(\partial\mathbf{\omega}/\partial[\tau]_{p}\) is the \(p\)-th row of \(\partial\mathbf{\omega}/\partial\mathbf{\gamma}^{\top}\) and can be calculated using Equation (1):  
\[\frac{\partial\mathbf{\omega}^{\top}}{\partial\mathbf{\gamma}}=\left[ \begin{array}{ cc}\frac{\partial\mathbf{\tau}^{\top}}{\partial\mathbf{p}}&\frac{\partial\mathbf{\tau}^{\top}}{ \partial\mathbf{v}}&\frac{\partial\mathbf{\tau}^{\top}}{\partial\mathbf{\delta}t}\\ \frac{\partial f_{d}^{\top}}{\partial\mathbf{p}}&\frac{\partial f_{d}^{\top}}{ \partial\mathbf{v}}&\frac{\partial f_{d}^{\top}}{\partial\mathbf{\delta}t}\end{array}  \right]. \tag{13}\]  
The FIM is completely defined by using Equations (10)-(13); therefore, the CRB for all of the parameters can be directly computed by inverting Equation (9). The DPE takes advantage of the fact that the signals emitted by the satellites are all received from the same PVT state, and the mutual information is shared across channels. It is recognized for its asymptotic efficiency and unbiasedness. This implies that the covariance matrix of the PVT state \(\mathbf{\gamma}\) approaches the lower error bound as determined by the inverse of the FIM.