# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 4 Asymptotic Performance Analysis for the 2 SP
This nonlinear equation can be linearized with respect to the initial PVT state guess \(\mathbf{\gamma}^{o}=[\mathbf{p}^{o},\delta t^{o},\mathbf{v}^{o}]^{\top}\) as follows:  
\[f_{d_{i}}\approx\dfrac{f_{c}}{c}\biggl{[}\left(\dfrac{\dot{x}_{i }-\dot{x}^{o}}{r_{i}^{o}}+\dfrac{x_{i}-x^{o}}{(r_{i}^{o})^{3}}\right)\delta x +\left(\dfrac{\dot{y}_{i}-\dot{y}^{o}}{r_{i}^{o}}+\dfrac{y_{i}-y^{o}}{(r_{i}^{ o})^{3}}\right)\delta y+\left(\dfrac{\dot{z}_{i}-\dot{z}^{o}}{r_{i}^{o}}+\dfrac{z_{i}-z^{o}}{(r _{i}^{o})^{3}}\right)\delta z\biggr{]}\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\dfrac{f _{c}}{c}\biggl{[}\dfrac{x_{i}-x^{o}}{r_{i}^{o}}\delta\dot{x}+\dfrac{y_{i}-y^{o} }{r_{i}^{o}}\delta\dot{y}+\dfrac{z_{i}-z^{o}}{r_{i}^{o}}\delta\dot{z}\biggr{]} +f_{d_{i}}^{o},\]  
where \(\mathbf{v}^{o}=[\dot{x}^{o},\dot{y}^{o},\dot{z}^{o}]\), \(\delta\dot{x}=\dot{x}^{o}-\dot{x}\), \(\delta\dot{y}=\dot{y}^{o}-\dot{y}\), and \(\delta\dot{z}=\dot{z}^{o}-\dot{z}\). \[f_{d_{i}}^{o}=-\dfrac{f_{c}}{c}(\mathbf{v}_{i}-\mathbf{v}^{o})^{\top}\dfrac{\mathbf{p}_{i}- \mathbf{p}^{o}}{\|\mathbf{p}_{i}-\mathbf{p}^{o}\|}\]  
represents the Doppler shift of the \(i\)-th satellite calculated from the initial state guess. As an improvement, each observation can be weighted according to the received signal quality using a weighted matrix \(\mathbf{W}\), which is real, positive definite, and symmetric. Therefore, the measurements and the desired parameter \(\mathbf{\gamma}\) can be linearized and formulated as a WLS problem:  
\[\Delta\hat{\mathbf{\gamma}}=\arg\min\Bigl{\{}\left(\delta\mathbf{\omega}-\mathbf{H} \Delta\mathbf{\gamma}\right)\mathbf{W}(\delta\mathbf{\omega}-\mathbf{H}\Delta\mathbf{ \gamma})^{\top}\Bigr{\}}, \tag{22}\]  
where  
\[\delta\mathbf{\omega} =[\delta\mathbf{\rho}^{\top},\delta f_{d}{}^{\top}]^{\top},\] \[\delta\mathbf{f}_{d} =\left[ \begin{array}{c}f_{d_{1}}-\epsilon_{f}-f_{d_{1}}^{o}\\ \vdots\\ f_{d_{M}}-\epsilon_{f}-f_{d_{M}}^{o}\end{array} \right],\] \[\mathbf{H} =\left[ \begin{array}{cc}\mathbf{H}_{1}&\mathbf{0}\\ \mathbf{H}_{2}&\mathbf{H}_{3}\end{array} \right],\] \[\Delta\mathbf{\gamma} =[\delta x,\delta y,\delta z,\delta z,\delta t_{1}\delta\dot{x}, \delta\dot{y},\delta\dot{z}]^{\top}.\]The definitions of matrices \(\mathbf{H}_{2}\) and \(\mathbf{H}_{3}\) are given by  
\[\mathbf{H}_{2}=\frac{f_{c}}{c}\left[ \begin{array}{ccc}\dfrac{ \dot{x}_{i}-\dot{x}^{o}}{r_{i}^{o}}+\dfrac{x_{i}-x^{o}}{(r_{i}^{o})^{3}}&\dfrac{ \dot{x}_{i}-\dot{x}^{o}}{r_{i}^{o}}+\dfrac{x_{i}-x^{o}}{(r_{i}^{o})^{3}}&\dfrac{ \dot{x}_{i}-\dot{x}^{o}}{r_{i}^{o}}+\dfrac{x_{i}-x^{o}}{(r_{i}^{o})^{3}}&0\\ \vdots&\vdots&\vdots&\vdots\\ \dfrac{\dot{x}_{i}-\dot{x}^{o}}{r_{i}^{o}}+\dfrac{x_{i}-x^{o}}{(r_{i}^{o})^{3}} &\dfrac{\dot{x}_{i}-\dot{x}^{o}}{r_{i}^{o}}+\dfrac{x_{i}-x^{o}}{(r_{i}^{o})^{3} }&\dfrac{\dot{x}_{i}-\dot{x}^{o}}{r_{i}^{o}}+\dfrac{x_{i}-x^{o}}{(r_{i}^{o})^{3 }}&0\end{array} \right],\] \[\mathbf{H}_{3}=\frac{f_{c}}{c}\left[ \begin{array}{ccc}\dfrac{x _{1}-x^{o}}{r_{1}^{o}}&\dfrac{y_{1}-y^{o}}{r_{1}^{o}}&\dfrac{z_{1}-z^{o}}{r_{ 1}^{o}}\\ \vdots&\vdots&\vdots\\ \dfrac{x_{M}-x^{o}}{r_{M}^{o}}&\dfrac{y_{M}-y^{o}}{r_{M}^{o}}&\dfrac{z_{M}-z^ {o}}{r_{M}^{o}}\end{array} \right],\]  
and the solution to the WLS problem is  
\[\Delta\hat{\gamma}=\left(\mathbf{H}^{\top}\mathbf{W}\mathbf{H}\right)^{-1} \mathbf{H}^{\top}\mathbf{W}\delta\mathbf{\omega}. \tag{23}\]  
Therefore, we have that \(\hat{\gamma}=\gamma^{o}+\Delta\hat{\gamma}\) is the estimation provided by the 2 SP. Following the derivation presented in [36], when \(\hat{\gamma}\) is located at a reasonable proximity to the ideal solution of \(\gamma\), we have \(\partial\mathbf{\omega}^{\top}/\partial\mathbf{\gamma}=\mathbf{H}|_{\gamma^{o}=\gamma}\).