# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 4 Asymptotic Performance Analysis for the 2 SP
Estimating \(\hat{\mathbf{\omega}}\) is equivalent to simultaneously maximizing \(M\) subproblems:  
\[\hat{\mathbf{\omega}}=\arg\max\sum_{i=1}^{M}\frac{1}{K}\left\|\hat{\kappa}_{i} \mathbf{d}^{\top}(\omega_{i})\mathbf{d}(\omega_{i})\hat{\kappa}_{i}^{\top}\right\|. \tag{19}\]  
It is important to note that no assumptions are made about the space in the estimation process of \(\hat{\mathbf{\omega}}\). However, the geometric relationship between the satellites and the receiver imposes constraints on the time delays and Doppler shifts of the received signals. It is crucial to recognize that not every vector comprising time delays and Doppler shifts can be uniquely associated with the PVT state \(\gamma\) through the inverse function of \(h\) alone. In other words, \(h\) operates as an injection function but not necessarily as a bijection function. This implies that the mapping defines a subset denoted as \(\Omega=\{\mathbf{\omega}:\mathbf{\omega}=h(\mathbf{\gamma}),\mathbf{\gamma}\in\Gamma\}\), for which a unique inverse mapping exists  
\[x_{i}(t_{k})=a_{i}c_{i}(t_{k})g_{i}(t_{k}-\tau_{i})\cos\big{(}2\pi(f_{c}+f_{d_ {i}})t_{k}+\phi_{i}\big{)}+n(t_{k}),\]0  
Obviously, since the estimate \(\hat{\omega}_{i}\) is obtained individually without knowing the other \(\{\hat{\omega}_{j}\}_{j\
eq i}\), there is no guarantee that the estimated vector \(\hat{\mathbf{\omega}}\) consisting of \(\hat{\omega}_{i}\) belongs to the subset \(\Omega\). Therefore, the desired parameter \(\mathbf{\gamma}\) cannot be obtained directly from \(\hat{\mathbf{\omega}}\) using the inverse function \(h^{-1}\). The common practice uses the LS method to compute the position and clock bias of the receiver based on the pseudorange model from Equation (1), which provides a nonlinear relation among the position, clock bias, and time delay estimates of each satellite:  
\[x_{i}(t_{k})=a_{i}c_{i}(t_{k})g_{i}(t_{k}-\tau_{i})\cos\big{(}2\pi(f_{c}+f_{d_ {i}})t_{k}+\phi_{i}\big{)}+n(t_{k}),\]1  
where \(r_{i}=\frac{1}{c}\|\mathbf{p}_{i}-\mathbf{p}\|\) denotes the propagation time for the geometric distance between the receiver and the \(i\)-th satellite. This results in a nonlinear and overdetermined system for \(M\geq 4\) that is usually solved with a linearized method and approximated with a Taylor series with respect to an initial position-time guess \([\mathbf{p}^{\rho},\delta t^{\rho}]^{\top}=[x^{\rho},y^{\rho},z^{\rho},\delta t^{ \rho}]^{\top}\). The linearized equation is  
\[x_{i}(t_{k})=a_{i}c_{i}(t_{k})g_{i}(t_{k}-\tau_{i})\cos\big{(}2\pi(f_{c}+f_{d_ {i}})t_{k}+\phi_{i}\big{)}+n(t_{k}),\]2  
where  
\[\delta\mathbf{\tau} =\left[ \begin{array}{cc}\tau_{1}+\delta t_{1}-\epsilon_{1}-\tau_{ 1}^{o}\\ \vdots\\ \tau_{M}+\delta t_{M}-\epsilon_{M}-\tau_{M}^{o}\end{array} \right],\] \[\mathbf{H}_{1} =\left[ \begin{array}{ccc}\dfrac{x_{1}-x^{o}}{r_{1}^{o}c}&\dfrac{ y_{1}-y^{o}}{r_{1}^{o}c}&\dfrac{z_{1}-z^{o}}{r_{1}^{o}c}&1\\ \vdots&\vdots&\vdots&\vdots\\ \dfrac{x_{M}-x^{o}}{r_{M}^{o}c}&\dfrac{y_{M}-y^{o}}{r_{M}^{o}c}&\dfrac{z_{M}- z^{o}}{r_{M}^{o}c}&1\end{array} \right],\] \[\Delta =\left[\delta x,\delta y,\delta z,\delta t_{1}\right]^{\top},\]  
and the solution of the problem in Equation (21) is given by  
\[\hat{\Delta}=\left(\mathbf{H}_{1}{}^{\top}\mathbf{H}_{1}\right)^{-1}\mathbf{H }_{1}{}^{\top}\delta\mathbf{\tau}.\]  
In this study, we consider the PVT state \(\mathbf{\gamma}\); both time delays and Doppler shifts are used in the estimation process. As shown in Equation (2), the Doppler shifts also provide information on the receiver position, i.e., through the pseudorange rate model.