# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 2 Signal Model  
A GNSS comprises a constellation of satellites with accurately known orbits transmitting predefined messages to their users. The primary purpose of a GNSS is navigation, as users employ receivers to compute correct position, velocity, and time estimates using the known positions of the satellites and the signals received by an antenna. This yields a PVT state representation:  
\[\gamma\triangleq[\mathbf{p},\mathbf{v},\delta t]\in\Gamma,\]  
where \(\mathbf{p}=[x,y,z]\) describes the 3-dimensional position vector of the receiver, which, in this study, is assumed to be a geographic and Cartesian coordinate system, i.e., Earth-centered and Earth-fixed. \(\mathbf{v}=[\dot{x},\dot{y},\dot{z}]\) denotes the velocity vector of the receiver, which is equal to the partial derivative of \(\mathbf{p}\), and \(\delta t\) is the clock bias of the receiver. Without loss of generality, we assume that \(\mathbf{\gamma}\) in the space of possible states \(\Gamma\in\mathbb{R}^{7}\). The received measurements are considered to be a superposition of plane waves with a known signal structure; they are interfered with by noise and are potentially corrupted by interference and multiple paths. Each plane wave corresponds to the line-of-sight signal of a visible satellite. Assuming that there are \(M\) satellites, the received GNSS signal sampled at time \(t_{k}\) can be written as [29]  
\[y(t_{k})=\sum_{i=1}^{M}a_{i}c_{i}(t_{k})g_{i}(t_{k}-\tau_{i})\cos\big{(}2\pi(f_ {c}+f_{d_{i}})t_{k}+\phi_{i}\big{)}+n(t_{k}),\]  
where the subindex \(i\in\{1,\cdots,M\}\) denotes the number of visible satellites. \(a_{i}\), \(c_{i}\), and \(g_{i}\) are the signal amplitude, the navigation bit, and the known PRN codes of the \(i\)-th satellite,respectively. \(f_{c}\) is the carrier frequency. \(\phi_{i}\) is the initial carrier phase shift of the \(i\)-th satellite, which is considered to be known from the carrier-phase loop. \(n(t_{k})\) is zero-mean additive white Gaussian noise (AWGN) with variance \(\sigma^{2}\). \(\tau_{i}\) and \(f_{d_{i}}\) denote the time delay and Doppler shift of the \(i\)-th satellite, respectively, which can be expressed as functions of the PVT state from the pseudorange and the pseudorange rate models [30]:  
\[\tau_{i} =\frac{\|\mathbf{p}_{i}-\mathbf{p}\|}{c}+\delta t-\delta t_{i}+\epsilon_{i}, \tag{1}\] \[f_{d_{i}} =-\frac{f_{c}}{c}(\mathbf{v}_{i}-\mathbf{v})\frac{(\mathbf{p}_{i}-\mathbf{p})^{ \top}}{\|\mathbf{p}_{i}-\mathbf{p}\|}+\epsilon_{f}, \tag{2}\]  
where \(\mathbf{p}_{i}=[x_{i},y_{i},z_{i}]\) and \(\mathbf{v}_{i}=[\dot{x}_{i},\dot{y}_{i},z_{i}]\) are the position and velocity of the \(i\)-th satellite, respectively, which can be computed from the known ephemeris. The operator \(\|\cdot\|\) denotes the Euclidean norm of a vector. \(\|\mathbf{p}_{i}-\mathbf{p}\|\) represents the Euclidean distance between the receiver and the \(i\)-th satellite. \(\delta t_{i}\) is the clock bias of the \(i\)-th satellite, and it is known from the navigation message. \(\epsilon_{i}\) contains errors arising from diverse sources, including atmospheric delays, multipath biases, ephemeris mismodeling, and relativistic effects, among other contributing factors. \(\epsilon_{f}\) refers to noise in the phase rate measurement due to non-modeled terms [31]. \(c\) is the speed of light.