# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 4 Asymptotic Performance Analysis for the 2 SP
Indeed, the approximation trick allows the relaxation of the relationship that exists between the different values of \(\omega_{i}\), enabling the estimation problem to work in a larger space, thus simplifying the procedure [26]. However, this simplification comes at the cost of a reduction in accuracy due to the deviation of \(\mathbf{yd}^{\top}(\omega_{i})\) from \(\mathbf{x}_{i}\mathbf{d}^{\top}(\omega_{i})\), which is denoted as  
\[\xi_{i}=\mathbf{yd}^{\top}(\omega_{i})-\mathbf{x}_{i}\mathbf{d}^{\top}( \omega_{i}),\]  
which represents the cross-correlation error and can be expressed as [37]  
\[\xi_{i}=\sum_{j=1,j\
eq i}^{M}\frac{a_{i}K}{2}c_{i}(t_{k})R_{i,j}(\Delta\tau_{ i,j})\cos\left(2\pi\Delta f_{d_{i,j}}k+\Delta\phi_{i,j}\right),\]  
where \(\Delta\tau_{i,j}=\tilde{\tau}_{i}-\tau_{j}\) denotes the relative delay between \(g_{i}\) and \(g_{j}\). \(\Delta f_{d_{i,j}}=\hat{f}_{d_{i}}-f_{d_{j}}\) denotes the relative Doppler shift between the \(i\)-th satellite and \(j\)-th satellite. \(\Delta\phi_{i,j}=\phi_{i}-\phi_{j}\) denotes the deviation of the carrier phase between the \(i\)-th satellite and the \(j\)-th satellite. \(R_{i,j}(\Delta\tau_{i,j})=\frac{1}{K}\sum_{k=1}^{K}g_{i}(t_{k}-\hat{\tau}_{i} )g_{j}(t_{k}-\tau_{j})\) denotes the cross-correlation function between \(g_{i}\) and \(g_{j}\). To quantify the magnitude of variance of \(\xi_{i}\), we investigate the cross-correlation function \(R_{i,j}\). As we know, GNSSs generally use Gold code sequences; there are three possible peak values for Gold codes of length \(2^{n}-1\) during synchronization [30]:  
\[R_{i,j}\in\Bigg{\{}\frac{2^{\lfloor\frac{n+2}{2}\rfloor}-1}{2^{n}-1},\frac{- 1}{2^{n}-1},\frac{-2^{\lfloor\frac{n+2}{2}\rfloor}-1}{2^{n}-1}\Bigg{\}},\]  
where \(n\) is the number of the shift register stage, and the operator \(\lfloor\cdot\rfloor\) denotes rounding to the nearest integer towards negative infinity. The probabilities corresponding to the three values of \(R_{i,j}\) are \(\{0.25,0.5,0.25\}\) and \(\{0.125,0.75,0.125\}\) for odd and even values of \(n\), respectively. In both cases, the variance of \(R_{i,j}\) has a uniform expression:  
\[\lim_{K\rightarrow+\infty}\mathbf{C}(\hat{\gamma})=\left[\dfrac{\partial^{2} \ell}{\partial\gamma^{2}}\right]^{-1}\dfrac{\partial\mathbf{\omega}}{\partial \gamma^{\top}}\mathbf{WC}(\hat{\mathbf{\omega}})\mathbf{W}\dfrac{\partial\mathbf{ \omega}^{\top}}{\partial\mathbf{\gamma}}\left[\dfrac{\partial^{2}\ell}{\partial \gamma^{2}}\right]^{-1}, \tag{24}\]0  
where \(\mathbb{D}(\cdot)\) denotes the variance. As the 2 SP processes each satellite channel individually, the differences between synchronization parameters from different satellites cannot be exploited. Therefore, a reasonable assumption is that the deviation of the time delay \(\Delta\tau_{i,j}\) and the deviation of the carrier phase \(\Delta\phi_{i,j}\) can be considered as random variables that are uniformly distributed on one period of the PRN code and \([0,2\pi]\), respectively. Neglecting the effect of the navigation bit \(c_{i}\), the variance of \(\xi_{i}\) can be expressed as \[\lim_{K\rightarrow+\infty}\mathbf{C}(\hat{\gamma})=\left[\dfrac{\partial^{2} \ell}{\partial\gamma^{2}}\right]^{-1}\dfrac{\partial\mathbf{\omega}}{\partial \gamma^{\top}}\mathbf{WC}(\hat{\mathbf{\omega}})\mathbf{W}\dfrac{\partial\mathbf{ \omega}^{\top}}{\partial\mathbf{\gamma}}\left[\dfrac{\partial^{2}\ell}{\partial \gamma^{2}}\right]^{-1}, \tag{24}\]1  
where \(P_{0}=2^{n-2}/(2^{n}-1)^{2}\) is a constant with respect to \(n\). The variance of \(\xi_{i}\) is positive with respect to the signal structure, received amplitude, and the processing length of the signal. In each of the independent estimation problems in Equation (18), using \(\mathbf{y}\) can be approximated as a degradation of the observation \(\hat{\mathbf{x}}_{i}\), indicating an increase in AWGN [38].