# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 4 Asymptotic Performance Analysis for the 2 SP  
The primary objective of the 2 SP is to establish a mapping that simplifies the process compared to that of the DPE. A natural reparameterization that comes to mind in GNSSs is to use the time delay and Doppler shift. By relaxing the constraints in Equation (6), this approach enables the decomposition of a single multivariate non-convex optimization problem into lower-dimensional counterparts by exploiting the near orthogonality of PRN codes. Receivers based on the 2 SP typically initiate the estimation of synchronization parameters through scalar tracking, followed by computing the PVT estimate using the principle of multilateration. Thanks to the independence provided by the reparameterization, the pdf \(p\big{(}\mathbf{y}|\mathbf{a},\mathbf{\omega}\big{)}\) can be decomposed into  
\[p\big{(}\mathbf{y}|\mathbf{a},\mathbf{\omega}\big{)}=\prod_{i=1}^{M}p\big{(} \mathbf{y}|a_{i},\omega_{i}\big{)}.\]  
In fact, \(a_{i}\) and \(\omega_{i}\) are only related to the signals from the \(i\)-th satellite. Let us define \(x_{i}(t_{k})\) as the observation at sampling time \(t_{k}\) while excluding signals from other satellites:  
\[x_{i}(t_{k})=a_{i}c_{i}(t_{k})g_{i}(t_{k}-\tau_{i})\cos\big{(}2\pi(f_{c}+f_{d_ {i}})t_{k}+\phi_{i}\big{)}+n(t_{k}),\]  
then, the pdf \(p\big{(}\mathbf{y}|a_{i},\omega_{i}\big{)}\) can be written in the form of a marginal distribution:  
\[p\big{(}\mathbf{y}|a_{i},\omega_{i}\big{)}=\int p\big{(}\mathbf{y}|\mathbf{x} _{i}\big{)}p\big{(}\mathbf{x}_{i}|a_{i},\omega_{i}\big{)}\mathrm{d}\mathbf{x }_{i},\]  
where \(\mathbf{x}_{i}=[x_{i}(t_{1}),\cdots,x_{i}(t_{K})]\). Due to the invariance principle of the ML estimate with the injective function in Equation (6) [35], the pdf \(p\big{(}\mathbf{y}|a_{i},\omega_{i}\big{)}\) can be decomposed into two estimation problems by first estimating \(\mathbf{x}_{i}\) using  
\[\hat{\mathbf{x}}_{i}=\arg\max\ln p\big{(}\mathbf{y}|\mathbf{x}_{i}\big{)}.\]  
Then, the pdf of \(\hat{\mathbf{x}}_{i}\) conditioned on the unknown parameters \(a_{i}\) and \(\omega_{i}\) is given by  
\[p(\hat{\mathbf{x}}_{i}|a_{i},\omega_{i})=\frac{1}{(2\pi\sigma^{2})^{\frac{K}{ 2}}}\exp\bigg{\{}-\frac{1}{2\sigma^{2}}\big{(}\hat{\mathbf{x}}_{i}-a_{i} \mathbf{d}(\omega_{i})\big{)}\big{(}\hat{\mathbf{x}}_{i}-a_{i}\mathbf{d}( \omega_{i})\big{)}^{\top}\bigg{\}}.\]  
When \(\mathcal{L}_{\hat{\mathbf{x}}_{i}}(a_{i},\omega_{i})=\ln p(\hat{\mathbf{x}}_{i }|a_{i},\omega_{i})\) is defined as the log-likelihood function, it follows that  
\[\mathcal{L}_{\hat{\mathbf{x}}_{i}}(a_{i},\omega_{i})=-\frac{K}{2}\ln(2\pi \sigma^{2})-\frac{1}{2\sigma^{2}}\big{(}\hat{\mathbf{x}}_{i}-a_{i}\mathbf{d}( \omega_{i})\big{)}\big{(}\hat{\mathbf{x}}_{i}-a_{i}\mathbf{d}(\omega_{i}) \big{)}^{\top}.\]Then, the ML estimates of \(a_{i}\) and \(\omega_{i}\) are defined by  
\[\hat{a}_{i},\hat{\omega}_{i}=\arg\max\mathcal{L}_{\hat{\kappa}_{i}}(a_{i},\omega_ {i}). \tag{17}\]  
This can be equated to solving a cost function of size \(1/M\) that is similar to that in Equation (7), yielding a solution of the same form as that of  
\[\hat{\omega}_{i}=\arg\max\left\|\hat{\kappa}_{i}\mathbf{d}^{\top}(\omega_{i}) \left(\mathbf{d}(\omega_{i})\mathbf{d}^{\top}(\omega_{i})\right)^{-1}\mathbf{ d}(\omega_{i})\hat{\kappa}_{i}^{\top}\right\|,\]  
where the elements obtained by multiplying \(\mathbf{d}(\omega_{i})\) and \(\mathbf{d}(\omega_{i})^{\top}\) are the sum of the corresponding elements in the two sampled signals. By applying \(\mathbf{d}(\omega_{i})\mathbf{d}(\omega_{i})^{\top}=K\), we obtain  
\[\hat{\omega}_{i}=\arg\max\frac{1}{K}\left\|\hat{\kappa}_{i}\mathbf{d}^{\top} (\omega_{i})\mathbf{d}(\omega_{i})\hat{\kappa}_{i}^{\top}\right\|. \tag{18}\]  
This corresponds to the maximization of the correlation involving the time delay and Doppler shift of each satellite. It represents the ideal correlation procedure employed in the processing of a GNSS receiver.