# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 2 Signal Model
If the receiver observes \(K\) snapshots, the signal model can be expressed in a compact form as [2]  
\[\mathbf{y}=\mathbf{a}\mathbf{D}(\mathbf{\omega})+\mathbf{n}, \tag{3}\]  
with the following definitions:  
* \(\mathbf{y}=[y(t_{1}),\cdots,y(t_{K})]\in\mathbb{R}^{1\times K}\) is the observed signal vector;
* \(\mathbf{a}=[a_{1},\cdots,a_{M}]\in\mathbb{R}^{1\times M}\) is a vector whose elements are the amplitudes of the \(M\) received signals;
* \(\mathbf{\omega}=[\mathbf{\tau},f_{d}]\in\mathbb{R}^{1\times 2M}\) is a vector containing the time delay and the Doppler shift of each satellite;
* \(\mathbf{D}(\mathbf{\omega})=[\mathbf{d}^{\top}(\omega_{1}),\cdots,\mathbf{d}^{ \top}(\omega_{M})]^{\top}\in\mathbb{R}^{M\times K}\) is referred to as the basis function matrix, where \(\omega_{i}=[\tau_{i},f_{d_{i}}]\) is the synchronization parameter for the \(i\)-th satellite, and \(\mathbf{d}(\omega_{i})=[d(t_{1},\omega_{i}),\cdots,d(t_{K},\omega_{i})]\); each component is defined by the delayed Doppler-shifted signal envelope \(d(t_{k},\omega_{i})=g_{i}(t_{k}-\tau_{i})\cos\left(2\pi(f_{c}+f_{d_{i}})t_{k}+ \phi_{i}\right)\);
* \(\mathbf{n}=[n(t_{1}),\cdots,n(t_{K})]\in\mathbb{R}^{1\times K}\) represents \(K\) snapshots of the AWGN vector, where \(\sigma^{2}\mathbf{1}_{K}\) is the invariable covariance matrix of \(\mathbf{n}\) during the observation interval, and \(\mathbf{1}_{K}\) is a \(K\times K\) identity matrix. Based on a collection of \(K\) snapshots, the probability density function (pdf) of the received signal conditioned on the unknown parameters \(\mathbf{a}\) and \(\mathbf{\omega}\) is given by \[p(\mathbf{y}|\mathbf{a},\mathbf{\omega})=\frac{1}{(2\pi\sigma^{2})^{\frac{K}{2}} }\exp\bigg{\{}-\frac{1}{2\sigma^{2}}\big{(}\mathbf{y}-\mathbf{a}\mathbf{D}( \mathbf{\omega})\big{)}\big{(}\mathbf{y}-\mathbf{a}\mathbf{D}(\mathbf{\omega})\big{)} ^{\top}\bigg{\}}.\] (4) Their log-likelihood function, \(\mathcal{L}_{\mathbf{y}}\), is defined as \[\mathcal{L}_{\mathbf{y}}(\mathbf{a},\mathbf{\omega})=\ln p(\mathbf{y}|\mathbf{a}, \mathbf{\omega}).\] It follows that [32] \[\mathcal{L}_{\mathbf{y}}(\mathbf{a},\mathbf{\omega})=-\frac{K}{2}\ln(2\pi\sigma^{ 2})-\frac{1}{2\sigma^{2}}\big{(}\mathbf{y}-\mathbf{a}\mathbf{D}(\mathbf{\omega}) \big{)}\big{(}\mathbf{y}-\mathbf{a}\mathbf{D}(\mathbf{\omega})\big{)}^{\top}.\] (5)  
In essence, the primary objective of positioning algorithms in GNSSs is to calculate the PVT state \(\mathbf{\gamma}\) by maximizing the log-likelihood function shown in Equation (5) according to the parameterized probability distribution model shown in Equation (4). The PVT state \(\mathbf{\gamma}\) can be directly estimated from the observation vectors \(\mathbf{y}\) using the DPE or, alternatively, can be estimated through the 2 SP by first determining the synchronization parameter \(\mathbf{\omega}\) and then using this estimate to compute \(\gamma\). The objective of this study is to assess and compare the asymptotic performance of these two methods. When \(K\) tends to infinity, the covariance matrices of \(\gamma\) for the DPE and 2 SP are denoted by \(\mathbf{C}_{1}\) and \(\mathbf{C}_{2}\), respectively.