# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 3 Asymptotic Performance Analysis for the DPE  
The DPE is based on the fact that the synchronization parameters of each satellite correspond uniquely to the receiver PVT state. Given that the number of visible satellites is generally larger than the dimension of the PVT state, the relationship between the synchronization parameter and the PVT state ensures that for every \(\gamma\), there is a unique \(\omega\). This relationship can be expressed as an injection function, signifying that a given PVT state can only be associated with a single pair consisting of a time delay and a Doppler shift for each satellite. It is easy to identify that  
\[\omega=h(\gamma),\forall\gamma\in\Gamma. \tag{6}\]  
The time-frequency parameterization model can be represented by the parameters \(\gamma\). It is worth noting that \(\mathbf{D}(\gamma)\) is used instead of \(\mathbf{D}\big{(}h(\gamma)\big{)}\) for simplicity in notation. The DPE counterpart of the model in Equation (3) is  
\[\mathbf{y}=\mathbf{a}\mathbf{D}(\gamma)+\mathbf{n},\]  
with the same definitions, including the constant assumption of \(\gamma\) and the remaining unknown parameters. The DPE incorporates the measurement model by obtaining the ML estimate of the PVT state from \(\mathbf{y}\) and reconstructs the signal by parameterizing the time delay and Doppler shift for each satellite. It is important to note that the DPE is a single-step procedure for estimating \(\gamma\) from all observations. This implies that the search is performed within the known space \(\Gamma\). The ML estimation appears as follows:  
\[\hat{\mathbf{a}},\hat{\gamma}=\operatorname*{arg\,max}_{\gamma\in\Gamma} \mathcal{L}_{\mathbf{y}}(\mathbf{a},\gamma),\]  
where the joint log-likelihood function \(\mathcal{L}_{\mathbf{y}}(\mathbf{a},\gamma)\) is given by  
\[\mathcal{L}_{\mathbf{y}}(\mathbf{a},\gamma)=\ln p\big{(}\mathbf{y}|\mathbf{a},h(\gamma)\big{)}.\]  
The ML estimate of \(\gamma\) is given by maximizing the likelihood function or, equivalently, minimizing the following nonlinear LS problem:  
\[\hat{\mathbf{a}},\hat{\gamma}=\operatorname*{arg\,max}_{\gamma\in\Gamma}\frac {1}{K}\|\mathbf{y}-\mathbf{a}\mathbf{D}(\gamma)\|^{2}, \tag{7}\]  
where the operator \(\|\cdot\|^{2}\) denotes the L\({}_{2}\)-norm of a vector. After applying the orthogonality principle to Equation (7), the ML estimate of the amplitude vector \(\mathbf{a}\) is  
\[\hat{\mathbf{a}}_{\mathrm{ML}}=\mathbf{y}\mathbf{D}^{\top}(\gamma)\big{(} \mathbf{D}(\gamma)\mathbf{D}^{\top}(\gamma)\big{)}^{-1}.\]  
It is intuitive that the ML estimate of \(\mathbf{a}\) is the LS solution of the difference between the reconstructed signal and the received signal \(\mathbf{y}\). By substituting this result into Equation (7) and expanding it, we obtain  
\[\hat{\gamma}_{ML}=\operatorname*{arg\,min}_{\gamma\in\Gamma}\|\mathbf{y}\|^{ 2}-\hat{\mathbf{a}}_{\mathrm{ML}}\mathbf{D}(\gamma)\mathbf{y}^{\top}.\]  
The resulting ML cost function can be expressed in relation to a projection onto the signal subspace as  
\[\hat{\gamma}_{\mathrm{ML}}=\operatorname*{arg\,max}_{\gamma\in\Gamma}\Big{\|} \mathbf{y}\mathbf{D}^{\top}(\gamma)\big{(}\mathbf{D}(\gamma)\mathbf{D}^{\top }(\gamma)\big{)}^{-1}\mathbf{D}(\gamma)\mathbf{y}^{\top}\Big{\|}. \tag{8}\]As the projection matrix is idempotent, the optimization problem of the DPE is actually finding a parameter \(\mathbf{\gamma}\) such that \(\mathbf{y}\) has the maximum effective projection onto the column space of \(\mathbf{D}(\mathbf{\gamma})\). Equation (8) is a well-known, intuitive, and conceptually simple method that is recognized as an ML estimate that is asymptotically efficient [33]. The multiple-parameter CRB indicates that the inverse of the FIM serves as a lower bound on the variance of any unbiased estimator. In this case, the FIM is a function of \(\mathbf{\mathrm{a}}\) and \(\mathbf{\gamma}\) with the \((M+7)\times(M+7)\) block matrix, and it is expressed in terms of submatrices as  
\[\left[