# Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects
## 1 Introduction
Indeed, while these methods are effective for lower-dimensional problems, they become impractical for high-dimensional scenarios due to the exponential growth of the grid of points required to evaluate the cost function. Another solution is to decompose multi-dimensional optimization into a series of recursive and simpler searches. Based on the expectation maximization principle, the space-alternating generalized expectation maximization optimization algorithm is introduced as an effective method in [20]. Nevertheless, despite these advancements, methods such as these still impose heavy computational costs and are challenging to implement in real-time applications. As mentioned by the authors of [21], more computationally efficient optimization algorithms still need to be developed. The theoretical results in [22] prove that DPE-based localization shows a performance improvement compared to that of 2 SP. For GNSS receivers, the Cramer-Rao bound (CRB), which is an essential tool for the analysis of the performance of localization systems, is used to show the asymptotic performance of each approach, as derived for both the 2 SP and DPE in [23], with the latter exhibiting superior performance. In particular, the 2 SP can achieve the same performance as that of the DPE only when the Fisher information matrix (FIM) of the time of arrival is known [24]. The results from [25] show that the DPE can obtain a lower root-mean-square error (RMSE) than that of the 2 SP. They also point out that if the received signal strength is a part of the estimation parameters and is used as a weighted matrix, then the 2 SP is asymptotically effectively the same as the DPE. In summary, these studies compare the asymptotic performance of the DPE and 2 SP by evaluating their covariance matrices. It is pointed out that the maximum likelihood (ML) invariant can not be satisfied when independently estimated synchronization parameters are used in the second estimation step of the 2 SP, and it is shown that the estimates remain ML-invariant only when the intermediate parameters are estimated with an appropriately weighted matrix for the weighted least square (WLS) [26]. The appropriately weighted matrix corresponds to the FIM of the considered model, which is calculated from the time delays and Doppler shifts. This explanation is related to the extended invariance principle (EXIP) [27], which aims to simplify the ML criterion by reparameterizing the estimation problem so that intermediate estimates are obtained, and their refinement through an appropriate WLS minimization achieves the same asymptotic performance as that of the initial model. In the context of GNSSs, the positioning problem is parameterized into a framework for the independent processing of the time delay and Doppler shift for each satellite, streamlining the PVT estimation process, and the EXIP also holds [28]. Overall, the EXIP provides a theoretical method for comparing the asymptotic performance of the 2 SP and DPE, demonstrating that the 2 SP can achieve the same asymptotic performance as that of the DPE through an appropriate WLS procedure. However, the previous works neglected the effect of the cross-correlation in GNSSs, leading to the false conjecture that the 2 SP is asymptotically efficient, i.e., the covariance matrix of the estimates tends toward the CRB. It is crucial to note that the pseudo-orthogonality of pseudo-random noise (PRN) codes causes the estimation problem of synchronization parameters to be unable to be mapped into a completely orthogonal space or be decomposed into independent subproblems for each satellite. This is due to the existence of cross-correlation errors, rendering the theory of ML invariance inapplicable. Instead, the DPE provides a way to process all signals jointly, which can be seen as equipping the receiver with abilities to mitigate cross-correlation errors [21; 23; 24], but corresponding theoretical analyses and quantitative comparisons are still missing.