Asymptotic Performance of GNSS Positioning Approaches under Cross-Correlation Effects

Abstract

Conventional global navigation satellite system receivers typically employ a two-step positioning procedure (2SP) by first independently estimating the synchronization parameters and then using these parameters to solve a system of superdeterministic equations derived from multilateration to accomplish positioning. Direct position estimation (DPE) has emerged as a promising alternative that utilizes a single-step procedure to obtain the maximum likelihood estimate of a position. This approach has been shown to effectively mitigate biases incurred by the second estimation step in 2SP. However, for code-division multiple-access systems, the pseudo-orthogonality of the spreading codes causes the estimation problem not to be mapped to a perfectly orthogonal space. Additionally, the cross-correlation interference between satellites renders the maximum likelihood invariant theory untenable in the first estimation step of the 2SP. This study presents the derivation of the Cramér–Rao bound constraint for both the 2SP and DPE, evaluating the performance degradation of the 2SP compared to that of the DPE with the consideration of cross-correlation. Furthermore, a more stringent result is proven, indicating that the 2SP is not as asymptotically efficient as the DPE in all scenarios. The derived bounds are validated using realistic scenarios, and the root-mean-square error performance of the respective maximum likelihood estimators is compared.

Keywords

• direct position estimation (DPE)
• Cramér–Rao bound (CRB)
• maximum likelihood estimation
• cross-correlation
• satellite navigation systems