Localization based on the reception of radio-frequency waveforms is a crucial problem in many civilian or military applications. It is also the main objective of all Global Navigation Satellite System (GNSS). Given delayed and Doppler shifted replicas of the satellites transmitted signals, the most widespread approach consists in a suboptimal two-step procedure. First, estimate the delays and Dopplers from each satellite independently, then estimate the user position and speed thanks to a Least Square (LS) minimization. More accurate and robust techniques, such as a direct Maximum Likelihood (ML) maximization, that exploit the links in between the different channels exist but suffer from an heavy computational burden that prevent their use in real time applications. Two-steps procedures with an appropriate Weighted LS (WLS) minimization are shown to be asymptotically equivalent to the ML procedure. In this paper, we develop a closed-form expression of this WLS asymptotically efficient solution. We show that this simple expression is the sum of two terms. The first one, depending on the pseudo-ranges is the widespread used WLS solution. The second one is a Doppler-aided corrective term that should be taken into account to improve the position estimation when the observation time increases.

Multipath remains the main source of error when using global navigation satellite systems (GNSS) in constrained environment, leading to biased measurements and thus to inaccurate estimated positions. This paper formulates the GNSS navigation problem as the resolution of an overdetermined system, which depends nonlinearly on the receiver position and linearly on the clock bias and drift, and possible biases affecting GNSS measurements. The extended Kalman filter is used to linearize the navigation problem whereas sparse estimation is considered to estimate multipath biases. We assume that only a part of the satellites are affected by multipath, i.e., that the unknown bias vector is sparse in the sense that several of its components are equal to zero. The natural way of enforcing sparsity is to introduce an 1 regularization associated with the bias vector. This leads to a least absolute shrinkage and selection operator (LASSO) problem that is solved using a reweighted-1 algorithm. The weighting matrix of this algorithm is designed carefully as functions of the satellite carrier to noise density ratio and the satellite elevations. The smooth variations of multipath biases versus time are enforced using a regularization based on total variation. An experiment conducted on real data allows the performance of the proposed method to be appreciated.

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. ABSTRACTA new sparse estimation method was recently introduced in a previous work to correct biases due to multipath (MP) in GNSS measurements. The proposed strategy was based on the resolution of a LASSO problem constructed from the navigation equations using the reweighted-1 method. This strategy requires to adjust the regularization parameters balancing the data fidelity term and the involved regularizations. This paper introduces a new Bayesian estimation method allowing the MP biases and the unknown model parameters and hyperparameters to be estimated directly from the GNSS measurements. The proposed method is based on BernoulliLaplacian priors, promoting sparsity of MP biases.