Asymptotically Efficient Solutions for TOA-FOA Localization With Clock Bias and Drift

Clock bias and drift between the unknown source and sensors affect the localization performance when using hybrid time of arrival (TOA) and frequency of arrival (FOA) measurements. A recent study proposed a semidefinite programming (SDP) solution for estimating motion parameters, including position and velocity. However, this approach is computationally intensive and suffers from numerical problems. This article introduces a projection onto the null space of the redundant variable vector, which alleviates the numerical issues and leads to a closed-form solution (CFS). Since the projection does not incorporate the constraints between the redundant variables and the unknown motion parameters, a correction stage is required to improve accuracy. Two correction approaches have been developed: one based on constraints and the other on first-order Taylor expansion. These result in two fast and effective solutions: projection with correction from constraints (Proj-C) and projection with correction from Taylor expansion (Proj-TE). The proposed solutions achieve Cramér-Rao lower bound (CRLB) performance levels in both theory and simulation, provided the noise is mild. We also consider the uncertainty of sensor positions and velocities, extending the proposed solutions to mitigate accuracy loss. Numerical results show that Proj-C has better mean-square-error (mse) performance for velocity and clock drift estimation than Proj-TE when the noise/error is relatively large, while the latter significantly reduces bias to the maximum likelihood estimation (MLE) level. Both the proposed solutions avoid degradation in the small noise/error region due to the relaxation in SDP.