TDOA Based Positioning in the Presence of Unknown Clock Skew

This paper studies the positioning problem of a single target node based on time-difference-of-arrival (TDOA) measurements in the presence of clock imperfections. Employing an affine model for the behaviour of a local clock, it is observed that TDOA based approaches suffer from a parameter of the model, called the clock skew. Modeling the clock skew as a nuisance parameter, this paper investigates joint clock skew and position estimation. The maximum likelihood estimator (MLE) is derived for this problem, which is highly nonconvex and difficult to solve. To avoid the difficulty in solving the MLE, we employ suitable approximations and relaxations and propose two suboptimal estimators based on semidefinite programming and linear estimation. To further improve the estimation accuracy, we also propose a refining step. In addition, the Cramér-Rao lower bound (CRLB) is derived for this problem as a benchmark. Simulation results show that the proposed suboptimal estimators can attain the CRLB for sufficiently high signal-to-noise ratios.