Reducing Bias for Multistatic Localization of a Moving Object by Transmitter At Unknown Position

Multistatic localization plays an important role in object localization. In this paper, we address the problem of multistatic localization of a moving object in the absence of transmitter position when the transmitter is not synchronized with the receivers. We propose to jointly estimate the unknowns, including the object position and velocity, the transmitter position, and the clock and frequency offsets, using both time delay and Doppler frequency shift measurements. To this end, we first formulate a constrained weighted least squares (CWLS) minimization problem, whose solution is found to have significant bias caused by approximations in transforming the model equations and solving the problem. To reduce the bias, we further formulate a non-convex bias reduced CWLS (BR-CWLS) problem by imposing a quadratic constraint, which is constructed by considering the second-order noise and errors of the matrices and vectors involved in the CWLS problem. One particular aspect of the proposed BR-CWLS method is that the errors in approximating the weighting matrix are taken into consideration for bias reduction. The non-convex BR-CWLS problem is solved by applying the semidefinite relaxation technique to relax it as a convex semidefinite program. In addition, we show through the mean square error (MSE) analysis that the performance of the BR-CWLS solution can approach the Cramer-Rao lower bound performance when the noise is not significant. We also derive the theoretical bias expression for evaluating the amount of bias. Simulation results demonstrate the good performance of the proposed method in terms of both MSE and bias.