A Semidefinite Relaxation Solution for Time Delay and Doppler Shift Localization Considering Sensor Location Errors and Its Bias Reduction Scheme

This article develops a robust source localization method using time delay and Doppler shift measurements, where the sensor motion effect accompanied by sensor location errors cannot be ignored. We begin by transforming the time delay and Doppler shift measurement models into a series of nonlinear equations that take sensor location errors into account, and then construct a constrained weighted least squares (CWLS) problem based on these equations. Because of the nonconvex nature of the problem, we relax it into a semidefinite programming (SDP) problem via convex relaxation and further propose a scheme to eliminate the influence of the additional estimation bias caused by the approximation applied in the transformation of measurement models. To perform the bias reduction, we derive the theoretical expression of the solution bias and then subtract it to obtain a bias-reduced solution. We also derive the closed-form expression of the hybrid Cramer–Rao lower bound (HCRLB) as the performance benchmark. Theoretical analysis and simulation results demonstrate that the mean-square error (MSE) performance of the proposed method can achieve the HCRLB accuracy and the bias can be significantly reduced with bias reduction.