Bias-Reduced SDR Method for Locating a Noncooperative Moving Source Using TOAs and FOAs

This work studies the localization of a noncooperative moving source based on the arrival time and frequency measurements. The signal transmission time stamp and carrier frequency are fully unknown in the absence of cooperation. To solve this challenging problem, we start by converting the observation model equations and establishing a constrained-weighted-least-squares (CWLS) optimization problem, for the estimation of the source position and velocity and the unknown timestamp and carrier frequency together. The CWLS problem is nonconvex and we solve it by utilizing semidefinite relaxation (SDR) so that the solution can be obtained by a semidefinite programming software package. The approximations imposed for measurement model transformation and the CWLS problem formulation can incur a large bias in the solution. For the purpose of reducing the bias, we further develop a different CWLS formulation having the capability of bias reduction, named as bias-reduced CWLS (BR-CWLS), and also solve it by applying SDR. The analysis for both problems reveals that their solutions can reach the Cramer–Rao lower bound performance. In addition, the theoretical remaining bias for the BR-CWLS solution is derived. Simulations provide the confirmation of the theoretical expectations and the ability of the developed method from BR-CWLS to reduce the solution bias.