Bias is a fundamental metric for evaluating the average error of an estimator, where lower bias indicates an estimator closer to the true value. Recent studies on time difference of arrival (TDOA) localization have introduced the modified polar representation (MPR) to unify near-field and far-field scenarios, enhancing robustness to the source range. However, existing closed-form solutions suffer from relatively high bias. This paper first reviews the current progress of MPR-based localization and then focuses on bias reduction (BR) in TDOA localization under MPR. To this end, a novel method, termed direct deviation refinement (DDR), is proposed, which refines a weighted spherical intersection to achieve lower bias. While DDR effectively lowers bias, its mean-square-error (MSE) is not optimal. To address this limitation, a robust iterative approach based on factor graph (FG) is further developed to alleviate the performance trade-off of the DDR. Simulation results demonstrate that the DDR significantly reduces bias compared to successive unconstrained minimization (SUM), generalized trust region subproblem (GTRS), and iterative Gauss–Newton (GN) solutions. While DDR lowers bias, it slightly increases MSE, whereas FG achieves improved MSE performance at the cost of a marginal bias increase.
Chen et al. (Thu,) studied this question.