Unified Model for Near and Far Field Localization: Subspace-Based Solution and Limitations Analysis

The modified polar representation (MPR) achieves a unified expression of near-field localization and far-field direction-finding models, overcoming the dependency on prior information of source range for either localization or direction-finding. It cleverly avoids the loss in localization accuracy caused by the range ambiguity and the coupling between range and angle in far field. However, existing MPR localization methods suffer from insufficient performance and robustness, as well as unclear boundary conditions, making them unable to satisfy the requirements of practical engineering applications. This paper addresses the time difference of arrival (TDOA) localization problem in MPR from the perspective of subspaces, separating the estimation of angles and inverse-range into two orthogonal spaces to enhance performance and robustness. We first employ nullspace projection to eliminate inverse-range and obtain the optimal estimation of angles, and then put the result back into the original equation to solve for inverse-range. When solving the inverse range, the matrix is rank-deficient due to the consideration of angle estimation errors. This problem is resolved by projecting the equations onto the subspace corresponding to the non-zero eigenvalues, so the optimal estimation of inverse-range is obtained straightforwardly through weighted least squares. Analysis and simulation experiments demonstrate that the proposed algorithm outperforms the existing best closed-form solution, generalized trust region sub-problem (GTRS), in terms of performance and robustness in high-noise scenarios. This article also analyzes the limitations of existing TDOA localization algorithms based on MPR, including the proposed algorithm. It clarifies the requirements of different algorithms concerning the minimum number of sensors and applicable scenarios, providing references for algorithm selection in engineering applications. © 2023 Chinese Institute of Electronics. All rights reserved.