Solution and Analysis of TDOA Localization of a Near or Distant Source in Closed Form

Point positioning and direction of arrival (DOA) localization require separate estimators, depending on whether the source is near or distant from the sensors. The use of modified polar representation (MPR) for the source location enables the integration of the two estimators together and eliminates the knowledge needed if the source is in the near-field or far-field. The previous work on MPR only provides an iterative implementation of the maximum likelihood estimator (MLE) initialized by a coarse semidefinite relaxation (SDR) solution. This paper proposes a formulation for time difference of arrival (TDOA) localization of a 2-D or 3-D source in MPR that would lead to a closed-form solution through the minimization of a quadratic function with a quadratic constraint. Two techniques, the successive unconstrained minimization (SUM) and the generalized trust region subproblem (GTRS), are applied to solve the optimization. Detailed analysis in the first order for mean-square and in the second order for estimation bias is performed for both methods. The theoretical results illustrate that the closed-form solution from each of the methods provides the Cramer-Rao lower bound performance for Gaussian noise. The GTRS solution has better accuracy than the SUM solution when the source signal is arriving at an azimuth or elevation angle close to zero, 90', or 180' or when the measurement noise level is large. They yield comparable performance with the MLE albeit having less complexity and avoiding possible divergence behavior. They outperform the closed-form solutions from the literature in angle estimation and have smaller bias.