Closed-Form Algebraic Solutions for 3-D Doppler-Only Source Localization

This paper presents new closed-form estimators for 3-D source localization using Doppler measurements collected by a moving sensor platform. Two estimators, based on bias compensation and instrumental variables, are developed with significant performance improvement over a previously proposed least-squares estimator (LSE). First, a refined variant of the LSE, called bias-compensated weighted LSE (BCWLSE), is presented utilizing a weighting matrix and a bias compensation step. A weighted instrumental variable estimator is then developed based on a modified pseudolinear equation for the LSE to avoid pseudolinear noise amplification that may deteriorate the BCWLSE performance at large noise. Comparative simulation studies are presented to demonstrate the performance advantages of the proposed estimators over the severely biased LSE and divergence-prone maximum likelihood estimator. The proposed estimators are observed to produce almost no bias and achieve a mean-squared-error performance close to the Cramér-Rao lower bound.