Accuracy of source localization based on squared-range least squares (SR-LS) criterion

An estimator for source localization from range measurements was recently proposed by Beck, Stoica, and Li. The method is based on minimizing a least squares criterion for the squared-ranges (SR-LS), and a key advantage is that the global minimum of the SR-LS criterion can be computed efficiently. If the range measurements are modeled with additive, Gaussian noise, then the maximum likelihood (ML) estimator minimizes the LS criterion based on the ranges (R-LS). However, no computationally efficient algorithm is currently known for finding the global minimum of R-LS, so it is useful to study the sensitivity of SR-LS to additive noise. We compare the asymptotic (large sample) accuracy of source location estimates based on the R-LS and SR-LS criteria to study the robustness of SR-LS to additive noise on the range measurements. The asymptotic covariance of SR-LS is equal to or greater than R-LS, and we provide compact matrix expressions for the difference. The covariance expressions are analyzed to gain insight regarding sensor geometries for which SR-LS is more sensitive to noise than R-LS. The case of three sensors is analyzed completely, and while sensor configurations exist for which the difference is unbounded, we show that these are peculiar limiting cases and that the difference is bounded in cases of practical interest.