Maximum likelihood estimators and Cramér–Rao bounds for the localization of an acoustical source with asynchronous arrays

Abstract

The performances of source localization using a microphone array can be improved by repeating the experiment with different array placements, assuming that the source position remains constant. In this article, two types of theoretical results on this setting are presented. The Maximum Likelihood Estimator (MLE) is derived, and Cramér–Rao bounds are computed, both for a strict model, where the power of the source is constant, and a relaxed model where the power of the source is allowed to change. Cramér–Rao bounds show that the performances (in terms of mean squared error of the estimation of the position and power of the source) of asynchronous arrays for the estimation of the position are, in some settings, significantly degraded compared to synchronous arrays. A particular example of such a setting is the case of two parallel arrays around the source. In contrast, the performances of power estimation are, in most cases, close to the performances of synchronous arrays. The obtained MLEs are compared to the state of the art in asynchronous array source localization using simulations, showing that the MLE for the strict model outperforms the state of the art. Experimental results illustrate the theoretical and numerical findings.