Approximate Maximum Likelihood DOA estimation in multiplicative noise environments

We consider the problem of localizing a source by means of a uniform linear array of sensors when the received signal is corrupted by multiplicative noise. Since the exact Maximum Likelihood (ML) estimator is computationally intensive, two approximate solutions are proposed, originating from the analysis of the high and low Signal to Noise Ratio (SNR) cases, respectively. First, starting with the no additive noise case, a very simple approximate ML (AML(1)) estimator is derived. A theoretical expression for its asymptotic variance in the presence of additive noise is derived. It shows that the AML1 estimator has a performance close to the Cramer-Rao bound (CRB) for moderate to high SNR. Next, the low SNR case is considered and the corresponding AML(2) solution is derived. It is shown that the approximate ML criterion can be concentrated with respect to (w.r.t.) both the multiplicative and additive noise powers, leaving out a 2-D minimization problem instead of a 4-D problem required by the exact ML. Numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis.