Approximate maximum likelihood estimators for array processing in multiplicative noise environments

In this paper, we consider the problem of localizing a source by means of a sensor array when the received signal is corrupted by multiplicative noise. This scenario is encountered, for example, in communications, owing to the presence of local scatterers in the vicinity of the mobile or due to wavefronts that propagate through random inhomogeneous media. 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 (AML1) estimator is derived. The performance of the AML1 estimator in the presence of additive noise is studied, and a theoretical expression for its asymptotic variance is derived. Its performance is shown to be close to the Cramér–Rao bound (CRB) for moderate to high SNR. Next, the low SNR case is considered, and the corresponding AML2 solution is derived. It is shown that the approximate ML criterion can be concentrated with respect to both the multiplicative and additive noise powers, leaving out a two-dimensional (2-D) minimization problem instead of a four-dimensional (4-D) problem required by the exact ML. Numerical results illustrate the performance of the estimators and confirm the validity of the theoretical analysis.