Maximum likelihood estimation of signal power in sensor array in the presence of unknown noise field

A simple approximate maximum likelihood (AML) estimator is derived for estimating a power of a single signal with a rank-one spatial covariance matrix known a priori except for a scaling. The noise field is assumed to be spatially uncorrelated and to have different unknown powers in each array sensor. The derivation of the AML estimator is carried out under the assumptions of a large number of samples and a weak signal. The variance of the introduced AML estimator is compared with the exact Cramer-Rao bound (CRB) of this estimation problem. It is shown analytically that the variance of the estimation errors and the corresponding CRB coincide asymptotically in the majority of practically important cases. The analogy between the presented AML estimator and the well-known ML estimator, which is derived under equal noise variance assumption (this estimator is based on the matched processing and is usually termed as conventional beamformer), is considered. This analogy enables straightforward extension of the AML estimator to the case of well-separated weak multiple sources with unknown locations and consideration of it as a variant of conventional beamformer for unknown noise field scenarios. The analytical results are verified by computer simulations.