Relationships between adaptive minimum variance beamforming and optimal source localization

For many years, the popular minimum variance (MV) adaptive beamformer has been well known for not having been derived as a maximum likelihood (ML) estimator. This paper demonstrates that by use of a judicious decomposition of the signal and noise, the log-likelihood function of source location is, in fact, directly proportional to the adaptive MV beamformer output power. In the proposed model, the measurement consists of an unknown temporal signal whose spatial wavefront is known as a function of its unknown location, which is embedded in complex Gaussian noise with unknown but positive definite covariance. Further, in cases where the available observation time is insufficient, a constrained ML estimator is derived here that is closely related to MV beamforming with a diagonally loaded data covariance matrix estimate. The performance of the constrained ML estimator compares favorably with robust MV techniques, giving slightly better root-mean-square error (RMSE) angle-of-arrival estimation of a plane-wave signal in interference. More importantly, however, the fact that such optimal ML techniques are closely related to conventional robust MV methods, such as diagonal loading, lends theoretical justification to the use of these practical approaches.