Augmenting the maximum likelihood delay estimator to give maximum likelihood direction

The maximum likelihood (ML) estimator was developed by Knapp and Carter [1] for determining time delay between two signals received at two spatially separated sensors in the presence of uncorrelated noise. This brief note shows that that estimator is readily extended to ML estimation of arrival angle even though the medium causes a non-linear phase-versus-frequency difference (in general, a filtering) between the two received signals.