A maximum likelihood approach to frequency-wavenumber analysis

This paper is concerned with the likelihood ratio detection and maximum likelihood estimation of plane waves traversing a small array on irregularly placed sensors. A new detection statistic is proposed. This statistic is related to the high resolution procedure of Capon, however it has a known null distribution in the case that no signal is present. Given that a wave is present, the asymptotic distribution of the maximum likelihood estimate of the wave parameters is derived. The problems of detection and estimation are distinguished. They are approached via the fundamental statistical procedures of likelihood ratio testing and maximum likelihood estimation. The solutions of both problems are found to be based upon the same quantity, the maximum likelihood statistic. The development leads to a clarification of the interrelationships and comparative properties of the conventional, the least squares, the high resolution and the new procedure. The approach is via Fourier inference.