Optimum and adaptive array processing in frequency-wavenumber space

In this paper a frequency-domain adaptive array structure is derived that processes data in selected regions of frequency-wave-number space to provide an MMSE estimate of a signal component of the observed acoustic field. The desired "optimal" processor structure is derived by appealing to the Sherman-Morrison inversion lemma to invert the sum of a diagonal noise matrix and several diadic matrices corresponding to plane-wave propagating signal and interference fields. The structure reduces to a parallel structure of beamformer-gain elements whose beamformer directions (phase delays) and gains are unknown because of a priori ignorance concerning interference directions and levels. The structure is adapted to these uncertainties by forming a stochastic approximation algorithm for the gains of several beams directed to a finite set of possible interference directions.