Abstract: Wave-number-frequency adaptive beamforming using a reduced rank k-.omega. matrix method. A spatial Fourier transform of the narrowband receiver outputs for the sensors in a linear array is taken to enter the wave number space and generate M independent spatial wave number beams w. The spatial FFT outputs are used to form the entries in a matrix of eigenvectors, wherein the k-th eigenvector represents a set of spatial convolution coefficients. The spatial wave number beams are weighted in an adaptation process by the spatial convolution coefficients to form a beam steered in the direction of a k-th signal source.