Compressive sensing for sparse arrays

We assess the usefulness of compressive sensing (CS) for processing signals from sparse or highly thinned receive arrays. Compressive sensing consists of a theory and related algorithms for solving under determined linear equations with the assumption that solutions have only a few non-zero or dominant values and the remaining values are zero or small. Sparse arrays are a natural application of CS when (a) there are only a few array observations with each expressed as a linear combination of the received plane wave signal amplitudes and (b) there are a large number of possible angles of arrival of signals but only a few signals are actually present. Our approach employs linear beamforming of the sparse array signals to create data for compressive sensing that are a linear combination of the plane wave signal amplitudes. The beamforming matrix serves to create coefficients in this linear combination that nearly orthonormal, a requirement for successful application of CS. We evaluate CS for the problem of a five element sparse array located on a 90° sector of a circular arc with large radius of curvature. Elements of the sparse array are themselves digitally beamformed phased arrays with main beams steered to a common direction. We found that compressive sensing works well for large signal to noise, reliably indicating the direction of arrival of a plane wave within the beamwidth of the phased array element and potentially reliable resolution of two equal intensity plane waves within one tenth of the beamwidth of the sparse array. False plane wave indications occur with significant probability when the signal to noise is small. However, an n or more of m indication algorithm with m statistically independent noise samples reduces significantly the probability of false indication.