Performance limits of a compressive sensing application to beamforming on a line array

Compressive sensing is a sampling theorem that exploits the sparsity of a signal in a domain Ψ, while being spread out in a sensing domain Φ. For example, a sinusoid time domain signal in Φ can be represented by one non-zero coefficient in the frequency domain Ψ. The time-frequency relationship is similar to the space-angle relationship that exists in underwater acoustics for an array of hydrophones. Wavefront curvatures that are spread out in the space domain can be represented in the angle domain by a sparse vector. This work investigates the performance limits of using compressive sensing to resolve signals in the angle domain, a task usually accomplished by beamforming. For compressive sensing, it has been shown that the performance of recovering a signal is related to the number of measurements, the number of non-zero coefficients, and the dimension of Ψ [E.J. Candes and M.B. Wakin IEEE Signal Processing Magazine 21-30 (March 2008)]. Typically, in underwater acoustics, the number of hydrophones and their locations are fixed, so that the performance is found to be dependent on the number of non-zero coefficients (signals in the water) and the dimension of the angle domain (beams). [Work supported by ARL:UT IRD]