Sub-Nyquist sampling of sparse and correlated signals in array processing

Abstract

This paper considers efficient sampling of simultaneously sparse and correlated (S&C) signals for automotive radar application. We propose an implementable sampling architecture for the acquisition of S&C at a sub-Nyquist rate. We prove a sampling theorem showing exact and stable reconstruction of the acquired signals even when the sampling rate is smaller than the Nyquist rate by orders of magnitude. Quantitatively, our results state that an ensemble M signals, composed of a-priori unknown latent R signals, each bandlimited to W/2 but only S-sparse in the Fourier domain, can be reconstructed exactly from compressive sampling only at a rate RSlogα⁡W samples per second. When R≪M and S≪W, this amounts to a significant reduction in sampling rate compared to the Nyquist rate of MW samples per second. This is the first result that presents an implementable sampling architecture and a sampling theorem for the compressive acquisition of S&C signals. We resort to a two-step algorithm to recover sparse and low-rank (S&L) matrix from a near optimal number of measurements. This result then translates into a signal reconstruction algorithm from a sub-Nyquist sampling rate.