Compressive Acquisition and Least-Squares Reconstruction of Correlated Signals

Abstract

This letter presents a framework for the compressive acquisition of correlated signals. We propose an implementable sampling architecture for the acquisition of ensembles of correlated (lying in an a priori unknown subspace) signals at a sub-Nyquist rate. The sampling architecture acquires structured compressive samples of the signals after preprocessing them with easy-to-implement components. Quantitatively, we show that an ensemble of $M$ correlated signals each of which is bandlimited to $W/2$ and can be decomposed as the linear combination of $R$ underlying signals can be acquired at roughly $RW$ (assuming without loss of generality $M ) samples per second. In the case, when $M \gg R$ , this results in significant reduction in the sampling rate compared to $MW$ samples per second dictated by the Shannon–Nyquist sampling theorem. The signal reconstruction is achieved by solving only a least-squares program, which is in stark contrast to the prohibitively computationally expensive semidefinite programming techniques suggested in the earlier literature. We also provide rigorous analytical results to support our claims.