Complex sparse projections for compressed sensing

Abstract

Sparse projections for compressed sensing have been receiving some attention recently. In this paper, we consider the problem of recovering a k-sparse signal (x) in an n-dimensional space from a limited number (m) of linear, noiseless compressive samples (y) using complex sparse projections. Our approach is based on constructing complex sparse projections using strategies rooted in combinatorial design and expander graphs. We are able to recover the non-zero coefficients of the k-sparse signal (x) iteratively using a low-complexity algorithm that is reminiscent of well-known iterative channel decoding methods. We show that the proposed framework is optimal in terms of sample requirements for signal recovery (m = O (k log(n/k))) and has a decoding complexity of O (m log(n/m)), which represents a tangible improvement over recent solvers. Moreover we prove that using the proposed complex-sparse framework, on average 2k ≪ m ≤ 4k real measurements (where each complex sample is counted as two real measurements) suffice to recover a k-sparse signal perfectly.