Deterministic construction of array QC CS measurement matrices based on Singer perfect difference sets

Abstract

Low-density parity-check (LDPC) codes and compressed sensing (CS) share many common environments. In this study, a novel approach for constructing a new class of deterministic sparse sensing matrices based on array quasi-cyclic (QC) LDPC codes via Singer perfect difference sets is proposed. In contrast to random and the other deterministic matrices, the proposed framework would be highly desirable as it is generated based on circulant permutation matrices, which requires less memory for storage and lower computational cost for sensing. Since the restricted isometric property is very difficult to verify, then the mutual coherence and the girth are two computationally tractable criteria that the authors used to assess the CS recovery capabilities of sensing matrices. In addition, inspired by LDPC codes, they extract a necessary condition for the proposed measurement matrix to have effective values for girth as large as g ≥ 6 and 8 . Comprehensive one-dimensional (1D) and 2D simulations verify that their proposed sensing matrix has minimum coherence and superior CS recovery abilities in comparison with the corresponding random Gaussian, Bernoulli, and the other deterministically generated matrices. Furthermore, the required physical storage space and the complexity of the hardware implementation are greatly reduced due to being sparse and QC in structure.