Constructions of quasi-cyclic measurement matrices based on array codes

Recently, Dimakis, Smarandache, and Vontobel indicated that the parity-check matrices of good LDPC codes can be used as provably good measurement matrices for compressed sensing (CS) under basis pursuit (BP). In this paper, we consider the parity-check matrix H(r, q) of the array codes, one of the most important kind of structured LDPC codes. The spark, i.e. the smallest number of linearly dependent columns in a matrix, of H(2, q) and H(3, q) are determined and two lower bounds of the sparks of H(r, q) are given for r ≥ 4. Moreover, we carry out numbers of simulations and show that many parity-check matrices of array codes and their submatrices perform better than the corresponding Gaussian random matrices. The proposed measurement matrices have perfect quasi-cyclic structures and can make the hardware realization convenient and easy, thus having great potentials in practice.