Quasi-block Matrices in Compressed Sensing

The conditions under which Compressed sensing (CS) succeeds depend on the structure of the measurement matrix. Researches indicated that matrices whose entries are drawn independently from certain probability distributions satisfy the restricted isometry property (RIP) and guarantee exact recovery of a sparse signal with high probability. Motivated by filter-based compressed sensing in multichannel sampling applications, block Toeplitz matrices were considered as measurement matrices and shown to also satisfy the RIP. In wideband communication systems, the output streams of random filters were periodically down-sampled to achieve reduction of sampling rate. The measurement matrix, however, change from block Toeplitz matrix to quasi-block Toeplitz matrix because the low-rate analog-to-digital converter (ADC) only collects a part of the measurements at the receiver. In this paper, the feasibility of quasi-block Toeplitz matrices as measurement matrices is discussed. It is shown that the quasi-block Toeplitz matrices with entries drawn from Gaussian distribution satisfy RIP with high probability and also ensures the exact reconstruction of the sparse signals. Simulation results validate their performance.