Measurement matrix in Compressed sensing of IR-UWB signal

Abstract

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 signal processing applications, random filtering with Toeplitz sensing matrices whose elements are drawn from the same distributions were considered and shown to also be sufficient to recover a sparse signal from reduced samples exactly with high probability. In order to achieve reduction of sampling rate in impulse radio Ultra-Wideband (IR-UWB) communication systems, the output streams of UWB channel were periodically down-sampled. Sub-sampling at the receiver leads to quasi-Toeplitz-structured measurement matrices, whose entries are UWB channel coefficients. In this paper, the feasibility of this kind of quasi-Toeplitz matrices as measurement matrices is discussed. It is shown that the quasi-Toeplitz matrices with entries drawn from logarithm normal distribution satisfy RIP with high probability and also ensures the exact reconstruction of the sparse signals. Simulation results validate their performance.