Cyclosparsity: A New Concept for Sparse Deconvolution

Abstract

Periodic random impulse signals are appropriate tools for several situations of interest and are a natural way for modeling highly localized events occuring randomly at given times Nevertheless the impulses are generally hidden and swallowed up in noise because of unwanted convolution Thus the resulting signal is not legible and may lead to erroneaous analysis and hence the need of deconvolution to restore the random periodic impulses The main purpose of this study is to introduce the concept of cyclic sparsity or cyclosparsity in deconvolution framework for signals that are jointly sparse and cyclostationary like periodic random impulses Indeed all related works in this area exploit only one property either sparsity or cyclostationarity and never both properties together Although the key feature of the cyclosparsity concept is that it gathers both properties to better characterize this kind of signals We show that deconvolution based on cyclic sparsity hypothesis increases the performances and reduces significantly the computation cost as well Finally we use computer simulations to investigate the behavior in deconvolution framework of the algorithms Matching Pursuit MP 13 Orthogonal Matching Pursuit OMP 14 Orthogonal Least Square OLS 15 Single Best Replacement SBR 19 20 21 and the proposed extensions to cyclic sparsity context Cyclo-MP Cyclo-OMP Cyclo-OLS and Cyclo-SBR