New Cyclic Sparsity Measures for Deconvolution Based on Convex Relaxation

Abstract

Periodic random impulse signals are powerful tools for various situations of interest and are a natural way for modeling highly localized events occurring randomly at given times. However, the measured impulses are often masked because of unwanted convolution and eventually drowned in noise. Thus, the resulting signal is not legible and may lead to poor or even erroneous analysis and, hence, the need of deconvolution to recover the random periodic impulses. As a matter of fact, periodic random impulse signals are sparse and the sparse coefficients are periodically correlated. It has thus emerged as to how to combine the data structure and the sparsity jointly for a best description. The originality of this study lies in the design of new measures of cyclic sparsity property for the deconvolution of signals that are simultaneously sparse and cyclostationary. To our knowledge, all related works in this field exploit only one property, either sparsity or cyclostationarity and never both properties together. The key feature of cyclic sparsity deconvolution is that it combines the cyclic structure and the sparsity together which implies a significantly enhanced performance. Finally, we include examples of computer simulations to illustrate the behavior in deconvolution context of the proposed algorithms against an $$\ell _1$$l1 sparse deconvolution through convex optimization. We show that deconvolution based on cyclic sparsity hypothesis increases the performance and reduces significantly the computation cost as well.