Sparse deconvolution of periodic random impulses based on convex relaxation

The framework of structured sparsity which is a natural extension of the standard sparsity concept is a useful tool for studying signals where the sparse coefficients are often dependent and structured. This structure can take different shapes, the interest is focused on signals with cyclic structure as periodic random impulses. Periodic random impulse signals are suitable tools for various situations of interest and are a natural way for modeling highly localized events occuring randomly at given times. Although, the impulses are generally hidden and swallowed up in noise because of undesired convolution. Thus, the resulting signal is not legible and may lead to erroneous analysis, and hence, the need of deconvolution to restore the random periodic impulses. The originality of the present contribution lies in the introduction of a measure of cyclic sparsity property for the deconvolution of signals that are jointly sparse and cyclostationary like periodic random impulses. Indeed, all related works in this area take advantage of 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 to obtain better results. Finally, we use computer simulations to investigate the behavior in deconvolution framework of the proposed algorithm against an 11-sparse deconvolution through convex optimization. We show that cyclic sparse deconvolution enhances the performances.