Cyclic sparse deconvolution through convex relaxation

It is well known that periodic random impulses are a natural way for modeling highly localized events occuring randomly at given times. Nevertheless, the measured impulses are usually masked due to unwanted convolution and generally drowned in noise. Thus, the resulting signal is not legible and may lead to bad or even erroneous analysis, and hence, the need of deconvolution to retrieve the random periodic impulses. Actually, periodic random impulses signals are sparse with periodically correlated coefficients. It is thus emerged how to combine the data structure and the sparsity simultaneously for a best description. The originality of this communications lies in the proposition of new measures of cyclic sparsity property for the deconvolution of signals that are jointly cyclostationary and sparse as periodic random impulses. As far as we know, all related studies in this field make use 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 which leads to an enhancement of performances. In the end, we presented examples of numerical simulations to show the behavior in deconvolution context of the proposed algorithms against an ℓ₁-sparse deconvolution based on convex optimization. We show that deconvolution based on cyclic sparsity hypothesis leads to satisfactory results.