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) , Orthogonal Matching Pursuit (OMP) , Orthogonal Least Square (OLS) , Single Best Replacement (SBR), [19, 20, 21] and the proposed extensions to cyclic sparsity context: Cyclo-MP, Cyclo-OMP, Cyclo-OLS and Cyclo-SBR.