A First Step to Convolutive Sparse Representation

In this paper an extension of the sparse decomposition problem is considered and an algorithm for solving it is presented. In this extension, it is known that one of the shifted versions of a signal s (not necessarily the original signal itself) has a sparse representation on an overcomplete dictionary, and we are looking for the sparsest representation among the representations of all the shifted versions of s. Then, the proposed algorithm finds simultaneously the amount of the required shift, and the sparse representation. Experimental results emphasize on the performance of our algorithm.