Extension of the Global Matched Filter to structured groups of atoms: Application to harmonic signals

When seeking a sparse representation of a signal on a redundant basis, one might want to convey available a priori information on the observations to the optimization criterion. If one observes a sum of harmonic signals in noise, taking into account the structure of each individual harmonic signal definitely improves the efficiency of an estimator of the fundamental frequencies. More or less efficient or elegant solutions to these problems have been proposed, modifying the penalty term in a sparse representation criterion is one of them. We show how to translate prior information by modifying the penalization term of the usual ℓ2 − ℓ1 regularized criterion, we indicate how to tune the corresponding hyper-parameters by forming the dual of these modified criterion and we evaluate the associated performance on the sum of harmonics example.