Sparsity in time-frequency representations

Abstract

We consider signals and operators in finite dimension which have sparse time-frequency representations. As main result we show that an $S$-sparse Gabor representation in ${\mathbb{C}}^n$ with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that $S\le Cn/\log (n)$. Our results are applicable to the channel estimation problem in wireless communications and they establish the usefulness of a class of measurement matrices for compressive sensing.