Sparsity in time-frequency representations

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 C with respect to a random unimodular window can be recovered by Basis Pursuit with high probability provided that S ≤ 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.