  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 $S\leq
Cn/\log(n)$. 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.
