The restricted isometry property for time-frequency structured random matrices

We establish the restricted isometry property for finite dimensional Gabor systems, that is, for families of time–frequency shifts of a randomly chosen window function. We show that the s-th order restricted isometry constant of the associated n×n Gabor synthesis matrix is small provided s ≤ c n/ log n. This improves on previous estimates that exhibit quadratic scaling of n in s. Our proof develops bounds for a corresponding chaos process.