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 $n\times n^2$ Gabor synthesis matrix is small provided $s \leq c , n^{2/3} / \log^2 n$. This improves on previous estimates that exhibit quadratic scaling of $n$ in $s$. Our proof develops bounds for a corresponding chaos process.