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×n2 Gabor synthesis matrix is small provided s ≤ c n2/3 / log2 n. This improves on previous estimates that exhibit quadratic scaling of n in s. Our proof develops bounds for a corresponding chaos process. Key Words: compressive sensing, restricted isometry property, Gabor system, time-frequency analysis, random matrix, chaos process. AMS Subject classification: 60B20, 42C40, 94A12

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Götz E. Pfander, Holger Rauhut, Joel A. Tropp
• Keywords: CORR 2011, Education, Isometry Property, Restricted Isometry Property, finite Dimensional Gabor