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Probability Theory and Related Fields

, Volume 156, Issue 3–4, pp 707–737 | Cite as

The restricted isometry property for time–frequency structured random matrices

  • Götz E. Pfander
  • Holger RauhutEmail author
  • Joel A. Tropp
Article

Abstract

This paper establishes the restricted isometry property for a Gabor system generated by n 2 time–frequency shifts of a random window function in n dimensions. The sth order restricted isometry constant of the associated n × n 2 Gabor synthesis matrix is small provided that sc n 2/3 / log2 n. This bound provides a qualitative improvement over previous estimates, which achieve only quadratic scaling of the sparsity s with respect to n. The proof depends on an estimate for the expected supremum of a second-order chaos.

Keywords

Compressed sensing Restricted isometry property Gabor system Time–frequency analysis Random matrix Chaos process 

Mathematics Subject Classification (2000)

60B20 42C40 94A12 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Götz E. Pfander
    • 1
  • Holger Rauhut
    • 2
    Email author
  • Joel A. Tropp
    • 3
  1. 1.School of Engineering and ScienceJacobs University BremenBremenGermany
  2. 2.Hausdorff Center for Mathematics and Institute for Numerical SimulationUniversity of BonnBonnGermany
  3. 3.California Institute of TechnologyPasadenaUSA

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