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Finite Frames pp 193-239 | Cite as

Gabor Frames in Finite Dimensions

  • Götz E. PfanderEmail author
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Gabor frames have been extensively studied in time-frequency analysis over the last 30 years. They are commonly used in science and engineering to synthesize signals from, or to decompose signals into, building blocks which are localized in time and frequency. This chapter contains a basic and self-contained introduction to Gabor frames on finite-dimensional complex vector spaces. In this setting, we give elementary proofs of the central results on Gabor frames in the greatest possible generality; that is, we consider Gabor frames corresponding to lattices in arbitrary finite Abelian groups. In the second half of this chapter, we review recent results on the geometry of Gabor systems in finite dimensions: the linear independence of subsets of its members, their mutual coherence, and the restricted isometry property for such systems. We apply these results to the recovery of sparse signals, and discuss open questions on the geometry of finite-dimensional Gabor systems.

Keywords

Gabor analysis on finite Abelian groups Linear independence Coherence Restricted isometry constants of Gabor frames Applications to compressed sensing Erasure channel error correction Channel identification 

Notes

Acknowledgements

The author acknowledges support under the Deutsche Forschungsgemeinschaft (DFG) grant 50292 DFG PF-4 (Sampling of Operators).

Parts of this paper were written during a sabbatical of the author at the Research Laboratory for Electronics and the Department of Mathematics at the Massachusetts Institut of Technology. He is grateful for the support and the stimulating research environment.

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Authors and Affiliations

  1. 1.School of Engineering and ScienceJacobs UniversityBremenGermany

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