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Finite Frames pp 193–239Cite as

Birkhäuser

Gabor Frames in Finite Dimensions

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Part of the book series: Applied and Numerical Harmonic Analysis ((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.

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Notes

  1. 1.

    Prior to the work of Gabor, von Neumann postulated that the function family which is now referred to as the Gaussian Gabor system is complete [70] (see the respective discussions in [46, 49]).

  2. 2.

    Our treatment is unit-free. The reader may assume that n counts seconds, then m counts hertz, or that n represents milliseconds, in which case m represents megahertz.

  3. 3.

    Here we assume that the receiver knows which coefficients have been erased and which coefficients have been received.

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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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  1. School of Engineering and Science, Jacobs University, 28759, Bremen, Germany

    Götz E. Pfander

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  1. , Department of Mathematics, University of Missouri, 202 Mathematical Sciences Building, Columbia, 65211, Missouri, USA

    Peter G. Casazza

  2. , Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, Berlin, 10623, Berlin, Germany

    Gitta Kutyniok

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Pfander, G.E. (2013). Gabor Frames in Finite Dimensions. In: Casazza, P., Kutyniok, G. (eds) Finite Frames. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8373-3_6

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