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Gabor Analysis and Algorithms

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Encyclopedia of Applied and Computational Mathematics

Mathematics Subject Classification

Primary 42C15; 42B35

Footnote 1 Footnote 2

Keywords and Phrases

Gabor frames; Janssen representation; Time-frequency analysis

Motivation

Abstract harmonic analysis explains how to describe the (global) Fourier transform (FT) of signals even over general LCA (locally compact Abelian) groups but typically requires square integrability or periodicity. For the analysis of time-variant signals, an alternative is needed, the so-called sliding window FT or the STFT, the short-time Fourier transform, defined over phase space, the Cartesian, and the product of the time domain with the frequency domain. Starting from a signal f it is obtained by first localizing f in time using a (typically bump-like) window function g followed by a Fourier analysis of the localized part [1]. Another important application of time-frequency analysis is in wireless communication where it helps to design reliable mobile communication systems.

This article presents the key ideas of

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Notes

  1. 1.

    FL was supported by an APART fellowship of the Austrian Academy of Sciences.

  2. 2.

    HGFei was active as a member of the UnlocX EU network when writing this note.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Wien, Vienna, Austria

    Hans Georg Feichtinger

  2. Department of Mathematics University of California, 969 Evans Hall, 94720-3840, Berkeley, CA, USA

    Franz Luef

Authors
  1. Hans Georg Feichtinger
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  2. Franz Luef
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Editor information

Editors and Affiliations

  1. University of Texas at Austin, Austin, TX, USA

    Björn Engquist

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Feichtinger, H.G., Luef, F. (2015). Gabor Analysis and Algorithms. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_354

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