Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Uncertainty Principles for Time-Frequency Representations

  • Chapter
Advances in Gabor Analysis

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

Abstract

We present a machine to produce new uncertainty principles from old ones. Traditionally an uncertainty principle is an inequality involving both a functionfand its Fourier transformf.To generate new uncertainty principles, we interpret the pair(f,f)as a time-frequency representation, replace it by a different time-frequency representation, and formulate a corresponding inequality. We discuss a few recent uncertainty principles for the short-time Fourier transform and the Wigner distribution that can be obtained in this way and suggest further problems.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. O. Amrein and A. M. BerthierOn support properties of LP-functions and their Fourier transformsJ. Functional Anal. 24 (1977), no. 3, 258–267.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. C. Bagchi and Swagato K. RayUncertainty principles like Hardy’s theorem on some Lie groupsJ. Austral. Math. Soc. Ser. A 65 (1998), no. 3, 289–302.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. J. Benedetto and H. HeinigFourier transform inequalities with measure weightsAdv. Math. 96 (1992), no. 2, 194–225.

    MathSciNet  MATH  Google Scholar 

  4. M. BenedicksOn Fourier transforms of functions supported on sets of finite Lebesgue measureJ. Math. Anal. Appl.106 (1985), 180–183.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Bonami, Demange B., and Jaming P.Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transform, Preprint (2001).

    Google Scholar 

  6. L. Cohen, Time-frequency distributions—a reviewProc. IEEE77 (1989), no. 7, 941 - 981.

    Google Scholar 

  7. L Cohen, The uncertainty principle for the short-time Fourier transformProc. Int. Soc. Opt. Eng.22563 (1995), 80–90.

    Google Scholar 

  8. M. G. Cowling and J. F. PriceBandwidth versus time concentration: the Heisenberg—Pauli—Weyl inequalitySIAM J. Math. Anal. 15 (1984), no. 1, 151–165.

    MathSciNet  Google Scholar 

  9. D. L. Donoho and P. B. StarkUncertainty principles and signal recoverySIAM J. Appl. Math. 49 (1989), no. 3, 906–931.

    MathSciNet  MATH  Google Scholar 

  10. H. Dym and H.P. McKeanFourier series and integralsProbab. Math. Statist., vol. 14, Academic Press, Boston, 1985.

    Google Scholar 

  11. H. G. FeichtingerOn a new Segal algebraMonatsh. Math. 92 (1981), no. 4, 269–289.

    MathSciNet  MATH  Google Scholar 

  12. H. G. Feichtinger and G. ZimmermannA Banach space of test functions for Gabor analysisGabor analysis and algorithms, Birkhäuser Boston, Boston, MA, 1998, pp. 123–170.

    Google Scholar 

  13. P. FlandrinTemps-fréquence2nd ed., Collection traitement du signal, Hermes, 1993.

    Google Scholar 

  14. G. B. FollandHarmonic analysis in phase spacePrinceton Univ. Press, Princeton, NJ, 1989.

    Google Scholar 

  15. G. B. Folland and A. SitaramThe uncertainty principle: a mathematical survey J. Fourier Anal. Appl. 3 (1997), no. 3, 207–238.

    MathSciNet  MATH  Google Scholar 

  16. E. GalperinUncertainty principles as embeddings of modulation spacesPh.D. thesis, Univ. of Connecticut, 2000.

    Google Scholar 

  17. K. GröchenigAn uncertainty principle related to the Poisson summation formulaStudia Math. 121 (1996), no. 1, 87–104.

    MathSciNet  MATH  Google Scholar 

  18. K. GröchenigFoundations of time-frequency analysis, BirkhäuserBoston, 2001.

    Google Scholar 

  19. K. Gröchenig and G. ZimmermannHardy’s theorem and the short-time Fourier transform of Schwartz functionsJ. London Math. Soc. 63 (2001), 205–214.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. H. HardyA theorem concerning Fourier transformsJ. London Math. Soc. 8 (1933), 227–231.

    Article  Google Scholar 

  21. V. Havin and B. JörickeThe uncertainty principle in harmonic analysisSpringer-Verlag, Berlin, 1994.

    Book  MATH  Google Scholar 

  22. F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and quadratic time-frequency signal representationsIEEE Signal Proc. Magazine9 (1992), no. 2, 21 - 67.

    Google Scholar 

  23. J. A. Hogan and J. D. LakeyEmbeddings and uncertainty principles for generalized modulation spacesModern Sampling Theory — Mathematics and Applications (J. J. Benedetto and P. Ferreira, eds.), Birkhäuser, 2001, pp. 73–106.

    Google Scholar 

  24. Lars Hörmander, A uniqueness theorem of Beurling for Fourier transform pairsArk. Mat.29 (1991), no. 2, 237–240.

    MATH  Google Scholar 

  25. P. JamingPrincipe d’incertitude qualitatif et reconstruction de phase pour la transformée de Wigner C. R. Acad. Sci. Paris Sér. I Math.327 (1998), 249–254.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. J. E. M. Janssen, Personal communication.

    Google Scholar 

  27. A. J. E. M. JanssenProof of a conjecture on the supports of Wigner distributionsJ. Fourier Anal. Appl.4 (1998), no. 6, 723–726.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Lakey, Personal communication.

    Google Scholar 

  29. E. H. LiebIntegral bounds for radar ambiguity functions and Wigner distributionsJ. Math. Phys.31 (1990), no. 3, 594–599.

    Google Scholar 

  30. E. WilczockZur Funktionalanalysis der Wavelet-und GabortransformationThesis, TU München, 1998.

    Google Scholar 

Download references

Authors
  1. Karlheinz Gröchenig
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Vienna, 1090, Vienna, Austria

    Hans G. Feichtinger

  2. Department of Mathematics, University of California—Davis, Davis, 95616, CA, USA

    Thomas Strohmer

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Science+Business Media New York

About this chapter

Cite this chapter

Gröchenig, K. (2003). Uncertainty Principles for Time-Frequency Representations. In: Feichtinger, H.G., Strohmer, T. (eds) Advances in Gabor Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0133-5_2

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-0133-5_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6627-3

  • Online ISBN: 978-1-4612-0133-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation