Skip to main content
Springer Nature Link
Log in

A Simple Observation on the Uncertainty Principle for the Fractional Fourier Transform

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

The aim of this letter is to show that uncertainty principles for the pair \(u,{\mathcal F}[u]\) (\({\mathcal F}\) the Fourier transform) can be transferred without much effort to the pair \({\mathcal F}_\alpha [u],{\mathcal F}_\beta [u]\) (\({\mathcal F}_\alpha \) the Fractional Fourier transform) provided \(\beta -\alpha \notin \pi {\mathbb {Z}}\). This letter is essentially of a tutorial nature and aims at avoiding that people waste their efforts on adaptations of the proofs from classical Fourier analysis to the fractional setting.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Notes

  1. Of course, not all results for the Fractional Fourier transform can be obtained this way. The work [14, 15] is a good example.

References

  1. Almeida, L.B.: The fractional Fourier transform and time–frequency representations. IEEE Trans. Sign. Proc. 42, 3084–3091 (1994)

    Article  Google Scholar 

  2. Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102, 159–182 (1975)

    Article  MathSciNet  Google Scholar 

  3. Bonami, A., Demange, B.: A survey on uncertainty principles related to quadratic forms. Collect. Math. Vol. Extra, 1–36 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Bonami, A., Demange, B., Jaming, P.: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19, 23–35 (2003)

    Article  MathSciNet  Google Scholar 

  5. Bultheel, A., Martínez-Sulbaran, H.: Recent developments in the theory of the fractional Fourier and linear canonical transforms. Bull. Belg. Math. Soc. Simon Stevin 13, 971–100 (2007)

    Article  MathSciNet  Google Scholar 

  6. Condon, N.U.: Immersion of the Fourier transform in a continuous group of functional transformations. Proc. Natl. Acad. Sci. USA 23, 158–164 (1937)

    Article  Google Scholar 

  7. Cowling, M.G., Price, J.F.: Bandwidth versus time concentration: The Heisenberg–Pauli–Weyl inequality. SIAM J. Math. Anal. 15, 151–165 (1984)

    Article  MathSciNet  Google Scholar 

  8. de Gosson, M.A., Luef, F.: Principe d’incertitude et positivité des opérateurs à trace; Applications à l’opérateur densité. Ann. Inst. H. Poincaré 9, 329–346 (2008)

    Article  Google Scholar 

  9. de Gosson, M.A., Luef, F.: Metaplectic group, symplectic Cayley transform, and fractional Fourier transforms. J. Math. Anal. Appl. 416, 947–968 (2014)

    Article  MathSciNet  Google Scholar 

  10. Faris, W.G.: Inequalities and uncertainty inequalities. J. Math. Phys. 19, 461–466 (1978)

    Article  Google Scholar 

  11. Folland, G.B., Sitaram, A.: The uncertainty principle: A mathematical survey. J. Fourier Anal. Appl. 3, 207–208 (1997)

    Article  MathSciNet  Google Scholar 

  12. Guanlei, X., Xiaotong, W., Xiaogang, X.: The logarithmic, Heisenberg’s and short-time uncertainty principles associated with fractional Fourier transform. Signal Process. 89, 339–343 (2009)

    Article  Google Scholar 

  13. Hirschman, I.I., Jr.: A note on entropy. Am. J. Math. 79, 152–156 (1957)

    Article  MathSciNet  Google Scholar 

  14. Hogan, J.A., Lakey, J.D.: On uncertainty bounds and growth estimates for fractional Fourier transforms. Appl. Anal. 85, 891–899 (2006)

    Article  MathSciNet  Google Scholar 

  15. Hogan, J.A., Lakey, J.D.: Hardy’s theorem and rotations. Proc. Am. Math. Soc. 134, 1459–1466 (2006)

    Article  MathSciNet  Google Scholar 

  16. Jaming, P.: Uniqueness results in an extension of Pauli’s phase retrieval. Appl. Comput. Harm. Anal. 37, 413–441 (2014)

    Article  MathSciNet  Google Scholar 

  17. Namias, V.: The fractional order Fourier transform and its applications to quantum mechanics. J. Inst. Math. Appl. 25, 241–265 (1980)

    Article  MathSciNet  Google Scholar 

  18. Nazarov, F.L.: Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type. Algebra i Analiz 5(1994), 663–717 (1993). (Russian)

    MathSciNet  Google Scholar 

  19. Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, Hoboken (2001)

    Google Scholar 

  20. Price, J.F.: Inequalities and local uncertainty principles. J. Math. Phys. 24, 1711–1714 (1983)

    Article  MathSciNet  Google Scholar 

  21. Price, J.F.: Sharp local uncertainty inequalities. Stud. Math. 7, 37–45 (1987)

    Article  MathSciNet  Google Scholar 

  22. Ricaud, B., Torrésani, B.: A survey of uncertainty principles and some signal processing applications. Adv. Comput. Math. 40, 629–650 (2013)

    Article  MathSciNet  Google Scholar 

  23. Ricaud, B., Torrésani, B.: Refined support and entropic uncertainty inequalities. IEEE Trans. Inform. Theory 59, 4272–4279 (2013)

    Article  MathSciNet  Google Scholar 

  24. Shi, J., Liu, X., Zhang, N.: On uncertainty principle for signal concentrations with fractional Fourier transform. Signal Process. 92, 2830–2836 (2012)

    Article  Google Scholar 

  25. Shinde, S., Gadre, V.M.: On uncertainty principle for real signals in the fractional Fourier transform domain. IEEE Trans. Signal Process. 49, 2545–2548 (2001)

    Article  MathSciNet  Google Scholar 

  26. Wiener, N.: Hermitian polynomials and Fourier analysis. J. Math. Phys. 8, 70–73 (1929)

    Article  Google Scholar 

  27. Yang, X.-J., Baleanu, D., Tenreiro, M.: Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. Bound. Value Probl. 132, 16 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Zayed, A.I.: On the relationship between the Fourier and fractional Fourier transforms. IEEE Signal Process. Lett. 3, 310–11 (1996)

    Article  Google Scholar 

  29. Zhang, Y., et al.: A comprehensive survey on fractional Fourier transform. Fund. Inform. 151, 1–48 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Univ. Bordeaux, IMB, UMR 5251, 33400, Talence, France

    Philippe Jaming

  2. CNRS, IMB, UMR 5251, 33400, Talence, France

    Philippe Jaming

Authors
  1. Philippe Jaming
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Jaming.

Additional information

Communicated by Veluma Thangavelu.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jaming, P. A Simple Observation on the Uncertainty Principle for the Fractional Fourier Transform. J Fourier Anal Appl 28, 51 (2022). https://doi.org/10.1007/s00041-022-09946-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00041-022-09946-2

Search

Navigation