Skip to main content
SpringerLink
Log in

Uncertainty Principles for The Quaternion Linear Canonical Transform

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

Abstract

The (right-sided) Quaternion Linear Canonical transform (QLCT) satisfies some uncertainty principles in a similar way to the Euclidean Fourier transform. The aim of this paper is to prove a quantitative uncertainty inequality about the essential supports of a nonzero function. We also extend signal recovery by using local uncertainty principle to the Quaternion Linear Canonical transform.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bahri, M., Ashino, R.: A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transforms. Abstr. Appl. Anal. 11 (2017). https://doi.org/10.1155/2017/3795120(Article ID 3795120)

    Article  MathSciNet  Google Scholar 

  2. Bahri, M., Ashino, R.: A Simplified proof of uncertainty principle for quaternion linear canonical transform. Abstr. Appl. Anal. 2016. https://doi.org/10.1155/2016/5874930(Article ID 5874930)

    Article  MathSciNet  Google Scholar 

  3. Bahri, M., Resnawati, Musdalifah, S.: A version of uncertainty principle for quaternion linear canonical transform. Abstr. Appl. Anal. 2018 (2018). https://doi.org/10.1155/2018/8732457(Article ID 8732457)

    Article  MathSciNet  Google Scholar 

  4. Bahri, M.: Quaternion linear canonical transform application. Glob. J. Pure Appl. Math. 11(1), 19–24 (2015)

    MathSciNet  Google Scholar 

  5. Cheng, D., Kou, K.I.: Properties of quaternion Fourier transforms. (2016). arXiv:1607.05100v1 [math.CA]

  6. Chen, L.P., Kou, K.I., Liu, M.S.: Pitt’s inequality and the uncertainty principle associated with the Quaternion Linear Canonical transform. J. Math. Anal. Appl. 423(1), 681–700 (2015)

    Article  MathSciNet  Google Scholar 

  7. Cheng, D., Kou, K.I.: Plancherel theorem and quaternion Fourier transform for square integrable functions. Complex Var. Elliptic Equ. 64(2), 223–242 (2019)

    Article  MathSciNet  Google Scholar 

  8. Collins, : Lens-system diffraction integral written in terms of matrix optics. J. Opt. Soc. Am. 60(9), 1168–1177 (1970)

    Article  ADS  Google Scholar 

  9. Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49, 906–931 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  10. Fan, X.L., Kou, K.I., Liu, M.S.: Quaternion Wigner–Ville distribution associated with the linear canonical transforms. Signal Process. 130, 129–141 (2017)

    Article  Google Scholar 

  11. Hitzer, E.M.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Cliff. Algebras 17(3), 497–517 (2007)

    Article  MathSciNet  Google Scholar 

  12. Hitzer, E.: Directional uncertainty principle for quaternion Fourier transforms. Adv. Appl. Clifford Algebras 20(2), 271–84 (2010)

    Article  MathSciNet  Google Scholar 

  13. Koc, A., Ozaktas, H.M., Candan, C., Kutay, M.A.: Digital computation of linear canonical transforms. IEEE Trans. Signal Process. 56(6), 2383–2394 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  14. Kou, K.I., Ou, J., Morais, J.: On uncertainty principle for quaternionic linear canonical transform. Abstr. Appl. Anal.2013 (2013). https://doi.org/10.1155/2013/725952(Article ID 725952)

    Article  MathSciNet  Google Scholar 

  15. Kou, K.I., Yang, Y., Zou, C.: Uncertainty principle for measurable sets and signal recovery in quaternion domains. Math. Methods Appl. Sci. 40(11), 3892–3900 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  16. Kou, K.I., Ou, J., Morais, J.: Uncertainty principles associated with quaternionic linear canonical transforms. Math. Methods Appl. Sci. 39(10), 2722–2736 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  17. Kou, K.I., Morais, J.: Asymptotic behaviour of the quaternion linear canonical transform and the Bochner-Minlos theorem. Appl. Math. Comput. 247, 675–688 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Kou, K.I., Ou, J.Y., Morais, J.: On uncertainty principle for quaternionic linear canonical transform. Abstr. Appl. Anal.2013 (2013). https://doi.org/10.1155/2013/725952(Article ID 725952)

    Article  MathSciNet  Google Scholar 

  19. Liu, Y.L., Kou, K.I., Ho, I.T.: New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms. Signal Process. 90(3), 933–945 (2010)

    Article  Google Scholar 

  20. Maassen, H., Uffink, J.B.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60(12), 1103–1106 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  21. Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representations. J. Math. Phys. 12(8), 1772–1780 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  22. Soltani, F., Ghazwani, J.: A variation of the \(L_p\) uncertainty principles for the Fourier transform. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerbaijan 42(1), 10–24 (2016)

    MathSciNet  MATH  Google Scholar 

  23. Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)

    Article  MathSciNet  Google Scholar 

  24. Wolf, K.B.: Chapter 9: Canonical transforms. In: Integral Transforms in Science and Engineering, vol. 11. Plenum Press, New York (1979)

    Chapter  Google Scholar 

  25. Yang, Y., Kou, K.I.: Novel uncertainty principles associated with 2D quaternion Fourier transforms. Integr. Transform Spec. Funct. 27(3), 213–226 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are deeply indebted to the referees and the editor for providing constructive comments and helps in improving the contents of this article.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences Ain Chock, University Hassan II Casablanca, 20100, Casablanca, Morocco

    A. Achak, A. Abouelaz, R. Daher & N. Safouane

Authors
  1. A. Achak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Abouelaz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Daher
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. Safouane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Achak.

Additional information

Communicated by Eckhard Hitzer

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

Achak, A., Abouelaz, A., Daher, R. et al. Uncertainty Principles for The Quaternion Linear Canonical Transform. Adv. Appl. Clifford Algebras 29, 99 (2019). https://doi.org/10.1007/s00006-019-1020-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-019-1020-x

Keywords

Mathematics Subject Classification

Search

Navigation