Skip to main content
SpringerLink
Log in

Inequalities Pertaining to Quaternion Ambiguity Function

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

Abstract

The quaternion ambiguity function is an expansion of the standard ambiguity function using quaternion algebra. Various properties such as linearity, translation, modulation, Moyal’s formula and inversion identity are studied in detail. In addition, an interesting interaction between the quaternion ambiguity function and the quaternion Fourier transform is demonstrated. Based on these facts, we seek for several versions of the uncertainty inequalities associated with the proposed quaternion ambiguity function.

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

Availability of data and materials

Data sharing is not applicable to this article, as no datasets were generated or analyzed during the current study.

References

  1. Bahri, M.: On two-dimensional quaternion Wigner–Ville distribution. J. Appl. Math. (2014). https://doi.org/10.1155/2014/139471

    Article  MathSciNet  Google Scholar 

  2. Bahri, M., Karim, S.A.A.: A variation on inequality for quaternion Fourier transform, modified convolution and correlation theorems for general quaternion linear canonical transform. Symmetry 14(7), 1303 (2022)

    Article  ADS  Google Scholar 

  3. Bau, M.A., Bahri, M., Bachtiar, N., Busrah, S.N., Nur, M.: One-dimensional quaternion Laplace transform: properties and its application to quaternion-valued differential equations. Partial Differ. Equ. Appl. Math. 8, 100547 (2023)

    Article  Google Scholar 

  4. Bracewell, R.: The Fourier Transform and Its Applications. McGraw Hill, Boston (2000)

    Google Scholar 

  5. Bülow, T.: Hypercomplex spectral signal representations for the processing and analysis of images. Ph.D. thesis, University of Kiel, Germany (1999)

  6. Cai, Z.F., Kou, K.I.: Laplace transform: a new approach in solving linear quaternion differential equation. Math. Methods Appl. Sci. 41(11), 4033–4048 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Debnath, L., Shah, F.A.: Wavelet Transform and Their Basic Properties. Birkhauser, Boston (2015)

    Book  Google Scholar 

  9. Dong 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 

  10. Dou, Y., Li, S.: Kernel function-based ambiguity function and its application on DOA estimation in impulsive noise. Sensors 22(18), 6996 (2022)

    Article  ADS  Google Scholar 

  11. Ekasasmita, W., Bahri, M., Bachtiar, N., Rahim, A., Nur, M.: One-dimensional quaternion Fourier transform with application to probability theory. Symmetry 15(4), 815 (2023)

    Article  ADS  Google Scholar 

  12. El Haoui, Y., Fahlaoui, S.: Beurling’s theorem for the quaternion Fourier transform. J. Pseudo Differ. Oper. Appl. 11, 187199 (2020). https://doi.org/10.1007/s11868-019-00281-7

    Article  MathSciNet  Google Scholar 

  13. El Haoui, Y., Fahlaoui, S.: Miyachi’s theorem for the quaternion Fourier transform. Circuits Syst. Signal Process. 39, 2193–2206 (2020)

    Article  Google Scholar 

  14. Ell, T.A.: Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceeding of the 32nd Conference on Decision and Control, IEEE Control Systems Society, pp. 1830–1841 (1993)

  15. Ell, T.A., Sangwine, S.J.: Hypercomplex Fourier transform of color images. IEEE Trans. Signal Process. 16(1), 22–35 (2007)

    ADS  MathSciNet  Google Scholar 

  16. Ell, T.A., Le Bihan, N., Sangwine, S.J.: Quaternion Fourier Transforms for Signal and Image Processing. Wiley, London (2014)

    Book  Google Scholar 

  17. Grigoryan, A.M., Jenkinson, J., Agaian, S.S.: Quaternion Fourier transform based alpha-rooting method for color image measurement and enhancement. Signal Process. 109, 269–289 (2015)

    Article  Google Scholar 

  18. Gröchenig, K.: Foundation of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  19. Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebras 20(3), 497–517 (2007)

    Article  MathSciNet  Google Scholar 

  20. Khalil, M.I.: Applying quaternion Fourier transforms for enhancing color images. Int. J. Image Graph. Signal Process. 2, 9–15 (2012)

    Article  Google Scholar 

  21. Lian, P.: Uncertainty principle for the quaternion Fourier transform. J. Math. Anal. Appl. 467(2), 1258–1269 (2018)

    Article  MathSciNet  Google Scholar 

  22. Lian, P.: Sharp Hausdorff–Young inequalities for the quaternion Fourier transforms. Proc. Am. Math. Soc. (2019). https://doi.org/10.1090/proc/14735

    Article  Google Scholar 

  23. Zhu, X., Zheng, S.: Uncertainty principles for the two-sided offset quaternion linear canonical transform. Math. Methods Appl. Sci. 44(18), 14236–14255 (2021)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This paper is supported by Ministry of Education, Culture, Research and Technology under WCR Scheme.

Author information

Authors and Affiliations

  1. Department of Mathematics, Hasanuddin University, Makassar, 90245, Indonesia

    Imanuel Agung Sembe, Mawardi Bahri, Nasrullah Bachtiar & Muhammad Zakir

Authors
  1. Imanuel Agung Sembe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mawardi Bahri
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nasrullah Bachtiar
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Muhammad Zakir
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mawardi Bahri.

Ethics declarations

Conflict of interest

The authors declare that there are no Conflict of interest.

Additional information

Communicated by Paula Cerejeiras.

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sembe, I.A., Bahri, M., Bachtiar, N. et al. Inequalities Pertaining to Quaternion Ambiguity Function. Adv. Appl. Clifford Algebras 34, 15 (2024). https://doi.org/10.1007/s00006-024-01320-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-024-01320-3

Keywords

Mathematics Subject Classification

Search

Navigation