Skip to main content
SpringerLink
Log in

A Novel Approach for Defining a Hilbert Number Transform

  • Short Paper
  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

In this brief, we introduce a new definition for the Hilbert number transform (HNT). Our approach employs concepts related to trigonometry over finite fields and uses as a starting point a specific definition for the Fourier number transform (FNT); one demonstrates that such an FNT allows to define an HNT which is analogous to the real-valued version of the corresponding transform. Additionally, we present a method to obtain the proposed HNT from the recently proposed steerable Fourier number 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

Data Availability

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

Notes

  1. In fact, an element \(\omega \) lying in an extension field \({\mathbb {F}}_q\), where q is a prime power, can be used to define a Fourier transform in a finite field. Such a possibility corresponds to a generalization which is not considered in this paper.

References

  1. R.E. Blahut, Fast Algorithms for Signal Processing (Cambridge University Press, Cambridge, 2010)

    Book  Google Scholar 

  2. R.M. Campello de Souza, H.M. de Oliveira, A.N. Kauffman, A.J.A. Paschoal, Trigonometry in finite fields and a new Hartley transform, in Proceedings of the IEEE International Symposium on Information Theory, p. 293 (1998). https://doi.org/10.1109/ISIT.1998.708898

  3. R.J. Cintra, V.S. Dimitrov, H.M. de Oliveira, R.M. Campello de Souza, Fragile watermarking using finite field trigonometrical transforms. Signal Process. Image Commun. 24(7), 587–597 (2009)

    Article  Google Scholar 

  4. G. Frigo, A. Derviskadic, M. Paolone, Reduced leakage synchrophasor estimation: Hilbert transform plus interpolated DFT. IEEE Trans. Instrum. Meas. 68(10), 1 (2019)

    Article  Google Scholar 

  5. M.A. Gondim, J.R. de Oliveira Neto, J.B. Lima, Steerable Fourier number transform with application to image encryption. Signal Process. Image Commun. 74, 89–95 (2019)

    Article  Google Scholar 

  6. S. Kak, The discrete finite Hilbert transform. Indian J. Pure Appl. Math. 8, 1385–1390 (1977)

    MathSciNet  MATH  Google Scholar 

  7. S. Kak, The number theoretic Hilbert transform. Circuits Syst. Signal Process. 33(8), 2539–2548 (2014)

    Article  MathSciNet  Google Scholar 

  8. J.B. Lima, Fast algorithm for computing cosine number transform. Electron. Lett. 51(20), 1570–1572 (2015)

    Article  Google Scholar 

  9. J.B. Lima, R.M. Campello de Souza, Tangent function and Chebyshev-like rational maps over finite fields. IEEE Trans. Circuits Syst. II Express Briefs 67(4), 775–779 (2020)

    Article  Google Scholar 

  10. J.B. Lima, R.M. Campello Souza, Finite field trigonometric transforms. Appl. Algebra Eng. Commun. Comput. 22(5–6), 393–411 (2011)

    Article  MathSciNet  Google Scholar 

  11. J.B. Lima, L.F.G. Novaes, Image encryption based on the fractional Fourier transform over finite fields. Signal Process. 94, 521–530 (2014)

    Article  Google Scholar 

  12. J.B. Lima, D. Panario, R.M. Campello de Souza, A trigonometric approach for Chebyshev polynomials over finite fields, in Applied Algebra and Number Theory, pp. 255–279. Cambridge University Press (2014)

  13. P.H.E.S. Lima, J.B. Lima, R.M. Campello de Souza, Fractional Fourier, Hartley, cosine and sine number-theoretic transforms based on matrix functions. Circuits Syst. Signal Process. 36(7), 2893–2916 (2016)

    Article  Google Scholar 

  14. M. Mikhail, Y. Abouelseoud, G. ElKobrosy, Two-phase image encryption scheme based on FFCT and fractals. Secur. Commun. Netw. 1–13 (2017)

  15. R. Onodera, Y. Yamamoto, Y. Ishii, Signal processing of interferogram using a two-dimensional discrete Hilbert transform, in Fringe 2005. ed. by W. Osten (Springer, Berlin, 2006), pp. 82–89

    Chapter  Google Scholar 

  16. A.V. Oppenheim, R.W. Schafer, Discrete-Time Signal Processing, 3rd edn. (Pearson, Hoboken, 2010)

    MATH  Google Scholar 

  17. A. Pedrouzo-Ulloa, J.R. Troncoso-Pastoriza, F. Pérez-González, Number theoretic transforms for secure signal processing. IEEE Trans. Inf. Forensics Secur. 12(5), 1125–1140 (2017)

    Article  Google Scholar 

  18. J.M. Pollard, The fast Fourier transform in a finite field. Math. Comput. 25(114), 365–374 (1971)

    Article  MathSciNet  Google Scholar 

  19. R. Ponuma, V. Aarthi, R. Amutha, Cosine number transform based hybrid image compression-encryption, in International Conference on Wireless Communications, Signal Processing and Networking (WiSPNET). (IEEE, Chennai, 2016), pp. 172–176

  20. N. Ramesh, S. Deepa, P.V. Ranjan, The mystery curve: a signal processing based power quality disturbance detection. IEEE Trans. Ind. Electron. Early Access, 1385–1390 (2020)

  21. I.S. Reed, T.K. Truong, The use of finite fields to compute convolutions. IEEE Trans. Inf. Theory 21(2), 208–213 (1975)

    Article  MathSciNet  Google Scholar 

  22. W. Shu, Y. Tianren, Algorithm for linear convolution using number theoretic transforms. Electron. Lett. 24(5), 249–250 (1988)

    Article  Google Scholar 

  23. T. Toivonen, J. Heikkila, Video filtering with Fermat number theoretic transforms using residue number system. IEEE Trans. Circuits Syst. Video Technol. 16(1), 92–101 (2006)

    Article  Google Scholar 

Download references

Acknowledgements

Juliano B. Lima is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq – under Grants 309598/2017-6 and 409543/2018-7.

Author information

Authors and Affiliations

  1. Department of Electronics and Systems, Federal University of Pernambuco, Recife, Brazil

    Juliano B. Lima, Marcos A. A. Gondim & Ricardo M. Campello de Souza

Authors
  1. Juliano B. Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcos A. A. Gondim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ricardo M. Campello de Souza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juliano B. Lima.

Additional information

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

Lima, J.B., Gondim, M.A.A. & Campello de Souza, R.M. A Novel Approach for Defining a Hilbert Number Transform. Circuits Syst Signal Process 41, 1776–1787 (2022). https://doi.org/10.1007/s00034-021-01834-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-021-01834-2

Keywords

Search

Navigation