Skip to main content
SpringerLink
Log in

Filter design based on the fractional Fourier transform associated with new convolutions and correlations

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

We introduce new convolutions and correlations associated with the Fractional Fourier Transform (FrFT) which present a significant simplicity in both the time and FrFT domains. This allows for several consequences and applications, among which we highlight the design of some multiplicative filters in the FrFT domain having a significant simplicity when compared with the already known ones. Thus, this has consequences, e.g.,  in signal filtering due to the need of modification of a calculated signal to remove undesirable aspects of the signal before it is used in a calculation or a controller. In special, we propose a new filter design implementation which exhibits advantages in comparison to other known ones. Concrete examples are presented to illustrate the theory.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Almeida, L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)

    Article  Google Scholar 

  2. Almeida, L.B.: Product and convolution theorems for the fractional Fourier transform. IEEE Signal Process. Lett. 4, 15–17 (1997)

    Article  Google Scholar 

  3. Anh, P.K., Castro, L.P., Thao, P.T., Tuan, N.M.: Two new convolutions for the fractional Fourier transform. Wireless Personal Communications 92(2), 623–637 (2017)

    Article  Google Scholar 

  4. Anh, P.K., Castro, L.P., Thao, P.T., Tuan, N.M.: Inequalities and consequences of new convolutions for the fractional Fourier transform with Hermite weights, American Institute of Physics, AIP Proceedings, 1798(1), 020006, 10-pp., NY (2017)

  5. Bahri, M., Amir, A.K., Ashino, R.: Correlation formulation using relationship between convolution and correlation in linear canonical transform domain. International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR) 2017, 177–182 (2017). https://doi.org/10.1109/ICWAPR.2017.8076685

    Article  Google Scholar 

  6. Bracewell, R.N.: The Fourier Transform and its Applications. McGraw-Hill Press, New York (1986)

    MATH  Google Scholar 

  7. Bhandari, A., Zayed, A.I.: Convolution and product theorems for the special affine Fourier transform. Nashed, M. Zuhair (ed.) et al., Frontiers in Orthogonal Polynomials and q-Series. Hackensack, NJ: World Scientific. Contemp. Math. Appl., Monogr. Expo. Lect. Notes 1, 119–137 (2018)

  8. Castro, L.P., Guerra, R.C., Tuan, N.M.: On integral operators generated by the Fourier transform and a reflection. Memoirs on Differential Equations and Mathematical Physics 66, 7–31 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Castro, L.P., Guerra, R.C., Tuan, N.M.: New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type. Math. Methods Appl. Sci. 43(7), 4835–4846 (2020)

    MathSciNet  MATH  Google Scholar 

  10. Castro, L.P., Haque, M.R., Murshed, M.M., Saitoh, S., Tuan, N.M.: Quadratic Fourier transforms. Ann. Funct. Anal. 5(1), 10–23 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castro, L.P., Minh, L.T., Tuan, N.M.: New convolutions for quadratic-phase Fourier integral operators and their applications. Mediterranean Journal of Mathematics 15(13), 1–17 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Castro, L.P., Saitoh, S.: New convolutions and norm inequalities. Math. Inequal. Appl 15(3), 707–716 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Castro, L.P., Silva, A.S., Tuan, N.M.: New convolutions with Hermite weight functions. Bull. Iran. Math. Soc. 47, 365–379 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Feng, Q., Li, B.Z.: Convolution theorem for fractional cosine-sine transform and its application. Math. Methods Appl. Sci. 40(10), 3651–3665 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feng, Q., Wang, R.B.: Fractional convolution, correlation theorem and its application in filter design. Signal, Image and Video Processing 14, 351–358 (2020)

    Article  Google Scholar 

  16. Goel, N., Singh, K.: Convolution and correlation theorems for the offset fractional Fourier transform and its application. AEU-Int J Electron Commun 70(2), 138–150 (2016)

    Article  Google Scholar 

  17. He, Z., Fan, Y., Zhuang, J., Dong, Y., Bai, H.: Correlation filters with weighted convolution responses, Proceedings of the IEEE International Conference on Computer Vision (ICCV), 1992–2000 (2017)

  18. Kamalakkannan, R., Roopkumar, R.: Multidimensional fractional Fourier transform and generalized fractional convolution. Integral Transforms Spec. Funct. 31(2), 152–165 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Marsi, S., Bhattacharya, J., Molina, R., Ramponi, G.: A non-linear convolution network for image processing. Electronics 10(2), 201 (2021)

    Article  Google Scholar 

  20. Moshinsky, M., Quesne, C.: Linear canonical transforms and their unitary representations. J. Math. Phys. 12, 1772–1783 (1971)

    Article  MATH  Google Scholar 

  21. Ozaktas, H.M., Barshan, B., Mendlovic, D., Onural, L.: Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms. J. Opt. Soc. Am. A 11, 547–559 (1994)

    Article  Google Scholar 

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

    Google Scholar 

  23. Pei, S.C., Ding, J.J.: Relations between fractional operations and time-frequency distributions and their applications. IEEE Trans. Signal Process. 49, 1638–1655 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pei, S.C., Ding, J.J.: Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing. IEEE Trans. Signal Process. 55, 4839–4850 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pei, S.C., Ding, J.J.: Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes. IEEE Trans. Signal Process. 58, 4079–4092 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Smith, S.W.: The Scientist and Engineer’s Guide to Digital Signal Processing. California Technical Publishing, California (1999)

    Google Scholar 

  27. Shim, S.K., Choi, J.G.: Generalized Fourier-Feynman transforms and generalized convolution products on Wiener space. II, Ann. Funct. Anal. 11, No. 2, 439–457 (2020)

  28. Tao, R., Deng, B., Wang, Y.: Fractional Fourier Transform and its Applications. Tsinghua University Press, Beijing (2009)

    Google Scholar 

  29. Wei, D.Y.: Novel convolution and correlation theorems for the fractional Fourier transform. Int. J. Light Electron Opt. 127(7), 3669–3675 (2016)

    Article  Google Scholar 

  30. Wei, D.Y., Ran, Q.W.: Multiplicative filtering in the fractional Fourier domain. Signal Image Video Process 7(3), 575–580 (2013)

    Article  Google Scholar 

  31. Wei, D., Ran, Q., Li, Y.: A convolution and correlation theorem for the linear canonical transform and its application. Circuits Syst. Signal Process 31, 301–312 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xiang, Q., Qin, K.: Convolution, correlation, and sampling theorems for the offset linear canonical transform. Signal, Image and Video Processing 8(3), 433–442 (2014)

    Article  Google Scholar 

  33. Zhang, X., Wei, K., Kang, X., Jinxiang, L.: Hybrid nonlinear convolution filters for image recognition. Appl Intell 51, 980–990 (2021)

    Article  Google Scholar 

  34. Zayed, A.I.: A product and convolution theorems for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 101–103 (1998)

    Article  Google Scholar 

  35. Zayed, A.I., García, A.G.: New sampling formulae for the fractional Fourier transform. Signal Processing 77, 111–114 (1999)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees who kindly reviewed the earlier version of this manuscript and provided valuable suggestions and comments.

Author information

Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, Portugal

    L. P. Castro

  2. Department of Mathematics, Hanoi Architectural University, Hanoi, Viet Nam

    L. T. Minh

  3. Department of Mathematics, VNU University of Education, Viet Nam National University, Hanoi, Viet Nam

    N. M. Tuan

Authors
  1. L. P. Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. T. Minh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. M. Tuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. P. Castro.

Additional information

Publisher's Note

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

L.P. Castro was supported in part by FCT – Portuguese Foundation for Science and Technology through the Center for Research and Development in Mathematics and Applications (CIDMA) of Universidade de Aveiro, within project UIDB/04106/2020.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Castro, L.P., Minh, L.T. & Tuan, N.M. Filter design based on the fractional Fourier transform associated with new convolutions and correlations. Math Sci 17, 445–454 (2023). https://doi.org/10.1007/s40096-022-00462-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-022-00462-4

Keywords

Mathematics Subject Classification

Search

Navigation