10.1007/s40096-022-00462-4

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

  • L. P. Castro*1,
  • L. T. Minh2,
  • N. M. Tuan3,
  1. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, PT
  2. Department of Mathematics, Hanoi Architectural University, Hanoi, VN
  3. Department of Mathematics, VNU University of Education, Viet Nam National University, Hanoi, VN

Published in Issue 2022-03-08

How to Cite

Castro, L. P., Minh, L. T., & Tuan, N. M. (2022). Filter design based on the fractional Fourier transform associated with new convolutions and correlations. Mathematical Sciences, 17(4 (December 2023). https://doi.org/10.1007/s40096-022-00462-4
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

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.

Keywords

  • Convolution,
  • Filter,
  • Signal,
  • Fractional Fourier transform,
  • Fourier transform

References

  1. Almeida (1994) The fractional Fourier transform and time-frequency representations 42(11) (pp. 3084-3091) https://doi.org/10.1109/78.330368
  2. Almeida (1997) Product and convolution theorems for the fractional Fourier transform (pp. 15-17) https://doi.org/10.1109/97.551689
  3. Anh et al. (2017) Two new convolutions for the fractional Fourier transform 92(2) (pp. 623-637) https://doi.org/10.1007/s11277-016-3567-3
  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 et al. (2017) Correlation formulation using relationship between convolution and correlation in linear canonical transform domain (pp. 177-182) https://doi.org/10.1109/ICWAPR.2017.8076685
  6. Bracewell (1986) McGraw-Hill Press
  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 et al. (2015) On integral operators generated by the Fourier transform and a reflection (pp. 7-31)
  9. Castro et al. (2020) New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type 43(7) (pp. 4835-4846)
  10. Castro et al. (2014) Quadratic Fourier transforms 5(1) (pp. 10-23) https://doi.org/10.15352/afa/1391614564
  11. Castro et al. (2018) New convolutions for quadratic-phase Fourier integral operators and their applications 15(13) (pp. 1-17)
  12. Castro and Saitoh (2012) New convolutions and norm inequalities 15(3) (pp. 707-716)
  13. Castro et al. (2021) New convolutions with Hermite weight functions (pp. 365-379) https://doi.org/10.1007/s41980-020-00496-1
  14. Feng and Li (2017) Convolution theorem for fractional cosine-sine transform and its application 40(10) (pp. 3651-3665) https://doi.org/10.1002/mma.4251
  15. Feng and Wang (2020) Fractional convolution, correlation theorem and its application in filter design (pp. 351-358) https://doi.org/10.1007/s11760-019-01563-9
  16. Goel and Singh (2016) Convolution and correlation theorems for the offset fractional Fourier transform and its application 70(2) (pp. 138-150) https://doi.org/10.1016/j.aeue.2015.10.009
  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 and Roopkumar (2020) Multidimensional fractional Fourier transform and generalized fractional convolution 31(2) (pp. 152-165) https://doi.org/10.1080/10652469.2019.1684486
  19. Marsi et al. (2021) A non-linear convolution network for image processing 10(2) https://doi.org/10.3390/electronics10020201
  20. Moshinsky and Quesne (1971) Linear canonical transforms and their unitary representations (pp. 1772-1783) https://doi.org/10.1063/1.1665805
  21. Ozaktas et al. (1994) Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms (pp. 547-559) https://doi.org/10.1364/JOSAA.11.000547
  22. Ozaktas et al. (2001) Wiley
  23. Pei and Ding (2001) Relations between fractional operations and time-frequency distributions and their applications (pp. 1638-1655) https://doi.org/10.1109/78.934134
  24. Pei and Ding (2007) Relations between Gabor transforms and fractional Fourier transforms and their applications for signal processing (pp. 4839-4850) https://doi.org/10.1109/TSP.2007.896271
  25. Pei and Ding (2010) Fractional Fourier transform, Wigner distribution, and filter design for stationary and nonstationary random processes (pp. 4079-4092) https://doi.org/10.1109/TSP.2010.2048206
  26. Smith (1999) California Technical Publishing
  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 et al. (2009) Tsinghua University Press
  29. Wei (2016) Novel convolution and correlation theorems for the fractional Fourier transform 127(7) (pp. 3669-3675) https://doi.org/10.1016/j.ijleo.2015.12.069
  30. Wei and Ran (2013) Multiplicative filtering in the fractional Fourier domain 7(3) (pp. 575-580) https://doi.org/10.1007/s11760-011-0261-5
  31. Wei et al. (2012) A convolution and correlation theorem for the linear canonical transform and its application (pp. 301-312) https://doi.org/10.1007/s00034-011-9319-4
  32. Xiang and Qin (2014) Convolution, correlation, and sampling theorems for the offset linear canonical transform 8(3) (pp. 433-442) https://doi.org/10.1007/s11760-012-0342-0
  33. Zhang et al. (2021) Hybrid nonlinear convolution filters for image recognition (pp. 980-990) https://doi.org/10.1007/s10489-020-01845-7
  34. Zayed (1998) A product and convolution theorems for the fractional Fourier transform 5(4) (pp. 101-103) https://doi.org/10.1109/97.664179
  35. Zayed and García (1999) New sampling formulae for the fractional Fourier transform (pp. 111-114) https://doi.org/10.1016/S0165-1684(99)00064-X