Filter design based on the fractional Fourier transform associated with new convolutions and correlations
Tóm tắt
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.
Tài liệu tham khảo
Almeida, L.B.: The fractional Fourier transform and time-frequency representations. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)
Almeida, L.B.: Product and convolution theorems for the fractional Fourier transform. IEEE Signal Process. Lett. 4, 15–17 (1997)
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)
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)
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
Bracewell, R.N.: The Fourier Transform and its Applications. McGraw-Hill Press, New York (1986)
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)
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)
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)
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)
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)
Castro, L.P., Saitoh, S.: New convolutions and norm inequalities. Math. Inequal. Appl 15(3), 707–716 (2012)
Castro, L.P., Silva, A.S., Tuan, N.M.: New convolutions with Hermite weight functions. Bull. Iran. Math. Soc. 47, 365–379 (2021)
Feng, Q., Li, B.Z.: Convolution theorem for fractional cosine-sine transform and its application. Math. Methods Appl. Sci. 40(10), 3651–3665 (2017)
Feng, Q., Wang, R.B.: Fractional convolution, correlation theorem and its application in filter design. Signal, Image and Video Processing 14, 351–358 (2020)
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)
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)
Kamalakkannan, R., Roopkumar, R.: Multidimensional fractional Fourier transform and generalized fractional convolution. Integral Transforms Spec. Funct. 31(2), 152–165 (2020)
Marsi, S., Bhattacharya, J., Molina, R., Ramponi, G.: A non-linear convolution network for image processing. Electronics 10(2), 201 (2021)
Moshinsky, M., Quesne, C.: Linear canonical transforms and their unitary representations. J. Math. Phys. 12, 1772–1783 (1971)
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)
Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing. Wiley, New York (2001)
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)
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)
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)
Smith, S.W.: The Scientist and Engineer’s Guide to Digital Signal Processing. California Technical Publishing, California (1999)
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)
Tao, R., Deng, B., Wang, Y.: Fractional Fourier Transform and its Applications. Tsinghua University Press, Beijing (2009)
Wei, D.Y.: Novel convolution and correlation theorems for the fractional Fourier transform. Int. J. Light Electron Opt. 127(7), 3669–3675 (2016)
Wei, D.Y., Ran, Q.W.: Multiplicative filtering in the fractional Fourier domain. Signal Image Video Process 7(3), 575–580 (2013)
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)
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)
Zhang, X., Wei, K., Kang, X., Jinxiang, L.: Hybrid nonlinear convolution filters for image recognition. Appl Intell 51, 980–990 (2021)
Zayed, A.I.: A product and convolution theorems for the fractional Fourier transform. IEEE Signal Process. Lett. 5(4), 101–103 (1998)
Zayed, A.I., García, A.G.: New sampling formulae for the fractional Fourier transform. Signal Processing 77, 111–114 (1999)