Advertisement

The multiple-parameter fractional Fourier transform

  • Jun Lang
  • Ran Tao
  • QiWen Ran
  • Yue Wang
Article

Abstract

The fractional Fourier transform (FRFT) has multiplicity, which is intrinsic in fractional operator. A new source for the multiplicity of the weight-type fractional Fourier transform (WFRFT) is proposed, which can generalize the weight coefficients of WFRFT to contain two vector parameters \( \mathfrak{M},\mathfrak{N} \in \mathbb{Z}^M \). Therefore a generalized fractional Fourier transform can be defined, which is denoted by the multiple-parameter fractional Fourier transform (MPFRFT). It enlarges the multiplicity of the FRFT, which not only includes the conventional FRFT and general multi-fractional Fourier transform as special cases, but also introduces new fractional Fourier transforms. It provides a unified framework for the FRFT, and the method is also available for fractionalizing other linear operators. In addition, numerical simulations of the MPFRFT on the Hermite-Gaussian and rectangular functions have been performed as a simple application of MPFRFT to signal processing.

Keywords

multiple-parameter fractional Fourier transform weight-type fractional Fourier transform multiplicity of the fractional Fourier transform signal processing 

Reference

  1. 1.
    Erden M F, Kutay M A, Ozaktas H M. Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration. IEEE Trans Signal Process, 1999, 47(5): 1458–1462CrossRefGoogle Scholar
  2. 2.
    Zalevsky Z, Mendlovic D. Fractional Wiener filter. Appl Opt, 1996, 35: 3930–3936CrossRefGoogle Scholar
  3. 3.
    Alieva T, Lopez V, F, Lopez A, et al. The angular Fourier transform in optical propagation problems. J Mod Opt, 1994, 41: 1037–1040CrossRefGoogle Scholar
  4. 4.
    Ozaktas H M, Barshan B, Mendlovic D, et al. Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms. J Opt Soc Amer A, 1994, 11: 547–559MathSciNetCrossRefGoogle Scholar
  5. 5.
    Unnikrishnan G, Joseph J, Singh K. Optical encryption by double-random phase encoding in the fractional Fourier domain. Opt Lett, 2000, 25: 887–889CrossRefGoogle Scholar
  6. 6.
    Qi L, Tao R, Zhou S Y, et al. Detection and parameter estimation of multicomponent LFM signal based on the fractional Fourier transform. Sci China Ser F-Inf Sci, 2004, 47(2): 184–198zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Tao R, Deng B, Wang Y. Research progress of the fractional Fourier transform in signal processing. Sci China Ser F-Inf Sci, 2006, 49(1): 1–25zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Deng B, Tao R, Wang Y. Convolution theorems for the linear canonical transform and their applications. Sci China Ser F-Inf Sci, 2006, 49(5): 592–603CrossRefMathSciNetGoogle Scholar
  9. 9.
    Tao R, Li B Z, Wang Y. Spectral analysis and reconstruction for periodic non-uniformly sampled signals in fractional Fourier domain. IEEE Trans Signal Process, 2007, 55(7): 3541–3547CrossRefMathSciNetGoogle Scholar
  10. 10.
    Li B Z, Tao R, Wang Y. New sampling formulae related to the linear canonical transform. Signal Process, 2007, 87: 983–990CrossRefGoogle Scholar
  11. 11.
    Meng X Y, Tao R, Wang Y. The fractional Fourier domain analysis of decimation and interpolation. Sci China Ser F-Inf Sci, 2007, 50(4): 521–538zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Tao R, Deng B, Zhang W Q, et al. Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain. IEEE Trans Signal Process, 2008, 56(1): 158–171CrossRefMathSciNetGoogle Scholar
  13. 13.
    Namias V. The fractional Fourier transform and its application to quantum mechanics. J Inst Math Appl, 1980, 25: 241–265zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    McBride A C, Kerr F H. On Namias fractional Fourier transforms. IMA J Appl Maths,1987, 39: 159–175zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Almeida L B. The fractional Fourier transform and time-frequency representations. IEEE Trans Signal Process, 1994, 42(11): 3084–3091CrossRefGoogle Scholar
  16. 16.
    Shih C. Fractionalization of Fourier transform. Opt Comm, 1995, 48: 495–498CrossRefGoogle Scholar
  17. 17.
    Liu S, Zhang J, Zhang Y. Properties of the fractionalization of a Fourier transform. Opt Comm, 1997, 133(1): 50–54CrossRefGoogle Scholar
  18. 18.
    Liu S, Jiang J, Zhang Y, et al. Generalized fractional Fourier transforms. J Phys A: Math Gen, 1997, 30: 973–981zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ran Q, Yueng D, Tseng E, et al. Multifractional Fourier transform method based on the generalized permutation matrix group. IEEE Trans Signal Process, 2005, 53(1): 83–98CrossRefMathSciNetGoogle Scholar
  20. 20.
    Cariolaro G, Erseghe T, Kraniauskas P, et al. A unified framework for the fractional Fourier transform. IEEE Trans Signal Process, 1998, 46(12): 3206–3212zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Erseghe T, Kraniauskas P, Cariolaro G. Unified fractional Fourier transform and sampling theorem. IEEE Trans Signal Process, 1999, 47(12): 3419–3423zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Cariolaro G, Erseghe T, Kraniauskas P, et al. Multiplicity of fractional Fourier transforms and their relationships. IEEE Trans Signal Process, 2000, 48(1): 227–241zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Tao R, Lang J, Wang Y. Optical image encryption based on the multiple-parameter fractional Fourier transform. Opt Lett, 2008, 33: 581–583CrossRefGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH 2008

Authors and Affiliations

  1. 1.Department of Electronic EngineeringBeijing Institute of TechnologyBeijingChina
  2. 2.Department of MathematicsHarbin Institute of TechnologyHarbinChina

Personalised recommendations