Advertisement

Science China Information Sciences

, Volume 53, Issue 11, pp 2287–2299 | Cite as

The discrete multiple-parameter fractional Fourier transform

  • Jun Lang
  • Ran Tao
  • Yue Wang
Research Papers

Abstract

As a generalization of the Fourier transform (FT), the fractional Fourier transform (FRFT) has many applications in the areas of optics, signal processing, information security, etc. Therefore, the efficient discrete computational method is the vital fundament for the application of the fractional Fourier transform. The multiple-parameter fractional Fourier transform (MPFRFT) is a generalized fractional Fourier transform, which not only includes FRFT as special cases, but also provides a unified framework for the study of FRFT. In this paper, we present in detail the discretization method of the MPFRFT and define the discrete multiple-parameter fractional Fourier transform (DMPFRFT). Then, we utilize the tensor product to define two-dimensional multiple-parameter fractional Fourier transform (2D-MPFRFT) and the corresponding two-dimensional discrete multiple-parameter fractional Fourier transform (2D-DMPFRFT). Finally, as an application, a novel image encryption method based on 2D-DMPFRFT is proposed. Numerical simulations are performed to demonstrate that the proposed method is reliable and more robust to blind decryption than several existing methods.

Keywords

multiple-parameter fractional Fourier transform fractional Fourier transform image encryption information security 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Namias V. The fractional Fourier transform and its application to quantum mechanics. J Inst Math Appl, 1980, 25: 241–265zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Almeida L B. The fractional Fourier transform and time-frequency representations. IEEE Trans Signal Process, 1994, 42: 3084–3091CrossRefGoogle Scholar
  3. 3.
    Shih C. Fractionalization of Fourier transform. Opt Comm, 1995, 48: 495–498CrossRefGoogle Scholar
  4. 4.
    Yang Q, Tao R, Wang Y. MIMO-OFDM system based on fractional Fourier transform and selecting algorithm for optimal order. Sci China Ser F-Inf Sci, 2008, 51: 1360–1371zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ran Q, Tan L. Introduction to Fractional Fourier Optics (in Chinese). Beijing: Science Press, 2004Google Scholar
  6. 6.
    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
  7. 7.
    Zhu B, Liu S, Ran Q. Optical image encryption based on multifractional Fourier transforms. Opt Lett, 2000, 25: 1159–1161CrossRefGoogle Scholar
  8. 8.
    Hennelly B, Sheridan J. Optical image encryption by random shifting in fractional Fourier domains. Opt Lett, 2003, 28: 269–271CrossRefGoogle Scholar
  9. 9.
    Tao R, Lang J, Wang Y. The multiple-parameter discrete fractional Hadamard transform. Opt Comm, 2009, 282: 1531–1535CrossRefGoogle Scholar
  10. 10.
    Tao R, Lang J, Wang Y. Optical image encryption based on the multiple-parameter fractional Fourier transform. Opt Lett, 2008, 33: 581–583CrossRefGoogle Scholar
  11. 11.
    Lang J, Tao R, Wang Y. Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function. Opt Comm, 2010, 283: 2092–2096CrossRefGoogle Scholar
  12. 12.
    Tao R, Zhang F, Wang Y. Research progress on discretization of fractional Fourier transform. Sci China Ser F-Inf Sci, 2008, 51: 859–880CrossRefMathSciNetGoogle Scholar
  13. 13.
    Long J, Tao R, Wang Y. The generalized weighed fractional Fourier transform and its application to image encryption. In: The 2nd International Congress on Image and Signal Processing (CISP’09). Tianjin, 2009. 1–5Google Scholar
  14. 14.
    Cariolaro G, Erseghe T, Kraniauskas P, et al. Multiplicity of fractional Fourier transforms and their relationships. IEEE Trans Signal Process, 2000, 48: 227–241zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lang J, Tao R, Wang Y, et al. The multiple-parameter fractional Fourier transform. Sci China Ser F-Inf Sci, 2008, 51: 1010–1024zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ozaydin M, Nemati S, Yeary M, et al. Orthogonal projections and discrete fractional Fourier transforms. In: IEEE 12th-Signal Processing Education Workshop, Teton National Park, Wyoming, 2006. 429–433Google Scholar
  17. 17.
    Candan C, Kutay M, Ozaktas H. The discrete fractional Fourier transform. IEEE Trans Signal Process, 2000, 48: 1329–1337zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Ozaktas H M, Arikan O, Kutay M A, et al. Digital computation of the fractional Fourier transform. IEEE Trans Signal Process, 1996, 44: 2141–2150CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of Electronic EngineeringBeijing Institute of TechnologyBeijingChina

Personalised recommendations