Circuits, Systems, and Signal Processing

, Volume 33, Issue 5, pp 1491–1505 | Cite as

On the Fast Fractional Jacket Transform

  • Yun Mao
  • Jun Peng
  • Ying GuoEmail author
  • Moon Ho Lee


Motivated by the center weighted Hadamard matrix, we propose an improved algorithm for the fast fractional jacket transform (FRJT) based on eigendecomposition of the fractional jacket matrix (FRJM). Employing a matrix diagonalization transformation that decomposes a matrix of large size into products of the matrices composed of eigenvectors and eigenvalues, an FRJM of large size can be fast factored into products of several sparse matrices in a recursive fashion. To generate an FRJM of large size, an algorithm for the factorable FRJM can be conveniently designated with a reduced computational complexity in terms of additions and multiplications. Since the proposed FRJM itself concerns interpretation as a suitable rotation in the time-frequency domain, it is applicable for optics and signal processing.


Fractional jacket transform Jacket matrix Matrix decomposition Fractional Hadamard transform Hadamard transform Signal processing 



This work was supported by the National Natural Science Foundation of China (61071096, 61379153), the bilateral cooperation of the science foundations between China and Korea (NSFC-NRF 61140391), and MEST 2012-0025-21, National Research Foundation, Korea.


  1. 1.
    N. Ahmed, K.R. Rao, Orthogonal Transforms for Digital Signal Processing (Springer, Berlin, 1975) zbMATHGoogle Scholar
  2. 2.
    L.B. Almeida, The fractional Fourier transform and time-frequency representation. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994) Google Scholar
  3. 3.
    C. Candan, M.A. Kutay, H.M. Ozaktas, The discrete fractional Fourier transform. IEEE Trans. Signal Process. 48(5), 1329–1337 (2000) zbMATHMathSciNetGoogle Scholar
  4. 4.
    R.A. Hom, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, New York, 1991) Google Scholar
  5. 5.
    M.H. Lee, The center weighted Hadamard transform. IEEE Trans. Circuits Syst. 36(9), 1247–1249 (1989) Google Scholar
  6. 6.
    M.H. Lee, A new reverse jacket transform and its fast algorithm. IEEE Trans. Circuits Syst. 47(1), 39–47 (2000) zbMATHGoogle Scholar
  7. 7.
    M.H. Lee, B.S. Rajan, J.Y. Park, A generalized reverse Jacket transform. IEEE Trans. Circuits Syst. 48(7), 684–688 (2001) zbMATHGoogle Scholar
  8. 8.
    M.H. Lee, X.-D. Zhang, W. Song, X.-G. Xia, Fast reciprocal Jacket transform with many parameters. IEEE Trans. Circuits Syst. 59(7), 1472–1481 (2012) MathSciNetGoogle Scholar
  9. 9.
    A.W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier domain. J. Opt. Soc. Am. A 10, 2181–2186 (1993) Google Scholar
  10. 10.
    D. Mendlovic, Z. Zalevsky, R.G. Dorsch, Y. Bitran, A.W. Lohmann, H. Ozaktas, New signal representation based on the fractional Fourier transform. J. Opt. Soc. Am. A 11, 2424–2431 (1995) MathSciNetGoogle Scholar
  11. 11.
    H. Neudecker, A note on Kronecker matrix products and matrix equation systems. SIAM J. Appl. Math. 17(3), 603–606 (1969) zbMATHMathSciNetGoogle Scholar
  12. 12.
    S.C. Pei, C.C. Tseng, M.H. Yeh, J.J. Shyu, Discrete fractional Hartley and Fourier transforms. IEEE Trans. Circuits Syst. 45(4), 665–675 (1998) zbMATHGoogle Scholar
  13. 13.
    S.C. Pei, M.H. Yeh, The discrete fractional cosine and sine transforms. IEEE Trans. Signal Process. 49(6), 1198–1207 (2001) MathSciNetGoogle Scholar
  14. 14.
    G. Zeng, M.H. Lee, A generalized reverse block Jacket transform. IEEE Trans. Circuits Syst. 55(7), 1589–1599 (2008) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Information Science & EngineeringCentral South UniversityChangshaChina
  2. 2.Institution of Information & Communication EngineeringChonbuk National UniversityChonjuKorea

Personalised recommendations