Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Ahmed, K.R. Rao, Orthogonal Transforms for Digital Signal Processing (Springer, Berlin, 1975)
L.B. Almeida, The fractional Fourier transform and time-frequency representation. IEEE Trans. Signal Process. 42(11), 3084–3091 (1994)
C. Candan, M.A. Kutay, H.M. Ozaktas, The discrete fractional Fourier transform. IEEE Trans. Signal Process. 48(5), 1329–1337 (2000)
R.A. Hom, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, New York, 1991)
M.H. Lee, The center weighted Hadamard transform. IEEE Trans. Circuits Syst. 36(9), 1247–1249 (1989)
M.H. Lee, A new reverse jacket transform and its fast algorithm. IEEE Trans. Circuits Syst. 47(1), 39–47 (2000)
M.H. Lee, B.S. Rajan, J.Y. Park, A generalized reverse Jacket transform. IEEE Trans. Circuits Syst. 48(7), 684–688 (2001)
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)
A.W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier domain. J. Opt. Soc. Am. A 10, 2181–2186 (1993)
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)
H. Neudecker, A note on Kronecker matrix products and matrix equation systems. SIAM J. Appl. Math. 17(3), 603–606 (1969)
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)
S.C. Pei, M.H. Yeh, The discrete fractional cosine and sine transforms. IEEE Trans. Signal Process. 49(6), 1198–1207 (2001)
G. Zeng, M.H. Lee, A generalized reverse block Jacket transform. IEEE Trans. Circuits Syst. 55(7), 1589–1599 (2008)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mao, Y., Peng, J., Guo, Y. et al. On the Fast Fractional Jacket Transform. Circuits Syst Signal Process 33, 1491–1505 (2014). https://doi.org/10.1007/s00034-013-9699-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-013-9699-8