Journal of Communications and Networks

The Korean Institute of Commucations and Information Sciences (한국통신학회)
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Fast DFT Matrices Transform Based on Generalized Prime Factor Algorithm

  • Guo, Ying (School of Information Science and Engineering, Central South University) ;
  • Mao, Yun (School of Information Science and Engineering, Central South University) ;
  • Park, Dong-Sun (Electronics and Information Engineering, Chonbuk National University) ;
  • Lee, Moon-Ho (Electronics and Information Engineering, Chonbuk National University)
  • Received : 2010.04.15
  • Accepted : 2010.09.17
  • Published : 2011.10.31
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Abstract

Inspired by fast Jacket transforms, we propose simple factorization and construction algorithms for the M-dimensional discrete Fourier transform (DFT) matrices underlying generalized Chinese remainder theorem (CRT) index mappings. Based on successive coprime-order DFT matrices with respect to the CRT with recursive relations, the proposed algorithms are presented with simplicity and clarity on the basis of the yielded sparse matrices. The results indicate that our algorithms compare favorably with the direct-computation approach.

Keywords

References

  1. N. Ahmed and K. R. Rao, Orthogonal Transform for Digital Signal Processing. Berlin, Germany: Springer Verlag, 1975.
  2. R. E. Blahut, Algebraic Codes for Data Transmission. Cambridge, UK: 2003.
  3. K. Y. Rao and J. E. Hershey, Hadamard Matrix Analysis and Synthesis. Norwell, MA: Kluwer, 1997.
  4. M. H. Lee, "The center weighted Hadamard transform," IEEE Trans. Circuits Syst., vol. 36, pp. 1247-1249, Sept. 1989. https://doi.org/10.1109/31.34673
  5. M. H. Lee, "A new reverse Jacket transform and its fast algorithm," IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process, vol. 47, no. 1, pp.39-47, Jan. 2000. https://doi.org/10.1109/82.818893
  6. M. H. Lee and Y. L. Borissov, "A proof of non-existence of bordered Jacket matrices of odd order over some field," Electron. Lett., vol. 46, pp. 349-351, Mar. 2010. https://doi.org/10.1049/el.2010.2991
  7. M. H. Lee, B. S. Rajan, and J. Y. Park, "A generalized reverse Jacket transform," IEEE Trans. Circuits Syst., vol. 48, pp. 684-688, July 2001. https://doi.org/10.1109/82.958338
  8. M. H. Lee and J. Hou, "Fast block inverse Jacket transform," IEEE Signal Process. Lett., vol. 13, no. 4, pp. 461-464, 2006. https://doi.org/10.1109/LSP.2006.873660
  9. M. H. Lee and G. Zeng, "Family of fast Jacket transform algorithms," Electron. Lett., vol. 43, no. 11, pp. 651-651, 2007. https://doi.org/10.1049/el:20070145
  10. G. Zeng and M. H. Lee, "A generalized reverse block Jacket transform," IEEE Trans. Circuits and Syst., vol. 55, pp. 1589-1599, July 2008. https://doi.org/10.1109/TCSI.2008.916620
  11. Z. Chen, M. H. Lee, and G. Zeng, "Fast cocyclic Jacket transform," IEEE Trans. Signal Process., vol. 56, no. 5, pp. 2143-2148, 2008. https://doi.org/10.1109/TSP.2007.912895
  12. M. H. Lee and K. Finlayson, "A simple element inverse Jacket transform coding," in Proc. IEEE ITW, New Zealand, vol. 14, 2006, pp. 1-4.
  13. X. J. Jiang and M. H. Lee, "Higher dimensional Jacket code for mobile communications," in Proc. APW, Sapporo, Japan, Aug. 2005, pp. 4-5.
  14. M. G. Parker and M. H. Lee, "Optimal bipolar sequence for the complex reverse-Jacket transform," in Proc. ISITA, Honolulu, USA, Nov. 2000, pp.5-8.
  15. J. Hou and M. H. Lee, Cocyclic Jacket Matrices and Its Application to Cryptography Systems. Berlin, Germany: Springer, vol. 3391, 2005.
  16. W. Song, M. H. Lee, and G. Zeng, "Orthogonal space-time block codes design using Jacket transform for MIMO transmission system," in Proc. IEEE ICC, May 2008, pp. 766-769.
  17. S. Winograd, "On computing the discrete Fourier transform," in Proc. Nat. Acad. Sci., USA, vol. 73, no. 4, Apr. 1976, pp. 1005-1006.
  18. C. Temperton, "Implementation of a self-sorting, in-place prime factor FFT algorithm," J. Computat. Physics, vol. 58, pp. 283-299, 1985. https://doi.org/10.1016/0021-9991(85)90164-0
  19. C. S. Burrus, "Index mapping for multidimensional formulation of the DFT and convolution," IEEE Trans. Acoust. Speech, Signal Process., vol. 25, no. 3, pp. 239-242, June 1977. https://doi.org/10.1109/TASSP.1977.1162938
  20. C. S. Burrus and P. W. Eschenbacher, "An in-place, in-order prime factor for FFT algorithm," IEEE Trans. Acoust. Speech, Signal Process., vol. 29, no. 4, pp. 806-817, Aug. 1981. https://doi.org/10.1109/TASSP.1981.1163645
  21. Z. Wang, "Index mapping for one to multidimensional," Electron. Lett., vol. 25, no. 12, pp. 781-782, June 1989. https://doi.org/10.1049/el:19890527
  22. J. C. Schatzman, "Index mapping for the fast Fourier transform," IEEE Trans. Signal Process., vol. 44, no. 3, pp. 717-719, Mar. 1996. https://doi.org/10.1109/78.489047
  23. H. Neudecker, "A note on Kronecker matrix products and matrix equation systems," SIAM J. Appl. Mathematics, vol. 17, no. 3, pp. 603-606, May 1969. https://doi.org/10.1137/0117057
  24. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. New York: Cambridge Univ. Press, 1991.