Advertisement

The Discrete Cosine Transform over Prime Finite Fields

  • M. M. C. de Souza
  • H. M. de Oliveira
  • R. M. C. de Souza
  • M. M. Vasconcelos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3124)

Abstract

This paper examines finite field trigonometry as a tool to construct trigonometric digital transforms. In particular, by using properties of the k-cosine function over GF(p), the Finite Field Discrete Cosine Transform (FFDCT) is introduced. The FFDCT pair in GF(p) is defined, having blocklengths that are divisors of (p+1)/2. A special case is the Mersenne FFDCT, defined when p is a Mersenne prime. In this instance blocklengths that are powers of two are possible and radix-2 fast algorithms can be used to compute the transform.

Keywords

Discrete Cosine Transform Discrete Fourier Transform Finite Field Inversion Formula Discrete Cosine Transform Coefficient 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Transform Coding: Past, Present and Future. IEEE SP Mag. 18, 6–93 (2001)Google Scholar
  2. 2.
    Pollard, J.M.: The Fast Fourier Transform in a Finite Field. Math. Comput. 25, 365–374 (1971)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Reed, I.S., Truong, T.K.: The Use of Finite Field to Compute Convolutions. IEEE Trans. Inform. Theory IT-21, 208–213 (1975)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Agarwal, R.C., Burrus, C.S.: Number Theoretic Transforms to Implement Fast Digital Convolution. Proc. IEEE 63, 550–560 (1975)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Reed, I.S., Truong, T.K., Kwoh, V.S., Hall, E.L.: Image Processing by Transforms over a Finite Field. IEEE Trans. Comput. C-26, 874–881 (1977)CrossRefGoogle Scholar
  6. 6.
    Blahut, R.E.: Transform Techniques for Error-Control Codes. IBM J. Res. Dev. 23, 299–315 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Massey, J.L.: The Discrete Fourier Transform in Coding and Cryptography. In: IEEE Information Theory Workshop, San Diego, CA (1998)Google Scholar
  8. 8.
    Campello de Souza, R.M., de Oliveira, H.M., Kauffman, A.N.: Trigonometry in Finite Fields and a New Hartley Transform. In: Proc. of the IEEE Int. Symp. on Info. Theory, p. 293 (1998)Google Scholar
  9. 9.
    de Oliveira, H.M., Campello de Souza, R.M., Kauffman, A.N.: Efficient Multiplex for Band-Limited Channels. In: Proc. of the Work. on Coding and Cryptography, pp. 235–241 (1999)Google Scholar
  10. 10.
    Miranda, J.P.C.L., De Oliveira, H.M.: On Galois-Division Multiple Access Systems: Figures of Merit and Performance Evaluation. In: Proc. of the 19 Braz. Telecom. Symp. (2001) (in English)Google Scholar
  11. 11.
    de Oliveira, H.M., Miranda, J.P.C.L., Campello de Souza, R.M.: Spread-Spectrum Based on Finite Field Fourier Transforms. In: Proc. of the ICSECIT - Int. Conf. on Systems Engineering, Communication and Information Technology, Punta Arenas, vol. 1 (2001)Google Scholar
  12. 12.
    de Oliveira, H.M., Campello de Souza, R.M.: Orthogonal Multilevel Spreading Sequence Design. In: Farrell, P.G., Darnell, M., Honary, B. (eds.) Coding, Communications and Broadcasting, pp. 291–303. Research Studies Press / John Wiley, Baldock (2000)Google Scholar
  13. 13.
    Lim, J.S.: Two-Dimensional Signal and Image Processing. Prentice-Hall, New Jersey (1990)Google Scholar
  14. 14.
    Burton, D.M.: Elementary Number Theory. McGraw Hill, New York (1997)Google Scholar
  15. 15.
    Blahut, R.E.: Fast Algorithms for Digital Signal Processing. Addison-Wesley, Reading (1985)zbMATHGoogle Scholar
  16. 16.
    Campello de Souza, R.M., de Oliveira, H.M.: The Complex Hartley Transform over a Finite Field. In: Farrell, P.G., Darnell, M., Honary, B. (eds.) Coding, Communications and Broadcasting, pp. 267–276. Research Studies Press / John Wiley, Baldock (2000)Google Scholar
  17. 17.
    Campello de Souza, R.M., de Oliveira, H.M., Campello de Souza, M.M.: Hartley Number- Theoretic Transforms. In: Proceedings of the 2001 IEEE International Symposium on Information Theory, p. 210 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • M. M. C. de Souza
    • 1
  • H. M. de Oliveira
    • 1
  • R. M. C. de Souza
    • 1
  • M. M. Vasconcelos
    • 1
  1. 1.Federal University of Pernambuco – UFPERecifeBrazil

Personalised recommendations