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.
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References
Transform Coding: Past, Present and Future. IEEE SP Mag. 18, 6–93 (2001)
Pollard, J.M.: The Fast Fourier Transform in a Finite Field. Math. Comput. 25, 365–374 (1971)
Reed, I.S., Truong, T.K.: The Use of Finite Field to Compute Convolutions. IEEE Trans. Inform. Theory IT-21, 208–213 (1975)
Agarwal, R.C., Burrus, C.S.: Number Theoretic Transforms to Implement Fast Digital Convolution. Proc. IEEE 63, 550–560 (1975)
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)
Blahut, R.E.: Transform Techniques for Error-Control Codes. IBM J. Res. Dev. 23, 299–315 (1979)
Massey, J.L.: The Discrete Fourier Transform in Coding and Cryptography. In: IEEE Information Theory Workshop, San Diego, CA (1998)
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)
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)
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)
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)
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)
Lim, J.S.: Two-Dimensional Signal and Image Processing. Prentice-Hall, New Jersey (1990)
Burton, D.M.: Elementary Number Theory. McGraw Hill, New York (1997)
Blahut, R.E.: Fast Algorithms for Digital Signal Processing. Addison-Wesley, Reading (1985)
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)
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)
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de Souza, M.M.C., de Oliveira, H.M., de Souza, R.M.C., Vasconcelos, M.M. (2004). The Discrete Cosine Transform over Prime Finite Fields. In: de Souza, J.N., Dini, P., Lorenz, P. (eds) Telecommunications and Networking - ICT 2004. ICT 2004. Lecture Notes in Computer Science, vol 3124. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27824-5_65
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DOI: https://doi.org/10.1007/978-3-540-27824-5_65
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