Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Existence theorems for transforms over finite rings with applications to $ 2$-D convolution

Author: David P. Maher
Journal: Math. Comp. 35 (1980), 757-765
MSC: Primary 10-04; Secondary 94B35
MathSciNet review: 572853
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An existence theorem for Fourier-like transforms over arbitrary finite commutative rings is proven in a simple fashion. Corollaries for the case of residue class rings over the integers and extensions of those rings follow directly. The theory is applied to construct very fast algorithms for the computation of two-dimensional convolutions over the integers $ \bmod\, M$.

References [Enhancements On Off] (What's this?)

  • [1] Ramesh C. Agarwal and C. Sidney Burrus, Number theoretic transforms to implement fast digital convolution, Proc. IEEE 63 (1975), no. 4, 550–560. MR 0451632
  • [2] Ramesh C. Agarwal and Charles S. Burrus, Fast convolution using Fermat number transforms with applications to digital filtering, IEEE Trans. Acoust. Speech Signal Processing ASSP-22 (1974), no. 2, 87–97. MR 0398650
  • [3] Ramesh C. Agarwal and Charles S. Burrus, Fast one-dimensional digital convolution by multidimensional techniques, IEEE Trans. Acoust. Speech Signal Processing ASSP-22 (1974), no. 1, 1–10. MR 0401340
  • [4] R. C. AGARWAL & J. W. COOLEY, "New algorithms for digital convolution," IEEE Trans. Acoust. Speech Signal Processing, v. 25, 1977, pp. 392-409.
  • [5] Rafael C. Gonzalez and Paul Wintz, Digital image processing, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. Applied Mathematics and Computation, No. 13. MR 0459042
  • [6] Jørn Justesen, On the complexity of decoding Reed-Solomon codes, IEEE Trans. Information Theory IT-22 (1976), no. 2, 237–238. MR 0465505
  • [7] D. Kibler, Necessary and sufficient conditions for the existence of the modular Fourier transform: Comments on “Number theoretic transforms to implement fast digital convolution” (Proc. IEEE 63 (1975), no. 4, 550–560) by R. C. Agarwal and C. S. Burrus, Proc. IEEE 65 (1977), no. 2, 265–267. With a reply by the authors. MR 0490402
  • [8] Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
  • [9] David P. Maher, Two-dimensional digital filtering using transforms over extensions of finite rings, Information linkage between applied mathematics and industry, II (Proc. Second Annual Workshop, Monterey, Calif., 1979) Academic Press, New York-London, 1980, pp. 269–278. MR 592274
  • [10] F. J. MacWILLIAMS &. N. J. A. SLOANE, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.
  • [11] Hideo Murakami, Irving S. Reed, and Lloyd R. Welch, A transform decoder for Reed-Solomon codes in multiple-user communication systems, IEEE Trans. Information Theory IT-23 (1977), no. 6, 675–683. MR 0469496
  • [12] Peter J. Nicholson, Algebraic theory of finite Fourier transforms, J. Comput. System Sci. 5 (1971), 524–547. MR 0286905
  • [13] H. J. NUSSBAUMER, "Digital filtering using pseudo Fermat number transforms," IEEE Trans. Acoust. Speech Signal Processing, v. 25, 1977, pp. 79-83.
  • [14] H. J. Nussbaumer and P. Quandalle, Computation of convolutions and discrete Fourier transforms by polynomial transforms, IBM J. Res. Develop. 22 (1978), no. 2, 134–144. MR 0471260
  • [15] J. M. Pollard, The fast Fourier transform in a finite field, Math. Comp. 25 (1971), 365–374. MR 0301966, 10.1090/S0025-5718-1971-0301966-0
  • [16] Charles M. Rader, Discrete convolutions via Mersenne transforms, IEEE Trans. Computers C-21 (1972), 1269–1273. MR 0438672
  • [17] Irving S. Reed and T. K. Truong, Complex integer convolutions over a direct sum of Galois fields, IEEE Trans. Information Theory IT-21 (1975), no. 6, 657–661. MR 0435049
  • [18] M. C. VANWORMHOUDT, "On number theoretic transforms in residue class rings," IEEE Trans. Acoust. Speech Signal Processing, v. 25, 1977, pp. 585-586.
  • [19] E. VEGH & L. M. LIEBOWITZ, Fast Complex Convolution Using Number Theoretic Transforms, NRL Report 7935, Naval Research Lab., Washington, D.C., 1975, pp. 1-13.
  • [20] S. Winograd, Some bilinear forms whose multiplicative complexity depends on the field of constants, Math. Systems Theory 10 (1976/77), no. 2, 169–180. Sixteenth Annual Symposium on Foundations of Computer Science (Berkeley, Calif., 1975), selected papers. MR 0468322

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10-04, 94B35

Retrieve articles in all journals with MSC: 10-04, 94B35

Additional Information

Article copyright: © Copyright 1980 American Mathematical Society