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Number Theoretic Transformation Techniques

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Digital Filtering in One and Two Dimensions

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Abstract

Processing signals with a digital computer or with dedicated digital hardware involves the implementation of computational schemes on sequences of numbers. Practically, it is not possible to process an infinitely long sequence, although it is common practice to analyze many processing systems as though this were the case. Thus, we deal with finite-length sequences which have finite values in the range 0 ≤ n ≤ N−1 and whose values are zero outside this range.

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References

  1. J. H. McClellan, Hardware realization of a Fermat number transform, IEEE Trans. Acoust., Speech, Signal Process. ASSP-24, 216–225 (1976).

    Article  Google Scholar 

  2. A. Baraniecka and G. A. Jullien, Hardware implementation of convolution using number theoretic transforms, Proc. Int. Conf. Acoust., Speech, Signal Process., Washington, D.C., 490-492 (1979).

    Google Scholar 

  3. R. C. Agarwal and C. S. Burras, Number theoretic transforms to implement fast digital convolution, Proc. IEEE 63, 550–560 (1975).

    Article  MathSciNet  Google Scholar 

  4. I. S. Reed and T. K. Traong, A fast DFT algorithm using complex integer transforms, Electron. Lett. 14, 191–193 (1978).

    Article  Google Scholar 

  5. H. J. Nussbaumer, Digital filtering using pseudo Fermat number transform, IEEE Jrans. Acoust., Speech, Signal Process. ASSP-25, 79–83 (1977).

    Article  Google Scholar 

  6. H. J. Nussbaumer, Linear filtering technique for computing Mersenne and Fermat number transforms, IBM J. Res. Dev. 21, 334–339 (1977).

    Article  MATH  Google Scholar 

  7. H. J. Nussbaumer, Relative evaluation of various number theoretic transforms for digital filtering applications, IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 88–93 (1978).

    Article  Google Scholar 

  8. H. J. Nussbaumer, Complex convolutions via Fermat number transforms, IBM J. Res. Dev. 20, 282–284 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. J. Nussbaumer, Digital filtering using complex Mersenne transforms, IBM J. Res. Dev. 20, 498–504 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  10. H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer-Verlag, Berlin and New York (1981).

    Book  MATH  Google Scholar 

  11. M. C. Vanwormhoudt, On number theoretic Fourier transforms in residue class rings, IEEE Trans. Acoust., Speech, Signal Process. ASSP-25, 585–586 (1977).

    Article  Google Scholar 

  12. M. C. Vanwormhoudt, Structural properties of complex residue rings applied to number theoretic Fourier transforms, IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 99–104 (1978).

    Article  Google Scholar 

  13. K. M. Henein, Number Theoretic Transforms in Digital Signal Processing, MSc. Thesis, Imperial College of Science and Technology, University of London (1983).

    Google Scholar 

  14. N. M. Nasrabadi, Orthogonal Transforms and their Applications to Image Coding, PhD. Thesis, Imperial College of Science and Technology, University of London (1984).

    Google Scholar 

  15. R. N. Gorgui-Naguib and A. Leboyer, Comment on “Determination of p-adic transform bases and lengths,” Electron. Lett. 21, 905–906 (1985).

    Article  Google Scholar 

  16. A. Leboyer, P-adic Numbers—P-adic Transform, MSc. Communications report, Imperial College of Science and Technology (1985).

    Google Scholar 

  17. V. Loahakosol and W. Surakampontorn, P-adic transforms, Electron. Lett. 20, 726–727 (1984).

    Article  Google Scholar 

  18. N. M. Nasrabadi and R. A. King, Fast digital convolution using p-adic transforms, Electron. Lett. 19, 266–267 (1983).

    Article  Google Scholar 

  19. N. M. Nasrabadi and R. A. King, Complex number theoretic transform in p-adic field, Proc. IEEE—Int. Conf. Acoust., Speech, Signal Process. 1984, pp. 28A.4.1–28A.4.3 (1984).

    Google Scholar 

  20. S.-C. Pei and J.-L. Wu, Determination of p-adic transform bases and lengths, Electron. Lett. 21, 431–432 (1985).

    Article  Google Scholar 

  21. W. W. Adams and L. J. Goldstein, Introduction to Number Theory, Prentice-Hall, Englewood Cliffs, NJ (1976).

    MATH  Google Scholar 

  22. T. A. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, Berlin (1976).

    Book  MATH  Google Scholar 

  23. Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York (1973).

    Google Scholar 

  24. W. J. Leveque, Fundamentals of Number Theory, Addison-Wesley, Reading, Massachusetts (1977).

    MATH  Google Scholar 

  25. I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, 4th edn., John Wiley and Sons, New York (1980).

    MATH  Google Scholar 

  26. I. N. Herstein, Topics in Algebra, 2nd edn., John Wiley and Sons, New York (1975).

    MATH  Google Scholar 

  27. J. H. McClellan and C. M. Rader, Number Theory in Digital Signal Processing, Prentice-Hall Signal Processing Series (1979).

    Google Scholar 

  28. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edn., Oxford University Press (1979).

    Google Scholar 

  29. C. M. Rader, Discrete convolutions via Mersenne transforms, IEEE Trans. Comput. C-21, 1269–1273 (1972).

    Article  MathSciNet  Google Scholar 

  30. R. C. Agarwal and C. S. Burrus, Fast convolution using Fermat number transforms with applications to digital filtering, IEEE Trans. Acoust, Speech, Signal Process. ASSP-22, 87–97 (1974).

    Article  MathSciNet  Google Scholar 

  31. S. Winograd, On computing the discrete Fourier transform, Math. Comput. 32, 175–199 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  32. D. P. Kolba and T. W. Parks, A prime factor FFT algorithm using high speed convolution, IEEE Trans. Acoust., Speech, Signal Process. ASSP-25, 90–103 (1977).

    Google Scholar 

  33. I. S. Reed and T. K. Truong, Convolutions over residue classes of quadratic integers, IEEE Trans. Inf. Theory IT-22, 468–475 (1976).

    Article  MathSciNet  Google Scholar 

  34. H. J. Nussbaumer and P. Quandalle, Fast computation of discrete Fourier transforms using polynomial transforms, IEEE Trans. Acoust., Speech, Signal Process. ASSP-27, 169–181 (1979).

    Article  MathSciNet  Google Scholar 

  35. H. J. Nussbaumer and P. Quandalle, Computation of convolutions and discrete Fourier transforms by polynomial transform, IBM J. Res. Dev. 22, 134–144 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  36. H. J. Nussbaumer, DFT computation by fast polynomial transform algorithms, Electron. Lett. 15, 701–702 (1979).

    Article  Google Scholar 

  37. B. Arambepola and P. J. W. Rayner, Discrete transform over polynomial rings with applications in computing multidimensional convolutions, IEEE Trans. Acoust., Speech, Signal Process. ASSP-28, 407–414 (1980).

    Article  MathSciNet  Google Scholar 

  38. H. J. Nussbaumer, Digital filtering using polynomial transforms, Electron. Lett. 13, 386–387 (1977).

    Article  Google Scholar 

  39. I. S. Reed, H. M. Shao, and T. K. Truong, Fast polynomial transform and its implementation by computer, IEE Proc. 128, Part E, No. 1, 50–60 (1981).

    Article  MathSciNet  Google Scholar 

  40. B. Arambepola and P. J. W. Rayner, Efficient transforms for multidimensional convolutions, Electron. Lett. 15, 189–190 (1979).

    Article  MathSciNet  Google Scholar 

  41. B. Arambepola and P. J. W. Rayner, Multidimensional fast Fourier transform algorithms, Electron. Lett. 15, 382–383 (1979).

    Article  MathSciNet  Google Scholar 

  42. H. J. Nussbaumer, New polynomial transform algorithms for multidimensional DFT’s and Convolutions, IEEE Trans. Acoust., Speech, Signal Process. ASSP-29, 761–783 (1981).

    Google Scholar 

  43. H. J. Nussbaumer, Inverse polynomial transform algorithms for DFT’s and convolutions, Proc. IEEE—Int. Conf. Acoust., Speech, Signal Process. 1981, 315–318 (1981).

    Google Scholar 

  44. K. Hensel, Theorie der Algebraischen Zahlen, Teubner, Leipzig (1908).

    MATH  Google Scholar 

  45. K. Hensel, Zahlentheorie, Goschen, Berlin and Leipzig (1913).

    Google Scholar 

  46. R. N. Gorgui-Naguib, P-adic Number Theory and its Applications in a Cryptographic System, PhD. Thesis, Imperial College of Science and Technology, University of London (1986).

    Google Scholar 

  47. G. Bachman, Introduction to P-adic Numbers and Valuation Theory, Academic Press, New York (1964).

    Google Scholar 

  48. E. V. Krishnamurthy, T. Mahadeva Rao, and K. Subramanian, Finite-segment p-adic number systems with applications to exact computation, Proc. Indian Acad. Sci. 81A, 58–79 (1975).

    MathSciNet  Google Scholar 

  49. E. V. Krishnamurthy, T. Mahadeva Rao, and K. Subramanian, P-adic arithmetic procedures for exact matrix computations, Proc. Indian Acad. Sci. 82A, 165–175 (1975).

    MathSciNet  Google Scholar 

  50. E. V. Krishnamurthy, Matrix processors using p-adic arithmetic for exact linear computations, IEEE Trans. Comput. C-26, 633–639 (1977).

    Article  MathSciNet  Google Scholar 

  51. R. N. Gorgui-Naguib and R. A. King, Comments on matrix processors using p-adic arithmetic for exact linear computation, IEEE Trans. Comput. C-35, 928–930 (1986).

    Article  MathSciNet  Google Scholar 

  52. N. Koblitz, P-adic Numbers, P-adic Analysis and Zeta-functions, Graduate Texts in Mathematics, Springer-Verlag, Berlin (1977).

    Book  MATH  Google Scholar 

  53. I. S. Reed and T. K. Truong, Fast Mersenne-prime transforms for digital filtering, Proc. IEEE 125, 433–440 (1978).

    MathSciNet  Google Scholar 

  54. K. Y. Liu, I. S. Reed, and T. K. Truong, Fast number theoretic transforms for digital filtering, Electron. Lett. 12, 644–646 (1976).

    Article  MathSciNet  Google Scholar 

  55. C. M. Rader, Discrete Fourier transforms when the number of data samples is prime, Proc. IEEE 56, 1107–1108 (1968).

    Article  Google Scholar 

  56. R. C. Agarwal and J. W. Cooley, New algorithms for digital convolution, IEEE Trans. Acoust, Speech, Signal Process. ASSP-25, 392–410 (1977).

    Article  Google Scholar 

  57. K. Mahler, P-adic Numbers and their Functions, Cambridge University Press (1980).

    Google Scholar 

  58. H. J. Nussbaumer, Fast polynomial algorithms for digital convolution, IEEE Trans. Acoust, Speech, Signal Process. ASSP-28, 205–215 (1980).

    Article  MathSciNet  Google Scholar 

  59. G. A. Jullien, R. Krichman, and W. C. Miller, Complex digital signal processing over finite rings, IEEE Trans. Circuits Syst. CAS-34, 365–377 (1987).

    Article  Google Scholar 

  60. G. A. Jullien, Implementation of multiplication modulo a prime number, with applications to number theoretic transforms, IEEE Trans. Comput. C-29, 899–905 (1980).

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Imperial College, London, England

    Robert King

  2. University of Windsor, Windsor, Ontario, Canada

    Majid Ahmadi

  3. University of Newcastle upon Tyne, Newcastle upon Tyne, England

    Raouf Gorgui-Naguib

  4. Imperial College and Bankers Trust Company, London, England

    Alan Kwabwe

  5. Colorado State University, Fort Collins, Colorado, USA

    Mahmood Azimi-Sadjadi

Authors
  1. Robert King
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  2. Majid Ahmadi
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  3. Raouf Gorgui-Naguib
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  4. Alan Kwabwe
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  5. Mahmood Azimi-Sadjadi
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© 1989 Springer Science+Business Media New York

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King, R., Ahmadi, M., Gorgui-Naguib, R., Kwabwe, A., Azimi-Sadjadi, M. (1989). Number Theoretic Transformation Techniques. In: Digital Filtering in One and Two Dimensions. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0918-3_9

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  • DOI: https://doi.org/10.1007/978-1-4899-0918-3_9

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4899-0920-6

  • Online ISBN: 978-1-4899-0918-3

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