New Approaches to Reduction of Computational Complexity in Signal Processing Systems

  • Thomas A. Kriz
Part of the NATO Conference Series book series (NATOCS, volume 5)


Mathematically, most computer-implemented signal processing activity can be represented as discrete form matrix-vector multiply or matrix inversion processes, wherein the matrix structure exhibits some form of symmetry with regard to a repetitious arrangement of elements in it. A direct computation approach to the implementation of these discrete mathematical processes, which ignores the nature of the symmetries, would require respectively O(n2) and O(n3) multiply operations for an n-order system. On the other hand, if one exploits the form of the matrix symmetry in the problem, it becomes possible to define partitioned solution approaches that significantly reduce the level of computation for these processes. Two well-known examples of such an exploitation are the use of the fast Fourier transform (FFT) method in implementing the generation of a discrete Fourier spectrum with O(n log2 n) multiply operations and the use of the Levinson recursive method in the solution of the Toeplitz equation problem with O(n2) multiply operations.


Fast Fourier Transform Computation Load Chinese Remainder Theorem Fermat Number Convolution Process 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Pollard, J. M. “The Fast Fourier Transform in a Finite Field,” Mathematics of Computation, Vol. 25, pp. 365–374, April 1971.CrossRefGoogle Scholar
  2. [2]
    Rader, C. M. “Discrete Convolution via Mersenne Transforms,” IEEE Transactions on Computations, Vol. C-21, pp. 1269–1273, December 1972.CrossRefGoogle Scholar
  3. [3]
    Agerwal, R. C. and Burrus, C. S. “Fast Convolution Using Fermat Number Transforms with Applications to Digital Filtering,” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-22, pp. 87–97, April 1974.CrossRefGoogle Scholar
  4. [4]
    Reed I. S. and Truong, T. K. “The Use of Finite Fields to Compute Convolutions,” IEEE Transactions on Information Theory, Vol, IT-21, pp. 208–213, March 1975.CrossRefGoogle Scholar
  5. [5]
    Winograd, S. “The Effects of a Field of Constants on the Number of Mulitplications,” Proceedings of the 16th Symposium on Foundations of Computer Science, pp, 1–2, 1975.Google Scholar
  6. [6]
    Winograd, S. “On Computing the Discrete Fourier Transform,” Proceeding of the National Academy of Sciences (USA), Vol. 73, pp. 1005–1006, April 1976.CrossRefGoogle Scholar
  7. [7]
    Nussbaumer, H. “Complex Convolution via Fermat Number Transforms,” IBM Journal of Research and Development, Vol. 20, pp. 282–284, May 1976.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • Thomas A. Kriz
    • 1
  1. 1.IBM Federal Systems DivisionOwegoUSA

Personalised recommendations