Agarwal-Cooley Convolution Algorithm

  • Richard Tolimieri
  • Chao Lu
  • Myoung An
Part of the Signal Processing and Digital Filtering book series (SIGNAL PROCESS)

Abstract

The cyclic convolution algorithms of chapter 6 are efficient for special small block lengths, but as the size of the block length increases, other methods are required. First, as discussed in chapter 6, these algorithms keep the number of required multiplications small, but they can require many additions. Also, each size requires a different algorithm. There is no uniform tructure that can be repeatedly called upon. In this chapter, a technique similar to the Good-Thomas PFA will be developed to decompose a large size cyclic convolution into several small size cyclic convolutions that in turn can be evaluated using the Winograd cyclic convolution algorithm. These ideas were introduced by Agarwal and Cooley [1] in 1977. As in the Good-Thomas PFA, the CRT is used to define an indexing of data. This indexing changes a one-dimensional cyclic convolution into a two-dimensional one. We will see how to compute a two-dimensional cyclic convolution by ‘nesting’ a fast algorithm for a one-dimensional case inside another fast algorithm for a one-dimensional cyclic convolution. There are several two-dimensional cyclic convolution algorithms that, although important, will not be discussed. These can be found in [2].

Keywords

Diagonal Matrix Discrete Fourier Transform Fast Algorithm Permutation Matrix Circulant Matrix 
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.

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References

  1. [1]
    Agarwal, R. C. and Cooley, J. W. “New Algorithms for Digital Convolution”, IEEE Trans. Acoust., Speech and Signal Proc., 25, 1977, pp. 392–410.MATHCrossRefGoogle Scholar
  2. [2]
    Blahut, R. E. Fast Algorithms for Digital Signal Processing, Addison-Wesley, 1985, Chapter 7.Google Scholar
  3. [3]
    Nussbaumer, H. J. Fast Fourier Transform and Convolution Algorithms, Second Edition, Springer-Verlag, 1981, Chapter 6.Google Scholar
  4. [4]
    Arambepola, B. and Rayner, P. J. “Efficient Transforms for Multidimensional Convolutions”, Electron. Lett., 15, 1979, pp. 189–190.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Richard Tolimieri
    • 1
  • Chao Lu
    • 2
  • Myoung An
    • 3
  1. 1.Department of Electrical EngineeringCity College of CUNYNew YorkUSA
  2. 2.Department of Computer and Information SciencesTowson State UniversityTowsonUSA
  3. 3.A.J. Devaney AssociatesAllstonUSA

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