International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 60   | 1 | 2 |

Section 1.3.3.3.2.3. Nesting of Winograd small FFTs

G. Bricognea

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1.3.3.3.2.3. Nesting of Winograd small FFTs

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Suppose that the CRT has been used as above to map an n-dimensional DFT to a μ-dimensional DFT. For each [\kappa = 1, \ldots, \mu] [κ runs over those pairs (i, j) such that [\nu (i,j) \,\gt\, 0]], the Rader/Winograd procedure may be applied to put the matrix of the κth 1D DFT in the CBA normal form of a Winograd small FFT. The full DFT matrix may then be written, up to permutation of data and results, as[\bigotimes_{\kappa = 1}^{\mu}({\bf C}_{\kappa} {\bf B}_{\kappa} {\bf A}_{\kappa}).]

A well known property of the tensor product of matrices allows this to be rewritten as[\left(\bigotimes_{\gamma = 1}^{\mu} {\bf C}_{\gamma}\right) \times \left(\bigotimes_{\beta = 1}^{\mu} {\bf B}_{\beta}\right) \times \left(\bigotimes_{\alpha = 1}^{\mu} {\bf A}_{\alpha}\right)]and thus to form a matrix in which the combined pre-addition, multiplication and post-addition matrices have been precomputed. This procedure, called nesting, can be shown to afford a reduction of the arithmetic operation count compared to the row–column method (Morris, 1978[link]).

Clearly, the nesting rearrangement need not be applied to all μ dimensions, but can be restricted to any desired subset of them.

References

Morris, R. L. (1978). A comparative study of time efficient FFT and WFTA programs for general purpose computers. IEEE Trans. Acoust. Speech Signal Process. 26, 141–150.








































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