International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 56-57
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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 [κ runs over those pairs (i, j) such that ], 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
A well known property of the tensor product of matrices allows this to be rewritten as 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).
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.Google Scholar