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4.8 What FFTW Really Computes

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In this section, we provide precise mathematical definitions for the Chris@82: transforms that FFTW computes. These transform definitions are fairly Chris@82: standard, but some authors follow slightly different conventions for the Chris@82: normalization of the transform (the constant factor in front) and the Chris@82: sign of the complex exponent. We begin by presenting the Chris@82: one-dimensional (1d) transform definitions, and then give the Chris@82: straightforward extension to multi-dimensional transforms. Chris@82:

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• The 1d Discrete Fourier Transform (DFT):   Chris@82:
• The 1d Real-data DFT:   Chris@82:
• 1d Real-even DFTs (DCTs):   Chris@82:
• 1d Real-odd DFTs (DSTs):   Chris@82:
• 1d Discrete Hartley Transforms (DHTs):   Chris@82:
• Multi-dimensional Transforms:   Chris@82:
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