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### 4.7 What FFTW Really Computes

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