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The forward (FFTW_FORWARD) discrete Fourier transform (DFT) of a
cannam@95: 1d complex array X of size n computes an array Y,
cannam@95: where:
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.FFTW_BACKWARD) DFT computes:
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.FFTW computes an unnormalized transform, in that there is no coefficient cannam@95: in front of the summation in the DFT. In other words, applying the cannam@95: forward and then the backward transform will multiply the input by cannam@95: n. cannam@95: cannam@95:
From above, an FFTW_FORWARD transform corresponds to a sign of
cannam@95: -1 in the exponent of the DFT. Note also that we use the
cannam@95: standard “in-order” output ordering—the k-th output
cannam@95: corresponds to the frequency k/n (or k/T, where T
cannam@95: is your total sampling period). For those who like to think in terms of
cannam@95: positive and negative frequencies, this means that the positive
cannam@95: frequencies are stored in the first half of the output and the negative
cannam@95: frequencies are stored in backwards order in the second half of the
cannam@95: output. (The frequency -k/n is the same as the frequency
cannam@95: (n-k)/n.)
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