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The forward (FFTW_FORWARD) discrete Fourier transform (DFT) of a
Chris@10: 1d complex array X of size n computes an array Y,
Chris@10: where:
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.FFTW_BACKWARD) DFT computes:
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.FFTW computes an unnormalized transform, in that there is no coefficient Chris@10: in front of the summation in the DFT. In other words, applying the Chris@10: forward and then the backward transform will multiply the input by Chris@10: n. Chris@10: Chris@10:
From above, an FFTW_FORWARD transform corresponds to a sign of
Chris@10: -1 in the exponent of the DFT. Note also that we use the
Chris@10: standard “in-order” output ordering—the k-th output
Chris@10: corresponds to the frequency k/n (or k/T, where T
Chris@10: is your total sampling period). For those who like to think in terms of
Chris@10: positive and negative frequencies, this means that the positive
Chris@10: frequencies are stored in the first half of the output and the negative
Chris@10: frequencies are stored in backwards order in the second half of the
Chris@10: output. (The frequency -k/n is the same as the frequency
Chris@10: (n-k)/n.)
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