fftw3: The 1d Discrete Fourier Transform (DFT)
4.8.1 The 1d Discrete Fourier Transform (DFT)
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The forward ('FFTW_FORWARD') discrete Fourier transform (DFT) of a 1d
complex array X of size n computes an array Y, where:
Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(-2 pi j k sqrt(-1)/n) .
The backward ('FFTW_BACKWARD') DFT computes:
Y[k] = sum for j = 0 to (n - 1) of X[j] * exp(2 pi j k sqrt(-1)/n) .
FFTW computes an unnormalized transform, in that there is no
coefficient in front of the summation in the DFT. In other words,
applying the forward and then the backward transform will multiply the
input by n.
From above, an 'FFTW_FORWARD' transform corresponds to a sign of -1
in the exponent of the DFT. Note also that we use the standard
"in-order" output ordering--the k-th output corresponds to the frequency
k/n (or k/T, where T is your total sampling period). For those who like
to think in terms of positive and negative frequencies, this means that
the positive frequencies are stored in the first half of the output and
the negative frequencies are stored in backwards order in the second
half of the output. (The frequency -k/n is the same as the frequency
(n-k)/n.)
Info Catalog
fftw3: What FFTW Really Computes
fftw3: What FFTW Really Computes
fftw3: The 1d Real-data DFT