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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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3 <title>The 1d Discrete Fourier Transform (DFT) - FFTW 3.3.3</title>
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49 <a name="The-1d-Discrete-Fourier-Transform-(DFT)"></a>
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51 <p>
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57
58 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
59
60 <p><a name="index-discrete-Fourier-transform-292"></a><a name="index-DFT-293"></a>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
61 1d complex array X of size n computes an array Y,
62 where:
63 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
64 <center><img src="equation-idft.png" align="top">.</center>
65
66 <p><a name="index-normalization-294"></a>FFTW computes an unnormalized transform, in that there is no coefficient
67 in front of the summation in the DFT. In other words, applying the
68 forward and then the backward transform will multiply the input by
69 n.
70
71 <p><a name="index-frequency-295"></a>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
72 -1 in the exponent of the DFT. Note also that we use the
73 standard &ldquo;in-order&rdquo; output ordering&mdash;the k-th output
74 corresponds to the frequency k/n (or k/T, where T
75 is your total sampling period). For those who like to think in terms of
76 positive and negative frequencies, this means that the positive
77 frequencies are stored in the first half of the output and the negative
78 frequencies are stored in backwards order in the second half of the
79 output. (The frequency -k/n is the same as the frequency
80 (n-k)/n.)
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