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author Chris Cannam
date Wed, 07 Sep 2016 10:40:32 +0100
parents 26056e866c29
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Chris@19 1 <html lang="en">
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Chris@19 3 <title>The 1d Discrete Fourier Transform (DFT) - FFTW 3.3.4</title>
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Chris@19 48 <div class="node">
Chris@19 49 <a name="The-1d-Discrete-Fourier-Transform-(DFT)"></a>
Chris@19 50 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
Chris@19 51 <p>
Chris@19 52 Next:&nbsp;<a rel="next" accesskey="n" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT">The 1d Real-data DFT</a>,
Chris@19 53 Previous:&nbsp;<a rel="previous" accesskey="p" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>,
Chris@19 54 Up:&nbsp;<a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
Chris@19 55 <hr>
Chris@19 56 </div>
Chris@19 57
Chris@19 58 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
Chris@19 59
Chris@19 60 <p><a name="index-discrete-Fourier-transform-295"></a><a name="index-DFT-296"></a>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
Chris@19 61 1d complex array X of size n computes an array Y,
Chris@19 62 where:
Chris@19 63 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
Chris@19 64 <center><img src="equation-idft.png" align="top">.</center>
Chris@19 65
Chris@19 66 <p><a name="index-normalization-297"></a>FFTW computes an unnormalized transform, in that there is no coefficient
Chris@19 67 in front of the summation in the DFT. In other words, applying the
Chris@19 68 forward and then the backward transform will multiply the input by
Chris@19 69 n.
Chris@19 70
Chris@19 71 <p><a name="index-frequency-298"></a>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
Chris@19 72 -1 in the exponent of the DFT. Note also that we use the
Chris@19 73 standard &ldquo;in-order&rdquo; output ordering&mdash;the k-th output
Chris@19 74 corresponds to the frequency k/n (or k/T, where T
Chris@19 75 is your total sampling period). For those who like to think in terms of
Chris@19 76 positive and negative frequencies, this means that the positive
Chris@19 77 frequencies are stored in the first half of the output and the negative
Chris@19 78 frequencies are stored in backwards order in the second half of the
Chris@19 79 output. (The frequency -k/n is the same as the frequency
Chris@19 80 (n-k)/n.)
Chris@19 81
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Chris@19 84