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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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+<html lang="en">
+<head>
+<title>The 1d Discrete Fourier Transform (DFT) - FFTW 3.3.3</title>
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+<link rel="next" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" title="The 1d Real-data DFT">
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+<!--
+This manual is for FFTW
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+Copyright (C) 2003 Matteo Frigo.
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+<div class="node">
+<a name="The-1d-Discrete-Fourier-Transform-(DFT)"></a>
+<a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
+<p>
+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>,
+Previous:&nbsp;<a rel="previous" accesskey="p" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>,
+Up:&nbsp;<a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
+<hr>
+</div>
+
+<h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
+
+<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
+1d complex array X of size n computes an array Y,
+where:
+<center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
+<center><img src="equation-idft.png" align="top">.</center>
+
+   <p><a name="index-normalization-294"></a>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.
+
+   <p><a name="index-frequency-295"></a>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
+-1 in the exponent of the DFT.  Note also that we use the
+standard &ldquo;in-order&rdquo; output ordering&mdash;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.)
+
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