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Current fftw source
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 18 Oct 2016 13:40:26 +0100
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3 <!-- This manual is for FFTW
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6 Copyright (C) 2003 Matteo Frigo.
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25 <title>FFTW 3.3.5: The 1d Discrete Fourier Transform (DFT)</title>
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35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
37 <link href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" rel="next" title="The 1d Real-data DFT">
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72 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
73 <div class="header">
74 <p>
75 Next: <a href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" accesskey="n" rel="next">The 1d Real-data DFT</a>, Previous: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="p" rel="prev">What FFTW Really Computes</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
76 </div>
77 <hr>
78 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029-1"></a>
79 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
80
81 <a name="index-discrete-Fourier-transform-1"></a>
82 <a name="index-DFT-2"></a>
83 <p>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
84 1d complex array <em>X</em> of size <em>n</em> computes an array <em>Y</em>,
85 where:
86 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
87 <center><img src="equation-idft.png" align="top">.</center></p>
88 <a name="index-normalization-8"></a>
89 <p>FFTW computes an unnormalized transform, in that there is no coefficient
90 in front of the summation in the DFT. In other words, applying the
91 forward and then the backward transform will multiply the input by
92 <em>n</em>.
93 </p>
94 <a name="index-frequency-1"></a>
95 <p>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
96 <em>-1</em> in the exponent of the DFT. Note also that we use the
97 standard &ldquo;in-order&rdquo; output ordering&mdash;the <em>k</em>-th output
98 corresponds to the frequency <em>k/n</em> (or <em>k/T</em>, where <em>T</em>
99 is your total sampling period). For those who like to think in terms of
100 positive and negative frequencies, this means that the positive
101 frequencies are stored in the first half of the output and the negative
102 frequencies are stored in backwards order in the second half of the
103 output. (The frequency <em>-k/n</em> is the same as the frequency
104 <em>(n-k)/n</em>.)
105 </p>
106
107
108
109 </body>
110 </html>