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Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam
date Tue, 19 Nov 2019 14:52:55 +0000
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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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3 <!-- This manual is for FFTW
4 (version 3.3.8, 24 May 2018).
5
6 Copyright (C) 2003 Matteo Frigo.
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8 Copyright (C) 2003 Massachusetts Institute of Technology.
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22 approved by the Free Software Foundation. -->
23 <!-- Created by GNU Texinfo 6.3, http://www.gnu.org/software/texinfo/ -->
24 <head>
25 <title>FFTW 3.3.8: The 1d Discrete Fourier Transform (DFT)</title>
26
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34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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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71 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
72 <div class="header">
73 <p>
74 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>
75 </div>
76 <hr>
77 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029-1"></a>
78 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
79
80 <a name="index-discrete-Fourier-transform-1"></a>
81 <a name="index-DFT-2"></a>
82 <p>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
83 1d complex array <em>X</em> of size <em>n</em> computes an array <em>Y</em>,
84 where:
85 <center><img src="equation-dft.png" align="top">.</center>
86 The backward (<code>FFTW_BACKWARD</code>) DFT computes:
87 <center><img src="equation-idft.png" align="top">.</center>
88 </p>
89 <a name="index-normalization-8"></a>
90 <p>FFTW computes an unnormalized transform, in that there is no coefficient
91 in front of the summation in the DFT. In other words, applying the
92 forward and then the backward transform will multiply the input by
93 <em>n</em>.
94 </p>
95 <a name="index-frequency-1"></a>
96 <p>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
97 <em>-1</em> in the exponent of the DFT. Note also that we use the
98 standard &ldquo;in-order&rdquo; output ordering&mdash;the <em>k</em>-th output
99 corresponds to the frequency <em>k/n</em> (or <em>k/T</em>, where <em>T</em>
100 is your total sampling period). For those who like to think in terms of
101 positive and negative frequencies, this means that the positive
102 frequencies are stored in the first half of the output and the negative
103 frequencies are stored in backwards order in the second half of the
104 output. (The frequency <em>-k/n</em> is the same as the frequency
105 <em>(n-k)/n</em>.)
106 </p>
107
108
109
110 </body>
111 </html>