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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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Chris@82 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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Chris@82 3 <!-- This manual is for FFTW
Chris@82 4 (version 3.3.8, 24 May 2018).
Chris@82 5
Chris@82 6 Copyright (C) 2003 Matteo Frigo.
Chris@82 7
Chris@82 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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Chris@82 24 <head>
Chris@82 25 <title>FFTW 3.3.8: The 1d Discrete Fourier Transform (DFT)</title>
Chris@82 26
Chris@82 27 <meta name="description" content="FFTW 3.3.8: The 1d Discrete Fourier Transform (DFT)">
Chris@82 28 <meta name="keywords" content="FFTW 3.3.8: The 1d Discrete Fourier Transform (DFT)">
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Chris@82 34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
Chris@82 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
Chris@82 36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
Chris@82 37 <link href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" rel="next" title="The 1d Real-data DFT">
Chris@82 38 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="prev" title="What FFTW Really Computes">
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Chris@82 67
Chris@82 68 </head>
Chris@82 69
Chris@82 70 <body lang="en">
Chris@82 71 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
Chris@82 72 <div class="header">
Chris@82 73 <p>
Chris@82 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>
Chris@82 75 </div>
Chris@82 76 <hr>
Chris@82 77 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029-1"></a>
Chris@82 78 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
Chris@82 79
Chris@82 80 <a name="index-discrete-Fourier-transform-1"></a>
Chris@82 81 <a name="index-DFT-2"></a>
Chris@82 82 <p>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
Chris@82 83 1d complex array <em>X</em> of size <em>n</em> computes an array <em>Y</em>,
Chris@82 84 where:
Chris@82 85 <center><img src="equation-dft.png" align="top">.</center>
Chris@82 86 The backward (<code>FFTW_BACKWARD</code>) DFT computes:
Chris@82 87 <center><img src="equation-idft.png" align="top">.</center>
Chris@82 88 </p>
Chris@82 89 <a name="index-normalization-8"></a>
Chris@82 90 <p>FFTW computes an unnormalized transform, in that there is no coefficient
Chris@82 91 in front of the summation in the DFT. In other words, applying the
Chris@82 92 forward and then the backward transform will multiply the input by
Chris@82 93 <em>n</em>.
Chris@82 94 </p>
Chris@82 95 <a name="index-frequency-1"></a>
Chris@82 96 <p>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
Chris@82 97 <em>-1</em> in the exponent of the DFT. Note also that we use the
Chris@82 98 standard &ldquo;in-order&rdquo; output ordering&mdash;the <em>k</em>-th output
Chris@82 99 corresponds to the frequency <em>k/n</em> (or <em>k/T</em>, where <em>T</em>
Chris@82 100 is your total sampling period). For those who like to think in terms of
Chris@82 101 positive and negative frequencies, this means that the positive
Chris@82 102 frequencies are stored in the first half of the output and the negative
Chris@82 103 frequencies are stored in backwards order in the second half of the
Chris@82 104 output. (The frequency <em>-k/n</em> is the same as the frequency
Chris@82 105 <em>(n-k)/n</em>.)
Chris@82 106 </p>
Chris@82 107
Chris@82 108
Chris@82 109
Chris@82 110 </body>
Chris@82 111 </html>