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Add null config files
author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
parents 7867fa7e1b6b
children
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cannam@127 3 <!-- This manual is for FFTW
cannam@127 4 (version 3.3.5, 30 July 2016).
cannam@127 5
cannam@127 6 Copyright (C) 2003 Matteo Frigo.
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cannam@127 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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cannam@127 24 <head>
cannam@127 25 <title>FFTW 3.3.5: The 1d Discrete Fourier Transform (DFT)</title>
cannam@127 26
cannam@127 27 <meta name="description" content="FFTW 3.3.5: The 1d Discrete Fourier Transform (DFT)">
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cannam@127 72 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
cannam@127 73 <div class="header">
cannam@127 74 <p>
cannam@127 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>
cannam@127 76 </div>
cannam@127 77 <hr>
cannam@127 78 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029-1"></a>
cannam@127 79 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
cannam@127 80
cannam@127 81 <a name="index-discrete-Fourier-transform-1"></a>
cannam@127 82 <a name="index-DFT-2"></a>
cannam@127 83 <p>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
cannam@127 84 1d complex array <em>X</em> of size <em>n</em> computes an array <em>Y</em>,
cannam@127 85 where:
cannam@127 86 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
cannam@127 87 <center><img src="equation-idft.png" align="top">.</center></p>
cannam@127 88 <a name="index-normalization-8"></a>
cannam@127 89 <p>FFTW computes an unnormalized transform, in that there is no coefficient
cannam@127 90 in front of the summation in the DFT. In other words, applying the
cannam@127 91 forward and then the backward transform will multiply the input by
cannam@127 92 <em>n</em>.
cannam@127 93 </p>
cannam@127 94 <a name="index-frequency-1"></a>
cannam@127 95 <p>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
cannam@127 96 <em>-1</em> in the exponent of the DFT. Note also that we use the
cannam@127 97 standard &ldquo;in-order&rdquo; output ordering&mdash;the <em>k</em>-th output
cannam@127 98 corresponds to the frequency <em>k/n</em> (or <em>k/T</em>, where <em>T</em>
cannam@127 99 is your total sampling period). For those who like to think in terms of
cannam@127 100 positive and negative frequencies, this means that the positive
cannam@127 101 frequencies are stored in the first half of the output and the negative
cannam@127 102 frequencies are stored in backwards order in the second half of the
cannam@127 103 output. (The frequency <em>-k/n</em> is the same as the frequency
cannam@127 104 <em>(n-k)/n</em>.)
cannam@127 105 </p>
cannam@127 106
cannam@127 107
cannam@127 108
cannam@127 109 </body>
cannam@127 110 </html>