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Merge build update
author Chris Cannam
date Thu, 31 Oct 2019 13:36:58 +0000
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Chris@42 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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Chris@42 3 <!-- This manual is for FFTW
Chris@42 4 (version 3.3.5, 30 July 2016).
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Chris@42 6 Copyright (C) 2003 Matteo Frigo.
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Chris@42 24 <head>
Chris@42 25 <title>FFTW 3.3.5: The 1d Discrete Fourier Transform (DFT)</title>
Chris@42 26
Chris@42 27 <meta name="description" content="FFTW 3.3.5: The 1d Discrete Fourier Transform (DFT)">
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Chris@42 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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Chris@42 72 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
Chris@42 73 <div class="header">
Chris@42 74 <p>
Chris@42 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>
Chris@42 76 </div>
Chris@42 77 <hr>
Chris@42 78 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029-1"></a>
Chris@42 79 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
Chris@42 80
Chris@42 81 <a name="index-discrete-Fourier-transform-1"></a>
Chris@42 82 <a name="index-DFT-2"></a>
Chris@42 83 <p>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
Chris@42 84 1d complex array <em>X</em> of size <em>n</em> computes an array <em>Y</em>,
Chris@42 85 where:
Chris@42 86 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
Chris@42 87 <center><img src="equation-idft.png" align="top">.</center></p>
Chris@42 88 <a name="index-normalization-8"></a>
Chris@42 89 <p>FFTW computes an unnormalized transform, in that there is no coefficient
Chris@42 90 in front of the summation in the DFT. In other words, applying the
Chris@42 91 forward and then the backward transform will multiply the input by
Chris@42 92 <em>n</em>.
Chris@42 93 </p>
Chris@42 94 <a name="index-frequency-1"></a>
Chris@42 95 <p>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
Chris@42 96 <em>-1</em> in the exponent of the DFT. Note also that we use the
Chris@42 97 standard &ldquo;in-order&rdquo; output ordering&mdash;the <em>k</em>-th output
Chris@42 98 corresponds to the frequency <em>k/n</em> (or <em>k/T</em>, where <em>T</em>
Chris@42 99 is your total sampling period). For those who like to think in terms of
Chris@42 100 positive and negative frequencies, this means that the positive
Chris@42 101 frequencies are stored in the first half of the output and the negative
Chris@42 102 frequencies are stored in backwards order in the second half of the
Chris@42 103 output. (The frequency <em>-k/n</em> is the same as the frequency
Chris@42 104 <em>(n-k)/n</em>.)
Chris@42 105 </p>
Chris@42 106
Chris@42 107
Chris@42 108
Chris@42 109 </body>
Chris@42 110 </html>