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100 lines
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100 lines
4.6 KiB
HTML
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<html>
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<!-- This manual is for FFTW
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(version 3.3.10, 10 December 2020).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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Permission is granted to make and distribute verbatim copies of this
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manual provided the copyright notice and this permission notice are
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preserved on all copies.
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Permission is granted to copy and distribute modified versions of this
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manual under the conditions for verbatim copying, provided that the
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entire resulting derived work is distributed under the terms of a
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permission notice identical to this one.
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Permission is granted to copy and distribute translations of this manual
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into another language, under the above conditions for modified versions,
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except that this permission notice may be stated in a translation
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approved by the Free Software Foundation. -->
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>The 1d Discrete Fourier Transform (DFT) (FFTW 3.3.10)</title>
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<meta name="description" content="The 1d Discrete Fourier Transform (DFT) (FFTW 3.3.10)">
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<link href="index.html" rel="start" title="Top">
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<link href="Concept-Index.html" rel="index" title="Concept Index">
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<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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<link href="What-FFTW-Really-Computes.html" rel="up" title="What FFTW Really Computes">
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<link href="The-1d-Real_002ddata-DFT.html" rel="next" title="The 1d Real-data DFT">
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</head>
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<body lang="en">
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<span id="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></span><div class="header">
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<p>
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Next: <a href="The-1d-Real_002ddata-DFT.html" accesskey="n" rel="next">The 1d Real-data DFT</a>, Previous: <a href="What-FFTW-Really-Computes.html" accesskey="p" rel="prev">What FFTW Really Computes</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<span id="The-1d-Discrete-Fourier-Transform-_0028DFT_0029-1"></span><h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
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<span id="index-discrete-Fourier-transform-1"></span>
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<span id="index-DFT-2"></span>
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<p>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
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1d complex array <em>X</em> of size <em>n</em> computes an array <em>Y</em>,
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where:
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<center><img src="equation-dft.png" align="top">.</center>
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The backward (<code>FFTW_BACKWARD</code>) DFT computes:
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<center><img src="equation-idft.png" align="top">.</center>
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</p>
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<span id="index-normalization-8"></span>
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<p>FFTW computes an unnormalized transform, in that there is no coefficient
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in front of the summation in the DFT. In other words, applying the
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forward and then the backward transform will multiply the input by
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<em>n</em>.
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</p>
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<span id="index-frequency-1"></span>
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<p>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
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<em>-1</em> in the exponent of the DFT. Note also that we use the
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standard “in-order” output ordering—the <em>k</em>-th output
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corresponds to the frequency <em>k/n</em> (or <em>k/T</em>, where <em>T</em>
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is your total sampling period). For those who like to think in terms of
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positive and negative frequencies, this means that the positive
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frequencies are stored in the first half of the output and the negative
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frequencies are stored in backwards order in the second half of the
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output. (The frequency <em>-k/n</em> is the same as the frequency
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<em>(n-k)/n</em>.)
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</p>
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</body>
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</html>
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