101 lines
4.6 KiB
HTML
101 lines
4.6 KiB
HTML
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
|
|
<html>
|
|
<!-- This manual is for FFTW
|
|
(version 3.3.10, 10 December 2020).
|
|
|
|
Copyright (C) 2003 Matteo Frigo.
|
|
|
|
Copyright (C) 2003 Massachusetts Institute of Technology.
|
|
|
|
Permission is granted to make and distribute verbatim copies of this
|
|
manual provided the copyright notice and this permission notice are
|
|
preserved on all copies.
|
|
|
|
Permission is granted to copy and distribute modified versions of this
|
|
manual under the conditions for verbatim copying, provided that the
|
|
entire resulting derived work is distributed under the terms of a
|
|
permission notice identical to this one.
|
|
|
|
Permission is granted to copy and distribute translations of this manual
|
|
into another language, under the above conditions for modified versions,
|
|
except that this permission notice may be stated in a translation
|
|
approved by the Free Software Foundation. -->
|
|
<!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
|
<title>The 1d Discrete Fourier Transform (DFT) (FFTW 3.3.10)</title>
|
|
|
|
<meta name="description" content="The 1d Discrete Fourier Transform (DFT) (FFTW 3.3.10)">
|
|
<meta name="keywords" content="The 1d Discrete Fourier Transform (DFT) (FFTW 3.3.10)">
|
|
<meta name="resource-type" content="document">
|
|
<meta name="distribution" content="global">
|
|
<meta name="Generator" content="makeinfo">
|
|
<link href="index.html" rel="start" title="Top">
|
|
<link href="Concept-Index.html" rel="index" title="Concept Index">
|
|
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
|
|
<link href="What-FFTW-Really-Computes.html" rel="up" title="What FFTW Really Computes">
|
|
<link href="The-1d-Real_002ddata-DFT.html" rel="next" title="The 1d Real-data DFT">
|
|
<link href="What-FFTW-Really-Computes.html" rel="prev" title="What FFTW Really Computes">
|
|
<style type="text/css">
|
|
<!--
|
|
a.summary-letter {text-decoration: none}
|
|
blockquote.indentedblock {margin-right: 0em}
|
|
div.display {margin-left: 3.2em}
|
|
div.example {margin-left: 3.2em}
|
|
div.lisp {margin-left: 3.2em}
|
|
kbd {font-style: oblique}
|
|
pre.display {font-family: inherit}
|
|
pre.format {font-family: inherit}
|
|
pre.menu-comment {font-family: serif}
|
|
pre.menu-preformatted {font-family: serif}
|
|
span.nolinebreak {white-space: nowrap}
|
|
span.roman {font-family: initial; font-weight: normal}
|
|
span.sansserif {font-family: sans-serif; font-weight: normal}
|
|
ul.no-bullet {list-style: none}
|
|
-->
|
|
</style>
|
|
|
|
|
|
</head>
|
|
|
|
<body lang="en">
|
|
<span id="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></span><div class="header">
|
|
<p>
|
|
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>
|
|
</div>
|
|
<hr>
|
|
<span id="The-1d-Discrete-Fourier-Transform-_0028DFT_0029-1"></span><h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
|
|
|
|
<span id="index-discrete-Fourier-transform-1"></span>
|
|
<span id="index-DFT-2"></span>
|
|
<p>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
|
|
1d complex array <em>X</em> of size <em>n</em> computes an array <em>Y</em>,
|
|
where:
|
|
<center><img src="equation-dft.png" align="top">.</center>
|
|
The backward (<code>FFTW_BACKWARD</code>) DFT computes:
|
|
<center><img src="equation-idft.png" align="top">.</center>
|
|
</p>
|
|
<span id="index-normalization-8"></span>
|
|
<p>FFTW computes an unnormalized transform, in that there is no coefficient
|
|
in front of the summation in the DFT. In other words, applying the
|
|
forward and then the backward transform will multiply the input by
|
|
<em>n</em>.
|
|
</p>
|
|
<span id="index-frequency-1"></span>
|
|
<p>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
|
|
<em>-1</em> in the exponent of the DFT. Note also that we use the
|
|
standard “in-order” output ordering—the <em>k</em>-th output
|
|
corresponds to the frequency <em>k/n</em> (or <em>k/T</em>, where <em>T</em>
|
|
is your total sampling period). For those who like to think in terms of
|
|
positive and negative frequencies, this means that the positive
|
|
frequencies are stored in the first half of the output and the negative
|
|
frequencies are stored in backwards order in the second half of the
|
|
output. (The frequency <em>-k/n</em> is the same as the frequency
|
|
<em>(n-k)/n</em>.)
|
|
</p>
|
|
|
|
|
|
|
|
</body>
|
|
</html>
|