mirror of
https://github.com/tildearrow/furnace.git
synced 2024-11-30 16:33:01 +00:00
115 lines
5.1 KiB
HTML
115 lines
5.1 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 Real-data DFT (FFTW 3.3.10)</title>
|
|
|
|
<meta name="description" content="The 1d Real-data DFT (FFTW 3.3.10)">
|
|
<meta name="keywords" content="The 1d Real-data 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="1d-Real_002deven-DFTs-_0028DCTs_0029.html" rel="next" title="1d Real-even DFTs (DCTs)">
|
|
<link href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html" rel="prev" title="The 1d Discrete Fourier Transform (DFT)">
|
|
<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-Real_002ddata-DFT"></span><div class="header">
|
|
<p>
|
|
Next: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" accesskey="n" rel="next">1d Real-even DFTs (DCTs)</a>, Previous: <a href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html" accesskey="p" rel="prev">The 1d Discrete Fourier Transform (DFT)</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-Real_002ddata-DFT-1"></span><h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>
|
|
|
|
<p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
|
|
<em>Y</em> of the size <code>n</code> real array <em>X</em>, exactly as defined
|
|
above, i.e.
|
|
<center><img src="equation-dft.png" align="top">.</center>
|
|
This output array <em>Y</em> can easily be shown to possess the
|
|
“Hermitian” symmetry
|
|
<span id="index-Hermitian-1"></span>
|
|
<i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,
|
|
where we take <em>Y</em> to be periodic so that
|
|
<i>Y<sub>n</sub> = Y</i><sub>0</sub>.
|
|
</p>
|
|
<p>As a result of this symmetry, half of the output <em>Y</em> is redundant
|
|
(being the complex conjugate of the other half), and so the 1d r2c
|
|
transforms only output elements <em>0</em>…<em>n/2</em> of <em>Y</em>
|
|
(<em>n/2+1</em> complex numbers), where the division by <em>2</em> is
|
|
rounded down.
|
|
</p>
|
|
<p>Moreover, the Hermitian symmetry implies that
|
|
<i>Y</i><sub>0</sub>
|
|
and, if <em>n</em> is even, the
|
|
<i>Y</i><sub><i>n</i>/2</sub>
|
|
element, are purely real. So, for the <code>R2HC</code> r2r transform, the
|
|
halfcomplex format does not store the imaginary parts of these elements.
|
|
<span id="index-r2r-2"></span>
|
|
<span id="index-R2HC"></span>
|
|
<span id="index-halfcomplex-format-2"></span>
|
|
</p>
|
|
|
|
<p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
|
|
<em>complex</em> array <em>X</em> with Hermitian symmetry, stored in the
|
|
r2c/<code>R2HC</code> output formats, respectively, where the backward
|
|
transform is defined exactly as for the complex case:
|
|
<center><img src="equation-idft.png" align="top">.</center>
|
|
The outputs <code>Y</code> of this transform can easily be seen to be purely
|
|
real, and are stored as an array of real numbers.
|
|
</p>
|
|
<span id="index-normalization-9"></span>
|
|
<p>Like FFTW’s complex DFT, these transforms are unnormalized. In other
|
|
words, applying the real-to-complex (forward) and then the
|
|
complex-to-real (backward) transform will multiply the input by
|
|
<em>n</em>.
|
|
</p>
|
|
|
|
|
|
|
|
</body>
|
|
</html>
|