mirror of
https://github.com/tildearrow/furnace.git
synced 2024-11-05 20:35:06 +00:00
156 lines
7.8 KiB
HTML
156 lines
7.8 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>1d Real-even DFTs (DCTs) (FFTW 3.3.10)</title>
|
||
|
|
|
||
|
|
<meta name="description" content="1d Real-even DFTs (DCTs) (FFTW 3.3.10)">
|
||
|
|
<meta name="keywords" content="1d Real-even DFTs (DCTs) (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_002dodd-DFTs-_0028DSTs_0029.html" rel="next" title="1d Real-odd DFTs (DSTs)">
|
||
|
|
<link href="The-1d-Real_002ddata-DFT.html" rel="prev" title="The 1d Real-data 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="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></span><div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html" accesskey="p" rel="prev">The 1d Real-data 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="g_t1d-Real_002deven-DFTs-_0028DCTs_0029-1"></span><h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
|
||
|
|
|
||
|
|
<p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
|
||
|
|
forward (and backward) DFTs as defined above, where the input array
|
||
|
|
<em>X</em> of length <em>N</em> is purely real and is also <em>even</em> symmetry. In
|
||
|
|
this case, the output array is likewise real and even symmetry.
|
||
|
|
<span id="index-real_002deven-DFT-1"></span>
|
||
|
|
<span id="index-REDFT-1"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<span id="index-REDFT00"></span>
|
||
|
|
<p>For the case of <code>REDFT00</code>, this even symmetry means that
|
||
|
|
<i>X<sub>j</sub> = X<sub>N-j</sub></i>,
|
||
|
|
where we take <em>X</em> to be periodic so that
|
||
|
|
<i>X<sub>N</sub> = X</i><sub>0</sub>.
|
||
|
|
Because of this redundancy, only the first <em>n</em> real numbers are
|
||
|
|
actually stored, where <em>N = 2(n-1)</em>.
|
||
|
|
</p>
|
||
|
|
<p>The proper definition of even symmetry for <code>REDFT10</code>,
|
||
|
|
<code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
|
||
|
|
because of the shifts by <em>1/2</em> of the input and/or output, although
|
||
|
|
the corresponding boundary conditions are given in <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html">Real even/odd DFTs (cosine/sine transforms)</a>. Because of the even symmetry, however,
|
||
|
|
the sine terms in the DFT all cancel and the remaining cosine terms are
|
||
|
|
written explicitly below. This formulation often leads people to call
|
||
|
|
such a transform a <em>discrete cosine transform</em> (DCT), although it is
|
||
|
|
really just a special case of the DFT.
|
||
|
|
<span id="index-discrete-cosine-transform-2"></span>
|
||
|
|
<span id="index-DCT-2"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>In each of the definitions below, we transform a real array <em>X</em> of
|
||
|
|
length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
|
||
|
|
</p>
|
||
|
|
<span id="REDFT00-_0028DCT_002dI_0029"></span><h4 class="subsubheading">REDFT00 (DCT-I)</h4>
|
||
|
|
<span id="index-REDFT00-1"></span>
|
||
|
|
<p>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
|
||
|
|
<center><img src="equation-redft00.png" align="top">.</center>
|
||
|
|
Note that this transform is not defined for <em>n=1</em>. For <em>n=2</em>,
|
||
|
|
the summation term above is dropped as you might expect.
|
||
|
|
</p>
|
||
|
|
<span id="REDFT10-_0028DCT_002dII_0029"></span><h4 class="subsubheading">REDFT10 (DCT-II)</h4>
|
||
|
|
<span id="index-REDFT10"></span>
|
||
|
|
<p>An <code>REDFT10</code> transform (type-II DCT, sometimes called “the” DCT) in FFTW is defined by:
|
||
|
|
<center><img src="equation-redft10.png" align="top">.</center>
|
||
|
|
</p>
|
||
|
|
<span id="REDFT01-_0028DCT_002dIII_0029"></span><h4 class="subsubheading">REDFT01 (DCT-III)</h4>
|
||
|
|
<span id="index-REDFT01"></span>
|
||
|
|
<p>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
|
||
|
|
<center><img src="equation-redft01.png" align="top">.</center>
|
||
|
|
In the case of <em>n=1</em>, this reduces to
|
||
|
|
<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
|
||
|
|
Up to a scale factor (see below), this is the inverse of <code>REDFT10</code> (“the” DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the “IDCT”.
|
||
|
|
<span id="index-IDCT-3"></span>
|
||
|
|
</p>
|
||
|
|
<span id="REDFT11-_0028DCT_002dIV_0029"></span><h4 class="subsubheading">REDFT11 (DCT-IV)</h4>
|
||
|
|
<span id="index-REDFT11"></span>
|
||
|
|
<p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
|
||
|
|
<center><img src="equation-redft11.png" align="top">.</center>
|
||
|
|
</p>
|
||
|
|
<span id="Inverses-and-Normalization"></span><h4 class="subsubheading">Inverses and Normalization</h4>
|
||
|
|
|
||
|
|
<p>These definitions correspond directly to the unnormalized DFTs used
|
||
|
|
elsewhere in FFTW (hence the factors of <em>2</em> in front of the
|
||
|
|
summations). The unnormalized inverse of <code>REDFT00</code> is
|
||
|
|
<code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
|
||
|
|
of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
|
||
|
|
in the original array multiplied by <em>N</em>, where <em>N</em> is the
|
||
|
|
<em>logical</em> DFT size. For <code>REDFT00</code>, <em>N=2(n-1)</em> (note that
|
||
|
|
<em>n=1</em> is not defined); otherwise, <em>N=2n</em>.
|
||
|
|
<span id="index-normalization-10"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>In defining the discrete cosine transform, some authors also include
|
||
|
|
additional factors of
|
||
|
|
√2
|
||
|
|
(or its inverse) multiplying selected inputs and/or outputs. This is a
|
||
|
|
mostly cosmetic change that makes the transform orthogonal, but
|
||
|
|
sacrifices the direct equivalence to a symmetric DFT.
|
||
|
|
</p>
|
||
|
|
<hr>
|
||
|
|
<div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html" accesskey="p" rel="prev">The 1d Real-data 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>
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
</body>
|
||
|
|
</html>
|