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Current fftw source
| author | Chris Cannam |
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| date | Tue, 18 Oct 2016 13:40:26 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.5/doc/html/1d-Real_002deven-DFTs-_0028DCTs_0029.html Tue Oct 18 13:40:26 2016 +0100 @@ -0,0 +1,164 @@ +<!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 3.3.5, 30 July 2016). + +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 5.2, http://www.gnu.org/software/texinfo/ --> +<head> +<title>FFTW 3.3.5: 1d Real-even DFTs (DCTs)</title> + +<meta name="description" content="FFTW 3.3.5: 1d Real-even DFTs (DCTs)"> +<meta name="keywords" content="FFTW 3.3.5: 1d Real-even DFTs (DCTs)"> +<meta name="resource-type" content="document"> +<meta name="distribution" content="global"> +<meta name="Generator" content="makeinfo"> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8"> +<link href="index.html#Top" rel="start" title="Top"> +<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> +<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> +<link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes"> +<link href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" rel="next" title="1d Real-odd DFTs (DSTs)"> +<link href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" rel="prev" title="The 1d Real-data DFT"> +<style type="text/css"> +<!-- +a.summary-letter {text-decoration: none} +blockquote.smallquotation {font-size: smaller} +div.display {margin-left: 3.2em} +div.example {margin-left: 3.2em} +div.indentedblock {margin-left: 3.2em} +div.lisp {margin-left: 3.2em} +div.smalldisplay {margin-left: 3.2em} +div.smallexample {margin-left: 3.2em} +div.smallindentedblock {margin-left: 3.2em; font-size: smaller} +div.smalllisp {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} +pre.smalldisplay {font-family: inherit; font-size: smaller} +pre.smallexample {font-size: smaller} +pre.smallformat {font-family: inherit; font-size: smaller} +pre.smalllisp {font-size: smaller} +span.nocodebreak {white-space:nowrap} +span.nolinebreak {white-space:nowrap} +span.roman {font-family:serif; font-weight:normal} +span.sansserif {font-family:sans-serif; font-weight:normal} +ul.no-bullet {list-style: none} +--> +</style> + + +</head> + +<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000"> +<a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a> +<div class="header"> +<p> +Next: <a href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" accesskey="p" rel="prev">The 1d Real-data DFT</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" 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#Concept-Index" title="Index" rel="index">Index</a>]</p> +</div> +<hr> +<a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029-1"></a> +<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. +<a name="index-real_002deven-DFT-1"></a> +<a name="index-REDFT-1"></a> +</p> + +<a name="index-REDFT00"></a> +<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_002fodd-DFTs-_0028cosine_002fsine-transforms_0029">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. +<a name="index-discrete-cosine-transform-2"></a> +<a name="index-DCT-2"></a> +</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> +<a name="REDFT00-_0028DCT_002dI_0029"></a> +<h4 class="subsubheading">REDFT00 (DCT-I)</h4> +<a name="index-REDFT00-1"></a> +<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> +<a name="REDFT10-_0028DCT_002dII_0029"></a> +<h4 class="subsubheading">REDFT10 (DCT-II)</h4> +<a name="index-REDFT10"></a> +<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> +<a name="REDFT01-_0028DCT_002dIII_0029"></a> +<h4 class="subsubheading">REDFT01 (DCT-III)</h4> +<a name="index-REDFT01"></a> +<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”. +<a name="index-IDCT-3"></a> +</p> +<a name="REDFT11-_0028DCT_002dIV_0029"></a> +<h4 class="subsubheading">REDFT11 (DCT-IV)</h4> +<a name="index-REDFT11"></a> +<p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by: +<center><img src="equation-redft11.png" align="top">.</center></p> +<a name="Inverses-and-Normalization"></a> +<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>. +<a name="index-normalization-10"></a> +</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#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" accesskey="p" rel="prev">The 1d Real-data DFT</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" 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#Concept-Index" title="Index" rel="index">Index</a>]</p> +</div> + + + +</body> +</html>