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diff src/fftw-3.3.3/doc/html/1d-Real_002deven-DFTs-_0028DCTs_0029.html @ 95:89f5e221ed7b
Add FFTW3
| author | Chris Cannam <cannam@all-day-breakfast.com> |
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| date | Wed, 20 Mar 2013 15:35:50 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.3/doc/html/1d-Real_002deven-DFTs-_0028DCTs_0029.html Wed Mar 20 15:35:50 2013 +0000 @@ -0,0 +1,127 @@ +<html lang="en"> +<head> +<title>1d Real-even DFTs (DCTs) - FFTW 3.3.3</title> +<meta http-equiv="Content-Type" content="text/html"> +<meta name="description" content="FFTW 3.3.3"> +<meta name="generator" content="makeinfo 4.13"> +<link title="Top" rel="start" href="index.html#Top"> +<link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes"> +<link rel="prev" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" title="The 1d Real-data DFT"> +<link rel="next" href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" title="1d Real-odd DFTs (DSTs)"> +<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> +<!-- +This manual is for FFTW +(version 3.3.3, 25 November 2012). + +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. + --> +<meta http-equiv="Content-Style-Type" content="text/css"> +<style type="text/css"><!-- + pre.display { font-family:inherit } + pre.format { font-family:inherit } + pre.smalldisplay { font-family:inherit; font-size:smaller } + pre.smallformat { font-family:inherit; font-size:smaller } + pre.smallexample { font-size:smaller } + pre.smalllisp { font-size:smaller } + span.sc { font-variant:small-caps } + span.roman { font-family:serif; font-weight:normal; } + span.sansserif { font-family:sans-serif; font-weight:normal; } +--></style> +</head> +<body> +<div class="node"> +<a name="1d-Real-even-DFTs-(DCTs)"></a> +<a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a> +<p> +Next: <a rel="next" accesskey="n" href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029">1d Real-odd DFTs (DSTs)</a>, +Previous: <a rel="previous" accesskey="p" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT">The 1d Real-data DFT</a>, +Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a> +<hr> +</div> + +<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 +X of length N is purely real and is also <dfn>even</dfn> symmetry. In +this case, the output array is likewise real and even symmetry. +<a name="index-real_002deven-DFT-301"></a><a name="index-REDFT-302"></a> + + <p><a name="index-REDFT00-303"></a>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 X to be periodic so that +<i>X<sub>N</sub> = X</i><sub>0</sub>. Because of this redundancy, only the first n real numbers are +actually stored, where N = 2(n-1). + + <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 1/2 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 <dfn>discrete cosine transform</dfn> (DCT), although it is +really just a special case of the DFT. +<a name="index-discrete-cosine-transform-304"></a><a name="index-DCT-305"></a> + + <p>In each of the definitions below, we transform a real array X of +length n to a real array Y of length n: + +<h5 class="subsubheading">REDFT00 (DCT-I)</h5> + +<p><a name="index-REDFT00-306"></a>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 n=1. For n=2, +the summation term above is dropped as you might expect. + +<h5 class="subsubheading">REDFT10 (DCT-II)</h5> + +<p><a name="index-REDFT10-307"></a>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> + +<h5 class="subsubheading">REDFT01 (DCT-III)</h5> + +<p><a name="index-REDFT01-308"></a>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 n=1, 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-309"></a> + +<h5 class="subsubheading">REDFT11 (DCT-IV)</h5> + +<p><a name="index-REDFT11-310"></a>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by: +<center><img src="equation-redft11.png" align="top">.</center> + +<h5 class="subsubheading">Inverses and Normalization</h5> + +<p>These definitions correspond directly to the unnormalized DFTs used +elsewhere in FFTW (hence the factors of 2 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 N, where N is the +<em>logical</em> DFT size. For <code>REDFT00</code>, N=2(n-1) (note that +n=1 is not defined); otherwise, N=2n. +<a name="index-normalization-311"></a> + + <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. + +<!-- =========> --> + </body></html> +