Mercurial > hg > batch-feature-extraction-tool
diff Lib/fftw-3.2.1/doc/html/.svn/text-base/1d-Real_002deven-DFTs-_0028DCTs_0029.html.svn-base @ 15:585caf503ef5 tip
Tidy up for ROLI
| author | Geogaddi\David <d.m.ronan@qmul.ac.uk> |
|---|---|
| date | Tue, 17 May 2016 18:50:19 +0100 |
| parents | 636c989477e7 |
| children |
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--- a/Lib/fftw-3.2.1/doc/html/.svn/text-base/1d-Real_002deven-DFTs-_0028DCTs_0029.html.svn-base Wed May 04 11:02:59 2016 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,124 +0,0 @@ -<html lang="en"> -<head> -<title>1d Real-even DFTs (DCTs) - FFTW 3.2.1</title> -<meta http-equiv="Content-Type" content="text/html"> -<meta name="description" content="FFTW 3.2.1"> -<meta name="generator" content="makeinfo 4.8"> -<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 3.2.1, 5 February 2009). - -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"> -<p> -<a name="1d-Real-even-DFTs-(DCTs)"></a> -<a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a> -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-291"></a><a name="index-REDFT-292"></a> -<a name="index-REDFT00-293"></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-294"></a><a name="index-DCT-295"></a> -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-REDFT00-296"></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-REDFT10-297"></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-REDFT01-298"></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-299"></a> - -<h5 class="subsubheading">REDFT11 (DCT-IV)</h5> - -<p><a name="index-REDFT11-300"></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-301"></a> -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> -