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Replace these with versions built using an older toolset (so as to avoid ABI compatibilities when linking on Ubuntu 14.04 for packaging purposes)
author Chris Cannam
date Fri, 07 Feb 2020 11:51:13 +0000
parents d0c2a83c1364
children
rev   line source
Chris@82 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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Chris@82 3 <!-- This manual is for FFTW
Chris@82 4 (version 3.3.8, 24 May 2018).
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Chris@82 6 Copyright (C) 2003 Matteo Frigo.
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Chris@82 24 <head>
Chris@82 25 <title>FFTW 3.3.8: 1d Real-even DFTs (DCTs)</title>
Chris@82 26
Chris@82 27 <meta name="description" content="FFTW 3.3.8: 1d Real-even DFTs (DCTs)">
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Chris@82 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
Chris@82 36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
Chris@82 37 <link href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" rel="next" title="1d Real-odd DFTs (DSTs)">
Chris@82 38 <link href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" rel="prev" title="The 1d Real-data DFT">
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Chris@82 68 </head>
Chris@82 69
Chris@82 70 <body lang="en">
Chris@82 71 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
Chris@82 72 <div class="header">
Chris@82 73 <p>
Chris@82 74 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> &nbsp; [<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>
Chris@82 75 </div>
Chris@82 76 <hr>
Chris@82 77 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029-1"></a>
Chris@82 78 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
Chris@82 79
Chris@82 80 <p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
Chris@82 81 forward (and backward) DFTs as defined above, where the input array
Chris@82 82 <em>X</em> of length <em>N</em> is purely real and is also <em>even</em> symmetry. In
Chris@82 83 this case, the output array is likewise real and even symmetry.
Chris@82 84 <a name="index-real_002deven-DFT-1"></a>
Chris@82 85 <a name="index-REDFT-1"></a>
Chris@82 86 </p>
Chris@82 87
Chris@82 88 <a name="index-REDFT00"></a>
Chris@82 89 <p>For the case of <code>REDFT00</code>, this even symmetry means that
Chris@82 90 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,
Chris@82 91 where we take <em>X</em> to be periodic so that
Chris@82 92 <i>X<sub>N</sub> = X</i><sub>0</sub>.
Chris@82 93 Because of this redundancy, only the first <em>n</em> real numbers are
Chris@82 94 actually stored, where <em>N = 2(n-1)</em>.
Chris@82 95 </p>
Chris@82 96 <p>The proper definition of even symmetry for <code>REDFT10</code>,
Chris@82 97 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
Chris@82 98 because of the shifts by <em>1/2</em> of the input and/or output, although
Chris@82 99 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,
Chris@82 100 the sine terms in the DFT all cancel and the remaining cosine terms are
Chris@82 101 written explicitly below. This formulation often leads people to call
Chris@82 102 such a transform a <em>discrete cosine transform</em> (DCT), although it is
Chris@82 103 really just a special case of the DFT.
Chris@82 104 <a name="index-discrete-cosine-transform-2"></a>
Chris@82 105 <a name="index-DCT-2"></a>
Chris@82 106 </p>
Chris@82 107
Chris@82 108 <p>In each of the definitions below, we transform a real array <em>X</em> of
Chris@82 109 length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
Chris@82 110 </p>
Chris@82 111 <a name="REDFT00-_0028DCT_002dI_0029"></a>
Chris@82 112 <h4 class="subsubheading">REDFT00 (DCT-I)</h4>
Chris@82 113 <a name="index-REDFT00-1"></a>
Chris@82 114 <p>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
Chris@82 115 <center><img src="equation-redft00.png" align="top">.</center>
Chris@82 116 Note that this transform is not defined for <em>n=1</em>. For <em>n=2</em>,
Chris@82 117 the summation term above is dropped as you might expect.
Chris@82 118 </p>
Chris@82 119 <a name="REDFT10-_0028DCT_002dII_0029"></a>
Chris@82 120 <h4 class="subsubheading">REDFT10 (DCT-II)</h4>
Chris@82 121 <a name="index-REDFT10"></a>
Chris@82 122 <p>An <code>REDFT10</code> transform (type-II DCT, sometimes called &ldquo;the&rdquo; DCT) in FFTW is defined by:
Chris@82 123 <center><img src="equation-redft10.png" align="top">.</center>
Chris@82 124 </p>
Chris@82 125 <a name="REDFT01-_0028DCT_002dIII_0029"></a>
Chris@82 126 <h4 class="subsubheading">REDFT01 (DCT-III)</h4>
Chris@82 127 <a name="index-REDFT01"></a>
Chris@82 128 <p>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
Chris@82 129 <center><img src="equation-redft01.png" align="top">.</center>
Chris@82 130 In the case of <em>n=1</em>, this reduces to
Chris@82 131 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
Chris@82 132 Up to a scale factor (see below), this is the inverse of <code>REDFT10</code> (&ldquo;the&rdquo; DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the &ldquo;IDCT&rdquo;.
Chris@82 133 <a name="index-IDCT-3"></a>
Chris@82 134 </p>
Chris@82 135 <a name="REDFT11-_0028DCT_002dIV_0029"></a>
Chris@82 136 <h4 class="subsubheading">REDFT11 (DCT-IV)</h4>
Chris@82 137 <a name="index-REDFT11"></a>
Chris@82 138 <p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
Chris@82 139 <center><img src="equation-redft11.png" align="top">.</center>
Chris@82 140 </p>
Chris@82 141 <a name="Inverses-and-Normalization"></a>
Chris@82 142 <h4 class="subsubheading">Inverses and Normalization</h4>
Chris@82 143
Chris@82 144 <p>These definitions correspond directly to the unnormalized DFTs used
Chris@82 145 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
Chris@82 146 summations). The unnormalized inverse of <code>REDFT00</code> is
Chris@82 147 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
Chris@82 148 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
Chris@82 149 in the original array multiplied by <em>N</em>, where <em>N</em> is the
Chris@82 150 <em>logical</em> DFT size. For <code>REDFT00</code>, <em>N=2(n-1)</em> (note that
Chris@82 151 <em>n=1</em> is not defined); otherwise, <em>N=2n</em>.
Chris@82 152 <a name="index-normalization-10"></a>
Chris@82 153 </p>
Chris@82 154
Chris@82 155 <p>In defining the discrete cosine transform, some authors also include
Chris@82 156 additional factors of
Chris@82 157 &radic;2
Chris@82 158 (or its inverse) multiplying selected inputs and/or outputs. This is a
Chris@82 159 mostly cosmetic change that makes the transform orthogonal, but
Chris@82 160 sacrifices the direct equivalence to a symmetric DFT.
Chris@82 161 </p>
Chris@82 162 <hr>
Chris@82 163 <div class="header">
Chris@82 164 <p>
Chris@82 165 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> &nbsp; [<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>
Chris@82 166 </div>
Chris@82 167
Chris@82 168
Chris@82 169
Chris@82 170 </body>
Chris@82 171 </html>