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Add null config files
author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
parents 7867fa7e1b6b
children
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cannam@127 3 <!-- This manual is for FFTW
cannam@127 4 (version 3.3.5, 30 July 2016).
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cannam@127 6 Copyright (C) 2003 Matteo Frigo.
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cannam@127 24 <head>
cannam@127 25 <title>FFTW 3.3.5: 1d Real-even DFTs (DCTs)</title>
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cannam@127 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)">
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cannam@127 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
cannam@127 72 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
cannam@127 73 <div class="header">
cannam@127 74 <p>
cannam@127 75 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>
cannam@127 76 </div>
cannam@127 77 <hr>
cannam@127 78 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029-1"></a>
cannam@127 79 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
cannam@127 80
cannam@127 81 <p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
cannam@127 82 forward (and backward) DFTs as defined above, where the input array
cannam@127 83 <em>X</em> of length <em>N</em> is purely real and is also <em>even</em> symmetry. In
cannam@127 84 this case, the output array is likewise real and even symmetry.
cannam@127 85 <a name="index-real_002deven-DFT-1"></a>
cannam@127 86 <a name="index-REDFT-1"></a>
cannam@127 87 </p>
cannam@127 88
cannam@127 89 <a name="index-REDFT00"></a>
cannam@127 90 <p>For the case of <code>REDFT00</code>, this even symmetry means that
cannam@127 91 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,where we take <em>X</em> to be periodic so that
cannam@127 92 <i>X<sub>N</sub> = X</i><sub>0</sub>.Because of this redundancy, only the first <em>n</em> real numbers are
cannam@127 93 actually stored, where <em>N = 2(n-1)</em>.
cannam@127 94 </p>
cannam@127 95 <p>The proper definition of even symmetry for <code>REDFT10</code>,
cannam@127 96 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
cannam@127 97 because of the shifts by <em>1/2</em> of the input and/or output, although
cannam@127 98 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,
cannam@127 99 the sine terms in the DFT all cancel and the remaining cosine terms are
cannam@127 100 written explicitly below. This formulation often leads people to call
cannam@127 101 such a transform a <em>discrete cosine transform</em> (DCT), although it is
cannam@127 102 really just a special case of the DFT.
cannam@127 103 <a name="index-discrete-cosine-transform-2"></a>
cannam@127 104 <a name="index-DCT-2"></a>
cannam@127 105 </p>
cannam@127 106
cannam@127 107 <p>In each of the definitions below, we transform a real array <em>X</em> of
cannam@127 108 length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
cannam@127 109 </p>
cannam@127 110 <a name="REDFT00-_0028DCT_002dI_0029"></a>
cannam@127 111 <h4 class="subsubheading">REDFT00 (DCT-I)</h4>
cannam@127 112 <a name="index-REDFT00-1"></a>
cannam@127 113 <p>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
cannam@127 114 <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>,
cannam@127 115 the summation term above is dropped as you might expect.
cannam@127 116 </p>
cannam@127 117 <a name="REDFT10-_0028DCT_002dII_0029"></a>
cannam@127 118 <h4 class="subsubheading">REDFT10 (DCT-II)</h4>
cannam@127 119 <a name="index-REDFT10"></a>
cannam@127 120 <p>An <code>REDFT10</code> transform (type-II DCT, sometimes called &ldquo;the&rdquo; DCT) in FFTW is defined by:
cannam@127 121 <center><img src="equation-redft10.png" align="top">.</center></p>
cannam@127 122 <a name="REDFT01-_0028DCT_002dIII_0029"></a>
cannam@127 123 <h4 class="subsubheading">REDFT01 (DCT-III)</h4>
cannam@127 124 <a name="index-REDFT01"></a>
cannam@127 125 <p>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
cannam@127 126 <center><img src="equation-redft01.png" align="top">.</center>In the case of <em>n=1</em>, this reduces to
cannam@127 127 <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> (&ldquo;the&rdquo; DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the &ldquo;IDCT&rdquo;.
cannam@127 128 <a name="index-IDCT-3"></a>
cannam@127 129 </p>
cannam@127 130 <a name="REDFT11-_0028DCT_002dIV_0029"></a>
cannam@127 131 <h4 class="subsubheading">REDFT11 (DCT-IV)</h4>
cannam@127 132 <a name="index-REDFT11"></a>
cannam@127 133 <p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
cannam@127 134 <center><img src="equation-redft11.png" align="top">.</center></p>
cannam@127 135 <a name="Inverses-and-Normalization"></a>
cannam@127 136 <h4 class="subsubheading">Inverses and Normalization</h4>
cannam@127 137
cannam@127 138 <p>These definitions correspond directly to the unnormalized DFTs used
cannam@127 139 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
cannam@127 140 summations). The unnormalized inverse of <code>REDFT00</code> is
cannam@127 141 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
cannam@127 142 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
cannam@127 143 in the original array multiplied by <em>N</em>, where <em>N</em> is the
cannam@127 144 <em>logical</em> DFT size. For <code>REDFT00</code>, <em>N=2(n-1)</em> (note that
cannam@127 145 <em>n=1</em> is not defined); otherwise, <em>N=2n</em>.
cannam@127 146 <a name="index-normalization-10"></a>
cannam@127 147 </p>
cannam@127 148
cannam@127 149 <p>In defining the discrete cosine transform, some authors also include
cannam@127 150 additional factors of
cannam@127 151 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
cannam@127 152 mostly cosmetic change that makes the transform orthogonal, but
cannam@127 153 sacrifices the direct equivalence to a symmetric DFT.
cannam@127 154 </p>
cannam@127 155 <hr>
cannam@127 156 <div class="header">
cannam@127 157 <p>
cannam@127 158 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>
cannam@127 159 </div>
cannam@127 160
cannam@127 161
cannam@127 162
cannam@127 163 </body>
cannam@127 164 </html>