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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 bd3cc4d1df30
children
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cannam@167 3 <!-- This manual is for FFTW
cannam@167 4 (version 3.3.8, 24 May 2018).
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cannam@167 6 Copyright (C) 2003 Matteo Frigo.
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cannam@167 24 <head>
cannam@167 25 <title>FFTW 3.3.8: 1d Real-odd DFTs (DSTs)</title>
cannam@167 26
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cannam@167 70 <body lang="en">
cannam@167 71 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></a>
cannam@167 72 <div class="header">
cannam@167 73 <p>
cannam@167 74 Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</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@167 75 </div>
cannam@167 76 <hr>
cannam@167 77 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029-1"></a>
cannam@167 78 <h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>
cannam@167 79
cannam@167 80 <p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
cannam@167 81 forward (and backward) DFTs as defined above, where the input array
cannam@167 82 <em>X</em> of length <em>N</em> is purely real and is also <em>odd</em> symmetry. In
cannam@167 83 this case, the output is odd symmetry and purely imaginary.
cannam@167 84 <a name="index-real_002dodd-DFT-1"></a>
cannam@167 85 <a name="index-RODFT-1"></a>
cannam@167 86 </p>
cannam@167 87
cannam@167 88 <a name="index-RODFT00"></a>
cannam@167 89 <p>For the case of <code>RODFT00</code>, this odd symmetry means that
cannam@167 90 <i>X<sub>j</sub> = -X<sub>N-j</sub></i>,
cannam@167 91 where we take <em>X</em> to be periodic so that
cannam@167 92 <i>X<sub>N</sub> = X</i><sub>0</sub>.
cannam@167 93 Because of this redundancy, only the first <em>n</em> real numbers
cannam@167 94 starting at <em>j=1</em> are actually stored (the <em>j=0</em> element is
cannam@167 95 zero), where <em>N = 2(n+1)</em>.
cannam@167 96 </p>
cannam@167 97 <p>The proper definition of odd symmetry for <code>RODFT10</code>,
cannam@167 98 <code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
cannam@167 99 because of the shifts by <em>1/2</em> of the input and/or output, although
cannam@167 100 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 odd symmetry, however,
cannam@167 101 the cosine terms in the DFT all cancel and the remaining sine terms are
cannam@167 102 written explicitly below. This formulation often leads people to call
cannam@167 103 such a transform a <em>discrete sine transform</em> (DST), although it is
cannam@167 104 really just a special case of the DFT.
cannam@167 105 <a name="index-discrete-sine-transform-2"></a>
cannam@167 106 <a name="index-DST-2"></a>
cannam@167 107 </p>
cannam@167 108
cannam@167 109 <p>In each of the definitions below, we transform a real array <em>X</em> of
cannam@167 110 length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
cannam@167 111 </p>
cannam@167 112 <a name="RODFT00-_0028DST_002dI_0029"></a>
cannam@167 113 <h4 class="subsubheading">RODFT00 (DST-I)</h4>
cannam@167 114 <a name="index-RODFT00-1"></a>
cannam@167 115 <p>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
cannam@167 116 <center><img src="equation-rodft00.png" align="top">.</center>
cannam@167 117 </p>
cannam@167 118 <a name="RODFT10-_0028DST_002dII_0029"></a>
cannam@167 119 <h4 class="subsubheading">RODFT10 (DST-II)</h4>
cannam@167 120 <a name="index-RODFT10"></a>
cannam@167 121 <p>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
cannam@167 122 <center><img src="equation-rodft10.png" align="top">.</center>
cannam@167 123 </p>
cannam@167 124 <a name="RODFT01-_0028DST_002dIII_0029"></a>
cannam@167 125 <h4 class="subsubheading">RODFT01 (DST-III)</h4>
cannam@167 126 <a name="index-RODFT01"></a>
cannam@167 127 <p>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
cannam@167 128 <center><img src="equation-rodft01.png" align="top">.</center>
cannam@167 129 In the case of <em>n=1</em>, this reduces to
cannam@167 130 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
cannam@167 131 </p>
cannam@167 132 <a name="RODFT11-_0028DST_002dIV_0029"></a>
cannam@167 133 <h4 class="subsubheading">RODFT11 (DST-IV)</h4>
cannam@167 134 <a name="index-RODFT11"></a>
cannam@167 135 <p>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
cannam@167 136 <center><img src="equation-rodft11.png" align="top">.</center>
cannam@167 137 </p>
cannam@167 138 <a name="Inverses-and-Normalization-1"></a>
cannam@167 139 <h4 class="subsubheading">Inverses and Normalization</h4>
cannam@167 140
cannam@167 141 <p>These definitions correspond directly to the unnormalized DFTs used
cannam@167 142 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
cannam@167 143 summations). The unnormalized inverse of <code>RODFT00</code> is
cannam@167 144 <code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
cannam@167 145 of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
cannam@167 146 in the original array multiplied by <em>N</em>, where <em>N</em> is the
cannam@167 147 <em>logical</em> DFT size. For <code>RODFT00</code>, <em>N=2(n+1)</em>;
cannam@167 148 otherwise, <em>N=2n</em>.
cannam@167 149 <a name="index-normalization-11"></a>
cannam@167 150 </p>
cannam@167 151
cannam@167 152 <p>In defining the discrete sine transform, some authors also include
cannam@167 153 additional factors of
cannam@167 154 &radic;2
cannam@167 155 (or its inverse) multiplying selected inputs and/or outputs. This is a
cannam@167 156 mostly cosmetic change that makes the transform orthogonal, but
cannam@167 157 sacrifices the direct equivalence to an antisymmetric DFT.
cannam@167 158 </p>
cannam@167 159 <hr>
cannam@167 160 <div class="header">
cannam@167 161 <p>
cannam@167 162 Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</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@167 163 </div>
cannam@167 164
cannam@167 165
cannam@167 166
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