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author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
parents 7867fa7e1b6b
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cannam@127 3 <!-- This manual is for FFTW
cannam@127 4 (version 3.3.5, 30 July 2016).
cannam@127 5
cannam@127 6 Copyright (C) 2003 Matteo Frigo.
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cannam@127 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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cannam@127 24 <head>
cannam@127 25 <title>FFTW 3.3.5: More DFTs of Real Data</title>
cannam@127 26
cannam@127 27 <meta name="description" content="FFTW 3.3.5: More DFTs of Real Data">
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cannam@127 37 <link href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT" rel="next" title="The Halfcomplex-format DFT">
cannam@127 38 <link href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" rel="prev" title="Multi-Dimensional DFTs of Real Data">
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cannam@127 69 </head>
cannam@127 70
cannam@127 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
cannam@127 72 <a name="More-DFTs-of-Real-Data"></a>
cannam@127 73 <div class="header">
cannam@127 74 <p>
cannam@127 75 Previous: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" accesskey="p" rel="prev">Multi-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</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="More-DFTs-of-Real-Data-1"></a>
cannam@127 79 <h3 class="section">2.5 More DFTs of Real Data</h3>
cannam@127 80 <table class="menu" border="0" cellspacing="0">
cannam@127 81 <tr><td align="left" valign="top">&bull; <a href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT" accesskey="1">The Halfcomplex-format DFT</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
cannam@127 82 </td></tr>
cannam@127 83 <tr><td align="left" valign="top">&bull; <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="2">Real even/odd DFTs (cosine/sine transforms)</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
cannam@127 84 </td></tr>
cannam@127 85 <tr><td align="left" valign="top">&bull; <a href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform" accesskey="3">The Discrete Hartley Transform</a>:</td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
cannam@127 86 </td></tr>
cannam@127 87 </table>
cannam@127 88
cannam@127 89 <p>FFTW supports several other transform types via a unified <em>r2r</em>
cannam@127 90 (real-to-real) interface,
cannam@127 91 <a name="index-r2r"></a>
cannam@127 92 so called because it takes a real (<code>double</code>) array and outputs a
cannam@127 93 real array of the same size. These r2r transforms currently fall into
cannam@127 94 three categories: DFTs of real input and complex-Hermitian output in
cannam@127 95 halfcomplex format, DFTs of real input with even/odd symmetry
cannam@127 96 (a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete
cannam@127 97 Hartley transforms (DHTs), all described in more detail by the
cannam@127 98 following sections.
cannam@127 99 </p>
cannam@127 100 <p>The r2r transforms follow the by now familiar interface of creating an
cannam@127 101 <code>fftw_plan</code>, executing it with <code>fftw_execute(plan)</code>, and
cannam@127 102 destroying it with <code>fftw_destroy_plan(plan)</code>. Furthermore, all
cannam@127 103 r2r transforms share the same planner interface:
cannam@127 104 </p>
cannam@127 105 <div class="example">
cannam@127 106 <pre class="example">fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
cannam@127 107 fftw_r2r_kind kind, unsigned flags);
cannam@127 108 fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out,
cannam@127 109 fftw_r2r_kind kind0, fftw_r2r_kind kind1,
cannam@127 110 unsigned flags);
cannam@127 111 fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2,
cannam@127 112 double *in, double *out,
cannam@127 113 fftw_r2r_kind kind0,
cannam@127 114 fftw_r2r_kind kind1,
cannam@127 115 fftw_r2r_kind kind2,
cannam@127 116 unsigned flags);
cannam@127 117 fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
cannam@127 118 const fftw_r2r_kind *kind, unsigned flags);
cannam@127 119 </pre></div>
cannam@127 120 <a name="index-fftw_005fplan_005fr2r_005f1d"></a>
cannam@127 121 <a name="index-fftw_005fplan_005fr2r_005f2d"></a>
cannam@127 122 <a name="index-fftw_005fplan_005fr2r_005f3d"></a>
cannam@127 123 <a name="index-fftw_005fplan_005fr2r"></a>
cannam@127 124
cannam@127 125 <p>Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional
cannam@127 126 transforms for contiguous arrays in row-major order, transforming (real)
cannam@127 127 input to output of the same size, where <code>n</code> specifies the
cannam@127 128 <em>physical</em> dimensions of the arrays. All positive <code>n</code> are
cannam@127 129 supported (with the exception of <code>n=1</code> for the <code>FFTW_REDFT00</code>
cannam@127 130 kind, noted in the real-even subsection below); products of small
cannam@127 131 factors are most efficient (factorizing <code>n-1</code> and <code>n+1</code> for
cannam@127 132 <code>FFTW_REDFT00</code> and <code>FFTW_RODFT00</code> kinds, described below), but
cannam@127 133 an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>) algorithm is used even for prime sizes.
cannam@127 134 </p>
cannam@127 135 <p>Each dimension has a <em>kind</em> parameter, of type
cannam@127 136 <code>fftw_r2r_kind</code>, specifying the kind of r2r transform to be used
cannam@127 137 for that dimension.
cannam@127 138 <a name="index-kind-_0028r2r_0029"></a>
cannam@127 139 <a name="index-fftw_005fr2r_005fkind"></a>
cannam@127 140 (In the case of <code>fftw_plan_r2r</code>, this is an array <code>kind[rank]</code>
cannam@127 141 where <code>kind[i]</code> is the transform kind for the dimension
cannam@127 142 <code>n[i]</code>.) The kind can be one of a set of predefined constants,
cannam@127 143 defined in the following subsections.
cannam@127 144 </p>
cannam@127 145 <p>In other words, FFTW computes the separable product of the specified
cannam@127 146 r2r transforms over each dimension, which can be used e.g. for partial
cannam@127 147 differential equations with mixed boundary conditions. (For some r2r
cannam@127 148 kinds, notably the halfcomplex DFT and the DHT, such a separable
cannam@127 149 product is somewhat problematic in more than one dimension, however,
cannam@127 150 as is described below.)
cannam@127 151 </p>
cannam@127 152 <p>In the current version of FFTW, all r2r transforms except for the
cannam@127 153 halfcomplex type are computed via pre- or post-processing of
cannam@127 154 halfcomplex transforms, and they are therefore not as fast as they
cannam@127 155 could be. Since most other general DCT/DST codes employ a similar
cannam@127 156 algorithm, however, FFTW&rsquo;s implementation should provide at least
cannam@127 157 competitive performance.
cannam@127 158 </p>
cannam@127 159 <hr>
cannam@127 160 <div class="header">
cannam@127 161 <p>
cannam@127 162 Previous: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" accesskey="p" rel="prev">Multi-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</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@127 163 </div>
cannam@127 164
cannam@127 165
cannam@127 166
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