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2 <head>
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3 <title>More DFTs of Real Data - FFTW 3.2.1</title>
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4 <meta http-equiv="Content-Type" content="text/html">
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5 <meta name="description" content="FFTW 3.2.1">
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7 <link title="Top" rel="start" href="index.html#Top">
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8 <link rel="up" href="Tutorial.html#Tutorial" title="Tutorial">
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9 <link rel="prev" href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" title="Multi-Dimensional DFTs of Real Data">
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10 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
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11 <!--
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12 This manual is for FFTW
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13 (version 3.2.1, 5 February 2009).
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14
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15 Copyright (C) 2003 Matteo Frigo.
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16
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17 Copyright (C) 2003 Massachusetts Institute of Technology.
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18
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19 Permission is granted to make and distribute verbatim copies of
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20 this manual provided the copyright notice and this permission
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21 notice are preserved on all copies.
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22
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23 Permission is granted to copy and distribute modified versions of
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24 this manual under the conditions for verbatim copying, provided
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25 that the entire resulting derived work is distributed under the
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26 terms of a permission notice identical to this one.
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27
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28 Permission is granted to copy and distribute translations of this
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29 manual into another language, under the above conditions for
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30 modified versions, except that this permission notice may be
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31 stated in a translation approved by the Free Software Foundation.
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32 -->
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44 --></style>
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45 </head>
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46 <body>
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47 <div class="node">
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48 <p>
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49 <a name="More-DFTs-of-Real-Data"></a>
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50 Previous: <a rel="previous" accesskey="p" href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>,
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51 Up: <a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a>
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52 <hr>
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53 </div>
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54
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55 <h3 class="section">2.5 More DFTs of Real Data</h3>
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56
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57 <ul class="menu">
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58 <li><a accesskey="1" href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>
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59 <li><a accesskey="2" 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>
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60 <li><a accesskey="3" href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>
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61 </ul>
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62
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63 <p>FFTW supports several other transform types via a unified <dfn>r2r</dfn>
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64 (real-to-real) interface,
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65 <a name="index-r2r-64"></a>so called because it takes a real (<code>double</code>) array and outputs a
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66 real array of the same size. These r2r transforms currently fall into
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67 three categories: DFTs of real input and complex-Hermitian output in
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68 halfcomplex format, DFTs of real input with even/odd symmetry
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69 (a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete
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70 Hartley transforms (DHTs), all described in more detail by the
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71 following sections.
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72
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73 <p>The r2r transforms follow the by now familiar interface of creating an
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74 <code>fftw_plan</code>, executing it with <code>fftw_execute(plan)</code>, and
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75 destroying it with <code>fftw_destroy_plan(plan)</code>. Furthermore, all
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76 r2r transforms share the same planner interface:
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77
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78 <pre class="example"> fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
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79 fftw_r2r_kind kind, unsigned flags);
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80 fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out,
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81 fftw_r2r_kind kind0, fftw_r2r_kind kind1,
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82 unsigned flags);
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83 fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2,
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84 double *in, double *out,
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85 fftw_r2r_kind kind0,
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86 fftw_r2r_kind kind1,
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87 fftw_r2r_kind kind2,
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88 unsigned flags);
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89 fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
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90 const fftw_r2r_kind *kind, unsigned flags);
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91 </pre>
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92 <p><a name="index-fftw_005fplan_005fr2r_005f1d-65"></a><a name="index-fftw_005fplan_005fr2r_005f2d-66"></a><a name="index-fftw_005fplan_005fr2r_005f3d-67"></a><a name="index-fftw_005fplan_005fr2r-68"></a>
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93 Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional
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94 transforms for contiguous arrays in row-major order, transforming (real)
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95 input to output of the same size, where <code>n</code> specifies the
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96 <em>physical</em> dimensions of the arrays. All positive <code>n</code> are
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97 supported (with the exception of <code>n=1</code> for the <code>FFTW_REDFT00</code>
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98 kind, noted in the real-even subsection below); products of small
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99 factors are most efficient (factorizing <code>n-1</code> and <code>n+1</code> for
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100 <code>FFTW_REDFT00</code> and <code>FFTW_RODFT00</code> kinds, described below), but
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101 an <i>O</i>(<i>n</i> log <i>n</i>) algorithm is used even for prime sizes.
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102
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103 <p>Each dimension has a <dfn>kind</dfn> parameter, of type
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104 <code>fftw_r2r_kind</code>, specifying the kind of r2r transform to be used
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105 for that dimension.
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106 <a name="index-kind-_0028r2r_0029-69"></a><a name="index-fftw_005fr2r_005fkind-70"></a>(In the case of <code>fftw_plan_r2r</code>, this is an array <code>kind[rank]</code>
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107 where <code>kind[i]</code> is the transform kind for the dimension
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108 <code>n[i]</code>.) The kind can be one of a set of predefined constants,
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109 defined in the following subsections.
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110
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111 <p>In other words, FFTW computes the separable product of the specified
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112 r2r transforms over each dimension, which can be used e.g. for partial
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113 differential equations with mixed boundary conditions. (For some r2r
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114 kinds, notably the halfcomplex DFT and the DHT, such a separable
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115 product is somewhat problematic in more than one dimension, however,
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116 as is described below.)
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117
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118 <p>In the current version of FFTW, all r2r transforms except for the
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119 halfcomplex type are computed via pre- or post-processing of
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120 halfcomplex transforms, and they are therefore not as fast as they
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121 could be. Since most other general DCT/DST codes employ a similar
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122 algorithm, however, FFTW's implementation should provide at least
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123 competitive performance.
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124
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125 <!-- =========> -->
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126 </body></html>
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127
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