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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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Chris@10 1 <html lang="en">
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Chris@10 3 <title>More DFTs of Real Data - FFTW 3.3.3</title>
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Chris@10 48 <a name="More-DFTs-of-Real-Data"></a>
Chris@10 49 <p>
Chris@10 50 Previous:&nbsp;<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>,
Chris@10 51 Up:&nbsp;<a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a>
Chris@10 52 <hr>
Chris@10 53 </div>
Chris@10 54
Chris@10 55 <h3 class="section">2.5 More DFTs of Real Data</h3>
Chris@10 56
Chris@10 57 <ul class="menu">
Chris@10 58 <li><a accesskey="1" href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>
Chris@10 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>
Chris@10 60 <li><a accesskey="3" href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>
Chris@10 61 </ul>
Chris@10 62
Chris@10 63 <p>FFTW supports several other transform types via a unified <dfn>r2r</dfn>
Chris@10 64 (real-to-real) interface,
Chris@10 65 <a name="index-r2r-65"></a>so called because it takes a real (<code>double</code>) array and outputs a
Chris@10 66 real array of the same size. These r2r transforms currently fall into
Chris@10 67 three categories: DFTs of real input and complex-Hermitian output in
Chris@10 68 halfcomplex format, DFTs of real input with even/odd symmetry
Chris@10 69 (a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete
Chris@10 70 Hartley transforms (DHTs), all described in more detail by the
Chris@10 71 following sections.
Chris@10 72
Chris@10 73 <p>The r2r transforms follow the by now familiar interface of creating an
Chris@10 74 <code>fftw_plan</code>, executing it with <code>fftw_execute(plan)</code>, and
Chris@10 75 destroying it with <code>fftw_destroy_plan(plan)</code>. Furthermore, all
Chris@10 76 r2r transforms share the same planner interface:
Chris@10 77
Chris@10 78 <pre class="example"> fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
Chris@10 79 fftw_r2r_kind kind, unsigned flags);
Chris@10 80 fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out,
Chris@10 81 fftw_r2r_kind kind0, fftw_r2r_kind kind1,
Chris@10 82 unsigned flags);
Chris@10 83 fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2,
Chris@10 84 double *in, double *out,
Chris@10 85 fftw_r2r_kind kind0,
Chris@10 86 fftw_r2r_kind kind1,
Chris@10 87 fftw_r2r_kind kind2,
Chris@10 88 unsigned flags);
Chris@10 89 fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
Chris@10 90 const fftw_r2r_kind *kind, unsigned flags);
Chris@10 91 </pre>
Chris@10 92 <p><a name="index-fftw_005fplan_005fr2r_005f1d-66"></a><a name="index-fftw_005fplan_005fr2r_005f2d-67"></a><a name="index-fftw_005fplan_005fr2r_005f3d-68"></a><a name="index-fftw_005fplan_005fr2r-69"></a>
Chris@10 93 Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional
Chris@10 94 transforms for contiguous arrays in row-major order, transforming (real)
Chris@10 95 input to output of the same size, where <code>n</code> specifies the
Chris@10 96 <em>physical</em> dimensions of the arrays. All positive <code>n</code> are
Chris@10 97 supported (with the exception of <code>n=1</code> for the <code>FFTW_REDFT00</code>
Chris@10 98 kind, noted in the real-even subsection below); products of small
Chris@10 99 factors are most efficient (factorizing <code>n-1</code> and <code>n+1</code> for
Chris@10 100 <code>FFTW_REDFT00</code> and <code>FFTW_RODFT00</code> kinds, described below), but
Chris@10 101 an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>) algorithm is used even for prime sizes.
Chris@10 102
Chris@10 103 <p>Each dimension has a <dfn>kind</dfn> parameter, of type
Chris@10 104 <code>fftw_r2r_kind</code>, specifying the kind of r2r transform to be used
Chris@10 105 for that dimension.
Chris@10 106 <a name="index-kind-_0028r2r_0029-70"></a><a name="index-fftw_005fr2r_005fkind-71"></a>(In the case of <code>fftw_plan_r2r</code>, this is an array <code>kind[rank]</code>
Chris@10 107 where <code>kind[i]</code> is the transform kind for the dimension
Chris@10 108 <code>n[i]</code>.) The kind can be one of a set of predefined constants,
Chris@10 109 defined in the following subsections.
Chris@10 110
Chris@10 111 <p>In other words, FFTW computes the separable product of the specified
Chris@10 112 r2r transforms over each dimension, which can be used e.g. for partial
Chris@10 113 differential equations with mixed boundary conditions. (For some r2r
Chris@10 114 kinds, notably the halfcomplex DFT and the DHT, such a separable
Chris@10 115 product is somewhat problematic in more than one dimension, however,
Chris@10 116 as is described below.)
Chris@10 117
Chris@10 118 <p>In the current version of FFTW, all r2r transforms except for the
Chris@10 119 halfcomplex type are computed via pre- or post-processing of
Chris@10 120 halfcomplex transforms, and they are therefore not as fast as they
Chris@10 121 could be. Since most other general DCT/DST codes employ a similar
Chris@10 122 algorithm, however, FFTW's implementation should provide at least
Chris@10 123 competitive performance.
Chris@10 124
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