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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual is for FFTW
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(version 3.3.10, 10 December 2020).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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Permission is granted to make and distribute verbatim copies of this
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manual provided the copyright notice and this permission notice are
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preserved on all copies.
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Permission is granted to copy and distribute modified versions of this
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manual under the conditions for verbatim copying, provided that the
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entire resulting derived work is distributed under the terms of a
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permission notice identical to this one.
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Permission is granted to copy and distribute translations of this manual
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into another language, under the above conditions for modified versions,
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except that this permission notice may be stated in a translation
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approved by the Free Software Foundation. -->
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<!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>More DFTs of Real Data (FFTW 3.3.10)</title>
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<meta name="description" content="More DFTs of Real Data (FFTW 3.3.10)">
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<meta name="keywords" content="More DFTs of Real Data (FFTW 3.3.10)">
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<meta name="resource-type" content="document">
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<link href="index.html" rel="start" title="Top">
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<link href="Concept-Index.html" rel="index" title="Concept Index">
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<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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<link href="Tutorial.html" rel="up" title="Tutorial">
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<link href="The-Halfcomplex_002dformat-DFT.html" rel="next" title="The Halfcomplex-format DFT">
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<link href="Multi_002dDimensional-DFTs-of-Real-Data.html" rel="prev" title="Multi-Dimensional DFTs of Real Data">
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</head>
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<body lang="en">
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<span id="More-DFTs-of-Real-Data"></span><div class="header">
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<p>
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Previous: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html" accesskey="p" rel="prev">Multi-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html" accesskey="u" rel="up">Tutorial</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<span id="More-DFTs-of-Real-Data-1"></span><h3 class="section">2.5 More DFTs of Real Data</h3>
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<table class="menu" border="0" cellspacing="0">
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<tr><td align="left" valign="top">• <a href="The-Halfcomplex_002dformat-DFT.html" accesskey="1">The Halfcomplex-format DFT</a></td><td> </td><td align="left" valign="top">
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</td></tr>
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<tr><td align="left" valign="top">• <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="2">Real even/odd DFTs (cosine/sine transforms)</a></td><td> </td><td align="left" valign="top">
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</td></tr>
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<tr><td align="left" valign="top">• <a href="The-Discrete-Hartley-Transform.html" accesskey="3">The Discrete Hartley Transform</a></td><td> </td><td align="left" valign="top">
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</td></tr>
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</table>
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<p>FFTW supports several other transform types via a unified <em>r2r</em>
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(real-to-real) interface,
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<span id="index-r2r"></span>
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so called because it takes a real (<code>double</code>) array and outputs a
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real array of the same size. These r2r transforms currently fall into
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three categories: DFTs of real input and complex-Hermitian output in
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halfcomplex format, DFTs of real input with even/odd symmetry
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(a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete
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Hartley transforms (DHTs), all described in more detail by the
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following sections.
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</p>
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<p>The r2r transforms follow the by now familiar interface of creating an
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<code>fftw_plan</code>, executing it with <code>fftw_execute(plan)</code>, and
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destroying it with <code>fftw_destroy_plan(plan)</code>. Furthermore, all
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r2r transforms share the same planner interface:
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</p>
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<div class="example">
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<pre class="example">fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
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fftw_r2r_kind kind, unsigned flags);
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fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out,
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fftw_r2r_kind kind0, fftw_r2r_kind kind1,
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unsigned flags);
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fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2,
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double *in, double *out,
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fftw_r2r_kind kind0,
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fftw_r2r_kind kind1,
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fftw_r2r_kind kind2,
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unsigned flags);
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fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
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const fftw_r2r_kind *kind, unsigned flags);
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</pre></div>
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<span id="index-fftw_005fplan_005fr2r_005f1d"></span>
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<span id="index-fftw_005fplan_005fr2r_005f2d"></span>
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<span id="index-fftw_005fplan_005fr2r_005f3d"></span>
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<span id="index-fftw_005fplan_005fr2r"></span>
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<p>Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional
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transforms for contiguous arrays in row-major order, transforming (real)
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input to output of the same size, where <code>n</code> specifies the
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<em>physical</em> dimensions of the arrays. All positive <code>n</code> are
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supported (with the exception of <code>n=1</code> for the <code>FFTW_REDFT00</code>
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kind, noted in the real-even subsection below); products of small
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factors are most efficient (factorizing <code>n-1</code> and <code>n+1</code> for
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<code>FFTW_REDFT00</code> and <code>FFTW_RODFT00</code> kinds, described below), but
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an <i>O</i>(<i>n</i> log <i>n</i>)
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algorithm is used even for prime sizes.
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</p>
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<p>Each dimension has a <em>kind</em> parameter, of type
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<code>fftw_r2r_kind</code>, specifying the kind of r2r transform to be used
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for that dimension.
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<span id="index-kind-_0028r2r_0029"></span>
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<span id="index-fftw_005fr2r_005fkind"></span>
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(In the case of <code>fftw_plan_r2r</code>, this is an array <code>kind[rank]</code>
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where <code>kind[i]</code> is the transform kind for the dimension
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<code>n[i]</code>.) The kind can be one of a set of predefined constants,
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defined in the following subsections.
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</p>
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<p>In other words, FFTW computes the separable product of the specified
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r2r transforms over each dimension, which can be used e.g. for partial
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differential equations with mixed boundary conditions. (For some r2r
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kinds, notably the halfcomplex DFT and the DHT, such a separable
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product is somewhat problematic in more than one dimension, however,
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as is described below.)
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</p>
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<p>In the current version of FFTW, all r2r transforms except for the
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halfcomplex type are computed via pre- or post-processing of
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halfcomplex transforms, and they are therefore not as fast as they
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could be. Since most other general DCT/DST codes employ a similar
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algorithm, however, FFTW’s implementation should provide at least
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competitive performance.
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</p>
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<hr>
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<div class="header">
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<p>
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Previous: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html" accesskey="p" rel="prev">Multi-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html" accesskey="u" rel="up">Tutorial</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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