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8.5 KiB
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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>Multi-Dimensional DFTs of Real Data (FFTW 3.3.10)</title>
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<meta name="description" content="Multi-Dimensional DFTs of Real Data (FFTW 3.3.10)">
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<meta name="keywords" content="Multi-Dimensional DFTs of Real Data (FFTW 3.3.10)">
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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="More-DFTs-of-Real-Data.html" rel="next" title="More DFTs of Real Data">
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<link href="One_002dDimensional-DFTs-of-Real-Data.html" rel="prev" title="One-Dimensional DFTs of Real Data">
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</head>
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<body lang="en">
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<span id="Multi_002dDimensional-DFTs-of-Real-Data"></span><div class="header">
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<p>
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Next: <a href="More-DFTs-of-Real-Data.html" accesskey="n" rel="next">More DFTs of Real Data</a>, Previous: <a href="One_002dDimensional-DFTs-of-Real-Data.html" accesskey="p" rel="prev">One-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="Multi_002dDimensional-DFTs-of-Real-Data-1"></span><h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3>
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<p>Multi-dimensional DFTs of real data use the following planner routines:
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</p>
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<div class="example">
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<pre class="example">fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
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double *in, fftw_complex *out,
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unsigned flags);
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fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
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double *in, fftw_complex *out,
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unsigned flags);
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fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
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double *in, fftw_complex *out,
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unsigned flags);
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</pre></div>
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<span id="index-fftw_005fplan_005fdft_005fr2c_005f2d"></span>
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<span id="index-fftw_005fplan_005fdft_005fr2c_005f3d"></span>
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<span id="index-fftw_005fplan_005fdft_005fr2c"></span>
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<p>as well as the corresponding <code>c2r</code> routines with the input/output
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types swapped. These routines work similarly to their complex
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analogues, except for the fact that here the complex output array is cut
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roughly in half and the real array requires padding for in-place
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transforms (as in 1d, above).
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</p>
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<p>As before, <code>n</code> is the logical size of the array, and the
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consequences of this on the the format of the complex arrays deserve
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careful attention.
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<span id="index-r2c_002fc2r-multi_002ddimensional-array-format"></span>
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Suppose that the real data has dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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(in row-major order).
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Then, after an r2c transform, the output is an n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × (n<sub>d-1</sub>/2 + 1)
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array of
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<code>fftw_complex</code> values in row-major order, corresponding to slightly
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over half of the output of the corresponding complex DFT. (The division
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is rounded down.) The ordering of the data is otherwise exactly the
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same as in the complex-DFT case.
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</p>
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<p>For out-of-place transforms, this is the end of the story: the real
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data is stored as a row-major array of size n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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and the complex
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data is stored as a row-major array of size n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × (n<sub>d-1</sub>/2 + 1)
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.
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</p>
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<p>For in-place transforms, however, extra padding of the real-data array
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is necessary because the complex array is larger than the real array,
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and the two arrays share the same memory locations. Thus, for
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in-place transforms, the final dimension of the real-data array must
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be padded with extra values to accommodate the size of the complex
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data—two values if the last dimension is even and one if it is odd.
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<span id="index-padding-1"></span>
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That is, the last dimension of the real data must physically contain
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2 * (n<sub>d-1</sub>/2+1)
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<code>double</code> values (exactly enough to hold the complex data).
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This physical array size does not, however, change the <em>logical</em>
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array size—only
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n<sub>d-1</sub>
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values are actually stored in the last dimension, and
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n<sub>d-1</sub>
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is the last dimension passed to the plan-creation routine.
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</p>
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<p>For example, consider the transform of a two-dimensional real array of
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size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a
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two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where
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the <code>y</code> dimension has been cut nearly in half because of
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redundancies in the output. Because <code>fftw_complex</code> is twice the
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size of <code>double</code>, the output array is slightly bigger than the
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input array. Thus, if we want to compute the transform in place, we
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must <em>pad</em> the input array so that it is of size <code>n0</code> by
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<code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding
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elements at the end of each row (which need not be initialized, as they
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are only used for output).
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</p>
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<p>The following illustration depicts the input and output arrays just
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described, for both the out-of-place and in-place transforms (with the
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arrows indicating consecutive memory locations):
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<img src="rfftwnd-for-html.png" alt="rfftwnd-for-html">
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</p>
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<p>These transforms are unnormalized, so an r2c followed by a c2r
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transform (or vice versa) will result in the original data scaled by
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the number of real data elements—that is, the product of the
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(logical) dimensions of the real data.
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<span id="index-normalization-1"></span>
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</p>
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<p>(Because the last dimension is treated specially, if it is equal to
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<code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional
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r2c/c2r transform. In that case, the last complex dimension also has
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size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the
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complex transforms.)
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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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Next: <a href="More-DFTs-of-Real-Data.html" accesskey="n" rel="next">More DFTs of Real Data</a>, Previous: <a href="One_002dDimensional-DFTs-of-Real-Data.html" accesskey="p" rel="prev">One-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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