124 lines
6.2 KiB
HTML
124 lines
6.2 KiB
HTML
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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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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>Real-data DFT Array Format (FFTW 3.3.10)</title>
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<meta name="description" content="Real-data DFT Array Format (FFTW 3.3.10)">
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<meta name="keywords" content="Real-data DFT Array Format (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="Basic-Interface.html" rel="up" title="Basic Interface">
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<link href="Real_002dto_002dReal-Transforms.html" rel="next" title="Real-to-Real Transforms">
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<link href="Real_002ddata-DFTs.html" rel="prev" title="Real-data DFTs">
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</style>
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</head>
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<body lang="en">
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<span id="Real_002ddata-DFT-Array-Format"></span><div class="header">
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<p>
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Next: <a href="Real_002dto_002dReal-Transforms.html" accesskey="n" rel="next">Real-to-Real Transforms</a>, Previous: <a href="Real_002ddata-DFTs.html" accesskey="p" rel="prev">Real-data DFTs</a>, Up: <a href="Basic-Interface.html" accesskey="u" rel="up">Basic Interface</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="Real_002ddata-DFT-Array-Format-1"></span><h4 class="subsection">4.3.4 Real-data DFT Array Format</h4>
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<span id="index-r2c_002fc2r-multi_002ddimensional-array-format-1"></span>
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<p>The output of a DFT of real data (r2c) contains symmetries that, in
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principle, make half of the outputs redundant (see <a href="What-FFTW-Really-Computes.html">What FFTW Really Computes</a>). (Similarly for the input of an inverse c2r transform.) In
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practice, it is not possible to entirely realize these savings in an
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efficient and understandable format that generalizes to
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multi-dimensional transforms. Instead, the output of the r2c
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transforms is <em>slightly</em> over half of the output of the
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corresponding complex transform. We do not “pack” the data in any
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way, but store it as an ordinary array of <code>fftw_complex</code> values.
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In fact, this data is simply a subsection of what would be the array in
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the corresponding complex transform.
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</p>
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<p>Specifically, for a real transform of <em>d</em> (= <code>rank</code>)
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dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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, the complex data 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 (with the division rounded
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down). That is, we only store the <em>lower</em> half (non-negative
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frequencies), plus one element, of the last dimension of the data from
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the ordinary complex transform. (We could have instead taken half of
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any other dimension, but implementation turns out to be simpler if the
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last, contiguous, dimension is used.)
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</p>
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<span id="index-out_002dof_002dplace-1"></span>
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<p>For an out-of-place transform, the real data is simply an array with
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physical 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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</p>
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<span id="index-in_002dplace-4"></span>
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<span id="index-padding-3"></span>
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<p>For an in-place transform, some complications arise since the complex data
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is slightly larger than the real data. In this case, the final
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dimension of the real data must be <em>padded</em> with extra values to
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accommodate the size of the complex data—two extra if the last
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dimension is even and one if it is odd. That is, the last dimension of
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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). This
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physical array size does not, however, change the <em>logical</em> array
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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 planner.
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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="Real_002dto_002dReal-Transforms.html" accesskey="n" rel="next">Real-to-Real Transforms</a>, Previous: <a href="Real_002ddata-DFTs.html" accesskey="p" rel="prev">Real-data DFTs</a>, Up: <a href="Basic-Interface.html" accesskey="u" rel="up">Basic Interface</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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