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125 lines
6.5 KiB
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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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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<title>Multi-dimensional Transforms (FFTW 3.3.10)</title>
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<meta name="description" content="Multi-dimensional Transforms (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="What-FFTW-Really-Computes.html" rel="up" title="What FFTW Really Computes">
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<link href="Multi_002dthreaded-FFTW.html" rel="next" title="Multi-threaded FFTW">
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<link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" rel="prev" title="1d Discrete Hartley Transforms (DHTs)">
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</head>
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<body lang="en">
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<span id="Multi_002ddimensional-Transforms"></span><div class="header">
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<p>
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Previous: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" accesskey="p" rel="prev">1d Discrete Hartley Transforms (DHTs)</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</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-Transforms-1"></span><h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4>
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<p>The multi-dimensional transforms of FFTW, in general, compute simply the
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separable product of the given 1d transform along each dimension of the
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array. Since each of these transforms is unnormalized, computing the
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forward followed by the backward/inverse multi-dimensional transform
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will result in the original array scaled by the product of the
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normalization factors for each dimension (e.g. the product of the
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dimension sizes, for a multi-dimensional DFT).
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</p>
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<span id="index-r2c-3"></span>
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<p>The definition of FFTW’s multi-dimensional DFT of real data (r2c)
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deserves special attention. In this case, we logically compute the full
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multi-dimensional DFT of the input data; since the input data are purely
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real, the output data have the Hermitian symmetry and therefore only one
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non-redundant half need be stored. More specifically, for an n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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multi-dimensional real-input DFT, the full (logical) complex output array
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<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
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<i>k</i><sub><i>d-1</i></sub>]
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has the symmetry:
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<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
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<i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
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<i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
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<i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>
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(where each dimension is periodic). Because of this symmetry, we only
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store the
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<i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1
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elements of the <em>last</em> dimension (division by <em>2</em> is rounded
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down). (We could instead have cut any other dimension in half, but the
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last dimension proved computationally convenient.) This results in the
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peculiar array format described in more detail by <a href="Real_002ddata-DFT-Array-Format.html">Real-data DFT Array Format</a>.
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</p>
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<p>The multi-dimensional c2r transform is simply the unnormalized inverse
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of the r2c transform. i.e. it is the same as FFTW’s complex backward
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multi-dimensional DFT, operating on a Hermitian input array in the
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peculiar format mentioned above and outputting a real array (since the
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DFT output is purely real).
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</p>
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<p>We should remind the user that the separable product of 1d transforms
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along each dimension, as computed by FFTW, is not always the same thing
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as the usual multi-dimensional transform. A multi-dimensional
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<code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the
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multi-dimensional DFT, requiring some post-processing to combine the
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requisite real and imaginary parts, as was described in <a href="The-Halfcomplex_002dformat-DFT.html">The Halfcomplex-format DFT</a>. Likewise, FFTW’s multidimensional
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<code>FFTW_DHT</code> r2r transform is not the same thing as the logical
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multi-dimensional discrete Hartley transform defined in the literature,
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as discussed in <a href="The-Discrete-Hartley-Transform.html">The Discrete Hartley Transform</a>.
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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="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" accesskey="p" rel="prev">1d Discrete Hartley Transforms (DHTs)</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</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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