126 lines
6.5 KiB
HTML
126 lines
6.5 KiB
HTML
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
|
||
|
|
<html>
|
||
|
|
<!-- This manual is for FFTW
|
||
|
|
(version 3.3.10, 10 December 2020).
|
||
|
|
|
||
|
|
Copyright (C) 2003 Matteo Frigo.
|
||
|
|
|
||
|
|
Copyright (C) 2003 Massachusetts Institute of Technology.
|
||
|
|
|
||
|
|
Permission is granted to make and distribute verbatim copies of this
|
||
|
|
manual provided the copyright notice and this permission notice are
|
||
|
|
preserved on all copies.
|
||
|
|
|
||
|
|
Permission is granted to copy and distribute modified versions of this
|
||
|
|
manual under the conditions for verbatim copying, provided that the
|
||
|
|
entire resulting derived work is distributed under the terms of a
|
||
|
|
permission notice identical to this one.
|
||
|
|
|
||
|
|
Permission is granted to copy and distribute translations of this manual
|
||
|
|
into another language, under the above conditions for modified versions,
|
||
|
|
except that this permission notice may be stated in a translation
|
||
|
|
approved by the Free Software Foundation. -->
|
||
|
|
<!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
|
||
|
|
<head>
|
||
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
||
|
|
<title>Multi-dimensional Transforms (FFTW 3.3.10)</title>
|
||
|
|
|
||
|
|
<meta name="description" content="Multi-dimensional Transforms (FFTW 3.3.10)">
|
||
|
|
<meta name="keywords" content="Multi-dimensional Transforms (FFTW 3.3.10)">
|
||
|
|
<meta name="resource-type" content="document">
|
||
|
|
<meta name="distribution" content="global">
|
||
|
|
<meta name="Generator" content="makeinfo">
|
||
|
|
<link href="index.html" rel="start" title="Top">
|
||
|
|
<link href="Concept-Index.html" rel="index" title="Concept Index">
|
||
|
|
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
|
||
|
|
<link href="What-FFTW-Really-Computes.html" rel="up" title="What FFTW Really Computes">
|
||
|
|
<link href="Multi_002dthreaded-FFTW.html" rel="next" title="Multi-threaded FFTW">
|
||
|
|
<link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" rel="prev" title="1d Discrete Hartley Transforms (DHTs)">
|
||
|
|
<style type="text/css">
|
||
|
|
<!--
|
||
|
|
a.summary-letter {text-decoration: none}
|
||
|
|
blockquote.indentedblock {margin-right: 0em}
|
||
|
|
div.display {margin-left: 3.2em}
|
||
|
|
div.example {margin-left: 3.2em}
|
||
|
|
div.lisp {margin-left: 3.2em}
|
||
|
|
kbd {font-style: oblique}
|
||
|
|
pre.display {font-family: inherit}
|
||
|
|
pre.format {font-family: inherit}
|
||
|
|
pre.menu-comment {font-family: serif}
|
||
|
|
pre.menu-preformatted {font-family: serif}
|
||
|
|
span.nolinebreak {white-space: nowrap}
|
||
|
|
span.roman {font-family: initial; font-weight: normal}
|
||
|
|
span.sansserif {font-family: sans-serif; font-weight: normal}
|
||
|
|
ul.no-bullet {list-style: none}
|
||
|
|
-->
|
||
|
|
</style>
|
||
|
|
|
||
|
|
|
||
|
|
</head>
|
||
|
|
|
||
|
|
<body lang="en">
|
||
|
|
<span id="Multi_002ddimensional-Transforms"></span><div class="header">
|
||
|
|
<p>
|
||
|
|
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>
|
||
|
|
</div>
|
||
|
|
<hr>
|
||
|
|
<span id="Multi_002ddimensional-Transforms-1"></span><h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4>
|
||
|
|
|
||
|
|
<p>The multi-dimensional transforms of FFTW, in general, compute simply the
|
||
|
|
separable product of the given 1d transform along each dimension of the
|
||
|
|
array. Since each of these transforms is unnormalized, computing the
|
||
|
|
forward followed by the backward/inverse multi-dimensional transform
|
||
|
|
will result in the original array scaled by the product of the
|
||
|
|
normalization factors for each dimension (e.g. the product of the
|
||
|
|
dimension sizes, for a multi-dimensional DFT).
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<span id="index-r2c-3"></span>
|
||
|
|
<p>The definition of FFTW’s multi-dimensional DFT of real data (r2c)
|
||
|
|
deserves special attention. In this case, we logically compute the full
|
||
|
|
multi-dimensional DFT of the input data; since the input data are purely
|
||
|
|
real, the output data have the Hermitian symmetry and therefore only one
|
||
|
|
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>
|
||
|
|
multi-dimensional real-input DFT, the full (logical) complex output array
|
||
|
|
<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
|
||
|
|
<i>k</i><sub><i>d-1</i></sub>]
|
||
|
|
has the symmetry:
|
||
|
|
<i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
|
||
|
|
<i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
|
||
|
|
<i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
|
||
|
|
<i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>
|
||
|
|
(where each dimension is periodic). Because of this symmetry, we only
|
||
|
|
store the
|
||
|
|
<i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1
|
||
|
|
elements of the <em>last</em> dimension (division by <em>2</em> is rounded
|
||
|
|
down). (We could instead have cut any other dimension in half, but the
|
||
|
|
last dimension proved computationally convenient.) This results in the
|
||
|
|
peculiar array format described in more detail by <a href="Real_002ddata-DFT-Array-Format.html">Real-data DFT Array Format</a>.
|
||
|
|
</p>
|
||
|
|
<p>The multi-dimensional c2r transform is simply the unnormalized inverse
|
||
|
|
of the r2c transform. i.e. it is the same as FFTW’s complex backward
|
||
|
|
multi-dimensional DFT, operating on a Hermitian input array in the
|
||
|
|
peculiar format mentioned above and outputting a real array (since the
|
||
|
|
DFT output is purely real).
|
||
|
|
</p>
|
||
|
|
<p>We should remind the user that the separable product of 1d transforms
|
||
|
|
along each dimension, as computed by FFTW, is not always the same thing
|
||
|
|
as the usual multi-dimensional transform. A multi-dimensional
|
||
|
|
<code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the
|
||
|
|
multi-dimensional DFT, requiring some post-processing to combine the
|
||
|
|
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
|
||
|
|
<code>FFTW_DHT</code> r2r transform is not the same thing as the logical
|
||
|
|
multi-dimensional discrete Hartley transform defined in the literature,
|
||
|
|
as discussed in <a href="The-Discrete-Hartley-Transform.html">The Discrete Hartley Transform</a>.
|
||
|
|
</p>
|
||
|
|
<hr>
|
||
|
|
<div class="header">
|
||
|
|
<p>
|
||
|
|
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>
|
||
|
|
</div>
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
</body>
|
||
|
|
</html>
|