Mercurial > hg > sv-dependency-builds
diff src/fftw-3.3.8/doc/html/Multi_002ddimensional-Transforms.html @ 82:d0c2a83c1364
Add FFTW 3.3.8 source, and a Linux build
| author | Chris Cannam |
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| date | Tue, 19 Nov 2019 14:52:55 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.8/doc/html/Multi_002ddimensional-Transforms.html Tue Nov 19 14:52:55 2019 +0000 @@ -0,0 +1,136 @@ +<!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 3.3.8, 24 May 2018). + +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 6.3, http://www.gnu.org/software/texinfo/ --> +<head> +<title>FFTW 3.3.8: Multi-dimensional Transforms</title> + +<meta name="description" content="FFTW 3.3.8: Multi-dimensional Transforms"> +<meta name="keywords" content="FFTW 3.3.8: Multi-dimensional Transforms"> +<meta name="resource-type" content="document"> +<meta name="distribution" content="global"> +<meta name="Generator" content="makeinfo"> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8"> +<link href="index.html#Top" rel="start" title="Top"> +<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> +<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> +<link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes"> +<link href="Multi_002dthreaded-FFTW.html#Multi_002dthreaded-FFTW" rel="next" title="Multi-threaded FFTW"> +<link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" rel="prev" title="1d Discrete Hartley Transforms (DHTs)"> +<style type="text/css"> +<!-- +a.summary-letter {text-decoration: none} +blockquote.indentedblock {margin-right: 0em} +blockquote.smallindentedblock {margin-right: 0em; font-size: smaller} +blockquote.smallquotation {font-size: smaller} +div.display {margin-left: 3.2em} +div.example {margin-left: 3.2em} +div.lisp {margin-left: 3.2em} +div.smalldisplay {margin-left: 3.2em} +div.smallexample {margin-left: 3.2em} +div.smalllisp {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} +pre.smalldisplay {font-family: inherit; font-size: smaller} +pre.smallexample {font-size: smaller} +pre.smallformat {font-family: inherit; font-size: smaller} +pre.smalllisp {font-size: smaller} +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"> +<a name="Multi_002ddimensional-Transforms"></a> +<div class="header"> +<p> +Previous: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="p" rel="prev">1d Discrete Hartley Transforms (DHTs)</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" 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#Concept-Index" title="Index" rel="index">Index</a>]</p> +</div> +<hr> +<a name="Multi_002ddimensional-Transforms-1"></a> +<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> + +<a name="index-r2c-3"></a> +<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_002ddata-DFT-Array-Format">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_002dformat-DFT">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">The Discrete Hartley Transform</a>. +</p> +<hr> +<div class="header"> +<p> +Previous: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="p" rel="prev">1d Discrete Hartley Transforms (DHTs)</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" 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#Concept-Index" title="Index" rel="index">Index</a>]</p> +</div> + + + +</body> +</html>