Mercurial > hg > batch-feature-extraction-tool
diff Lib/fftw-3.2.1/doc/html/.svn/text-base/Multi_002ddimensional-Transforms.html.svn-base @ 15:585caf503ef5 tip
Tidy up for ROLI
| author | Geogaddi\David <d.m.ronan@qmul.ac.uk> |
|---|---|
| date | Tue, 17 May 2016 18:50:19 +0100 |
| parents | 636c989477e7 |
| children |
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--- a/Lib/fftw-3.2.1/doc/html/.svn/text-base/Multi_002ddimensional-Transforms.html.svn-base Wed May 04 11:02:59 2016 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,101 +0,0 @@ -<html lang="en"> -<head> -<title>Multi-dimensional Transforms - FFTW 3.2.1</title> -<meta http-equiv="Content-Type" content="text/html"> -<meta name="description" content="FFTW 3.2.1"> -<meta name="generator" content="makeinfo 4.8"> -<link title="Top" rel="start" href="index.html#Top"> -<link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes"> -<link rel="prev" href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" title="1d Discrete Hartley Transforms (DHTs)"> -<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> -<!-- -This manual is for FFTW -(version 3.2.1, 5 February 2009). - -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. - --> -<meta http-equiv="Content-Style-Type" content="text/css"> -<style type="text/css"><!-- - pre.display { font-family:inherit } - pre.format { font-family:inherit } - pre.smalldisplay { font-family:inherit; font-size:smaller } - pre.smallformat { font-family:inherit; font-size:smaller } - pre.smallexample { font-size:smaller } - pre.smalllisp { font-size:smaller } - span.sc { font-variant:small-caps } - span.roman { font-family:serif; font-weight:normal; } - span.sansserif { font-family:sans-serif; font-weight:normal; } ---></style> -</head> -<body> -<div class="node"> -<p> -<a name="Multi-dimensional-Transforms"></a> -<a name="Multi_002ddimensional-Transforms"></a> -Previous: <a rel="previous" accesskey="p" href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029">1d Discrete Hartley Transforms (DHTs)</a>, -Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a> -<hr> -</div> - -<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-315"></a>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+1elements of the <em>last</em> dimension (division by 2 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>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>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>. - -<!-- ************************************************************ --> -</body></html> -