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| author | Chris Cannam <cannam@all-day-breakfast.com> |
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| date | Wed, 20 Mar 2013 15:35:50 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.3/doc/html/Multi_002ddimensional-Transforms.html Wed Mar 20 15:35:50 2013 +0000 @@ -0,0 +1,100 @@ +<html lang="en"> +<head> +<title>Multi-dimensional Transforms - FFTW 3.3.3</title> +<meta http-equiv="Content-Type" content="text/html"> +<meta name="description" content="FFTW 3.3.3"> +<meta name="generator" content="makeinfo 4.13"> +<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 3.3.3, 25 November 2012). + +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"> +<a name="Multi-dimensional-Transforms"></a> +<a name="Multi_002ddimensional-Transforms"></a> +<p> +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-325"></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> +