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Add null config files
author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
parents bd3cc4d1df30
children
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cannam@167 3 <!-- This manual is for FFTW
cannam@167 4 (version 3.3.8, 24 May 2018).
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cannam@167 6 Copyright (C) 2003 Matteo Frigo.
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cannam@167 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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cannam@167 24 <head>
cannam@167 25 <title>FFTW 3.3.8: Multi-dimensional Transforms</title>
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cannam@167 68 </head>
cannam@167 69
cannam@167 70 <body lang="en">
cannam@167 71 <a name="Multi_002ddimensional-Transforms"></a>
cannam@167 72 <div class="header">
cannam@167 73 <p>
cannam@167 74 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> &nbsp; [<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>
cannam@167 75 </div>
cannam@167 76 <hr>
cannam@167 77 <a name="Multi_002ddimensional-Transforms-1"></a>
cannam@167 78 <h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4>
cannam@167 79
cannam@167 80 <p>The multi-dimensional transforms of FFTW, in general, compute simply the
cannam@167 81 separable product of the given 1d transform along each dimension of the
cannam@167 82 array. Since each of these transforms is unnormalized, computing the
cannam@167 83 forward followed by the backward/inverse multi-dimensional transform
cannam@167 84 will result in the original array scaled by the product of the
cannam@167 85 normalization factors for each dimension (e.g. the product of the
cannam@167 86 dimension sizes, for a multi-dimensional DFT).
cannam@167 87 </p>
cannam@167 88
cannam@167 89 <a name="index-r2c-3"></a>
cannam@167 90 <p>The definition of FFTW&rsquo;s multi-dimensional DFT of real data (r2c)
cannam@167 91 deserves special attention. In this case, we logically compute the full
cannam@167 92 multi-dimensional DFT of the input data; since the input data are purely
cannam@167 93 real, the output data have the Hermitian symmetry and therefore only one
cannam@167 94 non-redundant half need be stored. More specifically, for an n<sub>0</sub>&nbsp;&times;&nbsp;n<sub>1</sub>&nbsp;&times;&nbsp;n<sub>2</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;n<sub>d-1</sub>
cannam@167 95 multi-dimensional real-input DFT, the full (logical) complex output array
cannam@167 96 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
cannam@167 97 <i>k</i><sub><i>d-1</i></sub>]
cannam@167 98 has the symmetry:
cannam@167 99 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
cannam@167 100 <i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
cannam@167 101 <i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
cannam@167 102 <i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>
cannam@167 103 (where each dimension is periodic). Because of this symmetry, we only
cannam@167 104 store the
cannam@167 105 <i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1
cannam@167 106 elements of the <em>last</em> dimension (division by <em>2</em> is rounded
cannam@167 107 down). (We could instead have cut any other dimension in half, but the
cannam@167 108 last dimension proved computationally convenient.) This results in the
cannam@167 109 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>.
cannam@167 110 </p>
cannam@167 111 <p>The multi-dimensional c2r transform is simply the unnormalized inverse
cannam@167 112 of the r2c transform. i.e. it is the same as FFTW&rsquo;s complex backward
cannam@167 113 multi-dimensional DFT, operating on a Hermitian input array in the
cannam@167 114 peculiar format mentioned above and outputting a real array (since the
cannam@167 115 DFT output is purely real).
cannam@167 116 </p>
cannam@167 117 <p>We should remind the user that the separable product of 1d transforms
cannam@167 118 along each dimension, as computed by FFTW, is not always the same thing
cannam@167 119 as the usual multi-dimensional transform. A multi-dimensional
cannam@167 120 <code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the
cannam@167 121 multi-dimensional DFT, requiring some post-processing to combine the
cannam@167 122 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&rsquo;s multidimensional
cannam@167 123 <code>FFTW_DHT</code> r2r transform is not the same thing as the logical
cannam@167 124 multi-dimensional discrete Hartley transform defined in the literature,
cannam@167 125 as discussed in <a href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>.
cannam@167 126 </p>
cannam@167 127 <hr>
cannam@167 128 <div class="header">
cannam@167 129 <p>
cannam@167 130 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> &nbsp; [<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>
cannam@167 131 </div>
cannam@167 132
cannam@167 133
cannam@167 134
cannam@167 135 </body>
cannam@167 136 </html>