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Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 19 Nov 2019 14:52:55 +0000
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3 <!-- This manual is for FFTW
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6 Copyright (C) 2003 Matteo Frigo.
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8 Copyright (C) 2003 Massachusetts Institute of Technology.
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25 <title>FFTW 3.3.8: Multi-dimensional Transforms</title>
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71 <a name="Multi_002ddimensional-Transforms"></a>
72 <div class="header">
73 <p>
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>
75 </div>
76 <hr>
77 <a name="Multi_002ddimensional-Transforms-1"></a>
78 <h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4>
79
80 <p>The multi-dimensional transforms of FFTW, in general, compute simply the
81 separable product of the given 1d transform along each dimension of the
82 array. Since each of these transforms is unnormalized, computing the
83 forward followed by the backward/inverse multi-dimensional transform
84 will result in the original array scaled by the product of the
85 normalization factors for each dimension (e.g. the product of the
86 dimension sizes, for a multi-dimensional DFT).
87 </p>
88
89 <a name="index-r2c-3"></a>
90 <p>The definition of FFTW&rsquo;s multi-dimensional DFT of real data (r2c)
91 deserves special attention. In this case, we logically compute the full
92 multi-dimensional DFT of the input data; since the input data are purely
93 real, the output data have the Hermitian symmetry and therefore only one
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>
95 multi-dimensional real-input DFT, the full (logical) complex output array
96 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
97 <i>k</i><sub><i>d-1</i></sub>]
98 has the symmetry:
99 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
100 <i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
101 <i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
102 <i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>
103 (where each dimension is periodic). Because of this symmetry, we only
104 store the
105 <i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1
106 elements of the <em>last</em> dimension (division by <em>2</em> is rounded
107 down). (We could instead have cut any other dimension in half, but the
108 last dimension proved computationally convenient.) This results in the
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>.
110 </p>
111 <p>The multi-dimensional c2r transform is simply the unnormalized inverse
112 of the r2c transform. i.e. it is the same as FFTW&rsquo;s complex backward
113 multi-dimensional DFT, operating on a Hermitian input array in the
114 peculiar format mentioned above and outputting a real array (since the
115 DFT output is purely real).
116 </p>
117 <p>We should remind the user that the separable product of 1d transforms
118 along each dimension, as computed by FFTW, is not always the same thing
119 as the usual multi-dimensional transform. A multi-dimensional
120 <code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the
121 multi-dimensional DFT, requiring some post-processing to combine the
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
123 <code>FFTW_DHT</code> r2r transform is not the same thing as the logical
124 multi-dimensional discrete Hartley transform defined in the literature,
125 as discussed in <a href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>.
126 </p>
127 <hr>
128 <div class="header">
129 <p>
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>
131 </div>
132
133
134
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