|
cannam@95
|
1 <html lang="en">
|
|
cannam@95
|
2 <head>
|
|
cannam@95
|
3 <title>Multi-dimensional Transforms - FFTW 3.3.3</title>
|
|
cannam@95
|
4 <meta http-equiv="Content-Type" content="text/html">
|
|
cannam@95
|
5 <meta name="description" content="FFTW 3.3.3">
|
|
cannam@95
|
6 <meta name="generator" content="makeinfo 4.13">
|
|
cannam@95
|
7 <link title="Top" rel="start" href="index.html#Top">
|
|
cannam@95
|
8 <link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
|
|
cannam@95
|
9 <link rel="prev" href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" title="1d Discrete Hartley Transforms (DHTs)">
|
|
cannam@95
|
10 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
|
|
cannam@95
|
11 <!--
|
|
cannam@95
|
12 This manual is for FFTW
|
|
cannam@95
|
13 (version 3.3.3, 25 November 2012).
|
|
cannam@95
|
14
|
|
cannam@95
|
15 Copyright (C) 2003 Matteo Frigo.
|
|
cannam@95
|
16
|
|
cannam@95
|
17 Copyright (C) 2003 Massachusetts Institute of Technology.
|
|
cannam@95
|
18
|
|
cannam@95
|
19 Permission is granted to make and distribute verbatim copies of
|
|
cannam@95
|
20 this manual provided the copyright notice and this permission
|
|
cannam@95
|
21 notice are preserved on all copies.
|
|
cannam@95
|
22
|
|
cannam@95
|
23 Permission is granted to copy and distribute modified versions of
|
|
cannam@95
|
24 this manual under the conditions for verbatim copying, provided
|
|
cannam@95
|
25 that the entire resulting derived work is distributed under the
|
|
cannam@95
|
26 terms of a permission notice identical to this one.
|
|
cannam@95
|
27
|
|
cannam@95
|
28 Permission is granted to copy and distribute translations of this
|
|
cannam@95
|
29 manual into another language, under the above conditions for
|
|
cannam@95
|
30 modified versions, except that this permission notice may be
|
|
cannam@95
|
31 stated in a translation approved by the Free Software Foundation.
|
|
cannam@95
|
32 -->
|
|
cannam@95
|
33 <meta http-equiv="Content-Style-Type" content="text/css">
|
|
cannam@95
|
34 <style type="text/css"><!--
|
|
cannam@95
|
35 pre.display { font-family:inherit }
|
|
cannam@95
|
36 pre.format { font-family:inherit }
|
|
cannam@95
|
37 pre.smalldisplay { font-family:inherit; font-size:smaller }
|
|
cannam@95
|
38 pre.smallformat { font-family:inherit; font-size:smaller }
|
|
cannam@95
|
39 pre.smallexample { font-size:smaller }
|
|
cannam@95
|
40 pre.smalllisp { font-size:smaller }
|
|
cannam@95
|
41 span.sc { font-variant:small-caps }
|
|
cannam@95
|
42 span.roman { font-family:serif; font-weight:normal; }
|
|
cannam@95
|
43 span.sansserif { font-family:sans-serif; font-weight:normal; }
|
|
cannam@95
|
44 --></style>
|
|
cannam@95
|
45 </head>
|
|
cannam@95
|
46 <body>
|
|
cannam@95
|
47 <div class="node">
|
|
cannam@95
|
48 <a name="Multi-dimensional-Transforms"></a>
|
|
cannam@95
|
49 <a name="Multi_002ddimensional-Transforms"></a>
|
|
cannam@95
|
50 <p>
|
|
cannam@95
|
51 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>,
|
|
cannam@95
|
52 Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
|
|
cannam@95
|
53 <hr>
|
|
cannam@95
|
54 </div>
|
|
cannam@95
|
55
|
|
cannam@95
|
56 <h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4>
|
|
cannam@95
|
57
|
|
cannam@95
|
58 <p>The multi-dimensional transforms of FFTW, in general, compute simply the
|
|
cannam@95
|
59 separable product of the given 1d transform along each dimension of the
|
|
cannam@95
|
60 array. Since each of these transforms is unnormalized, computing the
|
|
cannam@95
|
61 forward followed by the backward/inverse multi-dimensional transform
|
|
cannam@95
|
62 will result in the original array scaled by the product of the
|
|
cannam@95
|
63 normalization factors for each dimension (e.g. the product of the
|
|
cannam@95
|
64 dimension sizes, for a multi-dimensional DFT).
|
|
cannam@95
|
65
|
|
cannam@95
|
66 <p><a name="index-r2c-325"></a>The definition of FFTW's multi-dimensional DFT of real data (r2c)
|
|
cannam@95
|
67 deserves special attention. In this case, we logically compute the full
|
|
cannam@95
|
68 multi-dimensional DFT of the input data; since the input data are purely
|
|
cannam@95
|
69 real, the output data have the Hermitian symmetry and therefore only one
|
|
cannam@95
|
70 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
|
|
cannam@95
|
71 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
|
|
cannam@95
|
72 <i>k</i><sub><i>d-1</i></sub>]has the symmetry:
|
|
cannam@95
|
73 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
|
|
cannam@95
|
74 <i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
|
|
cannam@95
|
75 <i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
|
|
cannam@95
|
76 <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
|
|
cannam@95
|
77 store the
|
|
cannam@95
|
78 <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
|
|
cannam@95
|
79 down). (We could instead have cut any other dimension in half, but the
|
|
cannam@95
|
80 last dimension proved computationally convenient.) This results in the
|
|
cannam@95
|
81 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@95
|
82
|
|
cannam@95
|
83 <p>The multi-dimensional c2r transform is simply the unnormalized inverse
|
|
cannam@95
|
84 of the r2c transform. i.e. it is the same as FFTW's complex backward
|
|
cannam@95
|
85 multi-dimensional DFT, operating on a Hermitian input array in the
|
|
cannam@95
|
86 peculiar format mentioned above and outputting a real array (since the
|
|
cannam@95
|
87 DFT output is purely real).
|
|
cannam@95
|
88
|
|
cannam@95
|
89 <p>We should remind the user that the separable product of 1d transforms
|
|
cannam@95
|
90 along each dimension, as computed by FFTW, is not always the same thing
|
|
cannam@95
|
91 as the usual multi-dimensional transform. A multi-dimensional
|
|
cannam@95
|
92 <code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the
|
|
cannam@95
|
93 multi-dimensional DFT, requiring some post-processing to combine the
|
|
cannam@95
|
94 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
|
|
cannam@95
|
95 <code>FFTW_DHT</code> r2r transform is not the same thing as the logical
|
|
cannam@95
|
96 multi-dimensional discrete Hartley transform defined in the literature,
|
|
cannam@95
|
97 as discussed in <a href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>.
|
|
cannam@95
|
98
|
|
cannam@95
|
99 </body></html>
|
|
cannam@95
|
100
|
