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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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| 1 <html lang="en"> | |
| 2 <head> | |
| 3 <title>Multi-dimensional Transforms - FFTW 3.3.3</title> | |
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| 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)"> | |
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| 11 <!-- | |
| 12 This manual is for FFTW | |
| 13 (version 3.3.3, 25 November 2012). | |
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| 15 Copyright (C) 2003 Matteo Frigo. | |
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| 17 Copyright (C) 2003 Massachusetts Institute of Technology. | |
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| 48 <a name="Multi-dimensional-Transforms"></a> | |
| 49 <a name="Multi_002ddimensional-Transforms"></a> | |
| 50 <p> | |
| 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>, | |
| 52 Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a> | |
| 53 <hr> | |
| 54 </div> | |
| 55 | |
| 56 <h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4> | |
| 57 | |
| 58 <p>The multi-dimensional transforms of FFTW, in general, compute simply the | |
| 59 separable product of the given 1d transform along each dimension of the | |
| 60 array. Since each of these transforms is unnormalized, computing the | |
| 61 forward followed by the backward/inverse multi-dimensional transform | |
| 62 will result in the original array scaled by the product of the | |
| 63 normalization factors for each dimension (e.g. the product of the | |
| 64 dimension sizes, for a multi-dimensional DFT). | |
| 65 | |
| 66 <p><a name="index-r2c-325"></a>The definition of FFTW's multi-dimensional DFT of real data (r2c) | |
| 67 deserves special attention. In this case, we logically compute the full | |
| 68 multi-dimensional DFT of the input data; since the input data are purely | |
| 69 real, the output data have the Hermitian symmetry and therefore only one | |
| 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 | |
| 71 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ..., | |
| 72 <i>k</i><sub><i>d-1</i></sub>]has the symmetry: | |
| 73 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ..., | |
| 74 <i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> - | |
| 75 <i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ..., | |
| 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 | |
| 77 store the | |
| 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 | |
| 79 down). (We could instead have cut any other dimension in half, but the | |
| 80 last dimension proved computationally convenient.) This results in the | |
| 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>. | |
| 82 | |
| 83 <p>The multi-dimensional c2r transform is simply the unnormalized inverse | |
| 84 of the r2c transform. i.e. it is the same as FFTW's complex backward | |
| 85 multi-dimensional DFT, operating on a Hermitian input array in the | |
| 86 peculiar format mentioned above and outputting a real array (since the | |
| 87 DFT output is purely real). | |
| 88 | |
| 89 <p>We should remind the user that the separable product of 1d transforms | |
| 90 along each dimension, as computed by FFTW, is not always the same thing | |
| 91 as the usual multi-dimensional transform. A multi-dimensional | |
| 92 <code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the | |
| 93 multi-dimensional DFT, requiring some post-processing to combine the | |
| 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 | |
| 95 <code>FFTW_DHT</code> r2r transform is not the same thing as the logical | |
| 96 multi-dimensional discrete Hartley transform defined in the literature, | |
| 97 as discussed in <a href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>. | |
| 98 | |
| 99 </body></html> | |
| 100 |