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