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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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2 <html>
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3 <!-- This manual is for FFTW
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4 (version 3.3.8, 24 May 2018).
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5
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6 Copyright (C) 2003 Matteo Frigo.
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7
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8 Copyright (C) 2003 Massachusetts Institute of Technology.
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9
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10 Permission is granted to make and distribute verbatim copies of this
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11 manual provided the copyright notice and this permission notice are
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12 preserved on all copies.
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13
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14 Permission is granted to copy and distribute modified versions of this
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15 manual under the conditions for verbatim copying, provided that the
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16 entire resulting derived work is distributed under the terms of a
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17 permission notice identical to this one.
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18
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19 Permission is granted to copy and distribute translations of this manual
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22 approved by the Free Software Foundation. -->
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24 <head>
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25 <title>FFTW 3.3.8: Multi-dimensional Transforms</title>
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26
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27 <meta name="description" content="FFTW 3.3.8: Multi-dimensional Transforms">
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28 <meta name="keywords" content="FFTW 3.3.8: Multi-dimensional Transforms">
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33 <link href="index.html#Top" rel="start" title="Top">
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34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
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37 <link href="Multi_002dthreaded-FFTW.html#Multi_002dthreaded-FFTW" rel="next" title="Multi-threaded FFTW">
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38 <link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" rel="prev" title="1d Discrete Hartley Transforms (DHTs)">
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64 -->
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65 </style>
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66
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67
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68 </head>
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69
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70 <body lang="en">
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71 <a name="Multi_002ddimensional-Transforms"></a>
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72 <div class="header">
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73 <p>
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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> [<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>
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75 </div>
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76 <hr>
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77 <a name="Multi_002ddimensional-Transforms-1"></a>
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78 <h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4>
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79
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80 <p>The multi-dimensional transforms of FFTW, in general, compute simply the
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81 separable product of the given 1d transform along each dimension of the
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82 array. Since each of these transforms is unnormalized, computing the
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83 forward followed by the backward/inverse multi-dimensional transform
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84 will result in the original array scaled by the product of the
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85 normalization factors for each dimension (e.g. the product of the
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86 dimension sizes, for a multi-dimensional DFT).
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87 </p>
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88
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89 <a name="index-r2c-3"></a>
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90 <p>The definition of FFTW’s multi-dimensional DFT of real data (r2c)
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91 deserves special attention. In this case, we logically compute the full
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92 multi-dimensional DFT of the input data; since the input data are purely
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93 real, the output data have the Hermitian symmetry and therefore only one
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94 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>
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95 multi-dimensional real-input DFT, the full (logical) complex output array
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96 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
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97 <i>k</i><sub><i>d-1</i></sub>]
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98 has the symmetry:
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99 <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
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100 <i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
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101 <i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
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102 <i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>
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103 (where each dimension is periodic). Because of this symmetry, we only
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104 store the
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105 <i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1
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106 elements of the <em>last</em> dimension (division by <em>2</em> is rounded
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107 down). (We could instead have cut any other dimension in half, but the
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108 last dimension proved computationally convenient.) This results in the
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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>.
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110 </p>
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111 <p>The multi-dimensional c2r transform is simply the unnormalized inverse
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112 of the r2c transform. i.e. it is the same as FFTW’s complex backward
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113 multi-dimensional DFT, operating on a Hermitian input array in the
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114 peculiar format mentioned above and outputting a real array (since the
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115 DFT output is purely real).
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116 </p>
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117 <p>We should remind the user that the separable product of 1d transforms
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118 along each dimension, as computed by FFTW, is not always the same thing
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119 as the usual multi-dimensional transform. A multi-dimensional
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120 <code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the
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121 multi-dimensional DFT, requiring some post-processing to combine the
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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’s multidimensional
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123 <code>FFTW_DHT</code> r2r transform is not the same thing as the logical
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124 multi-dimensional discrete Hartley transform defined in the literature,
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125 as discussed in <a href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>.
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126 </p>
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127 <hr>
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128 <div class="header">
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129 <p>
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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> [<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>
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131 </div>
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132
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133
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134
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135 </body>
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136 </html>
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