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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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20 into another language, under the above conditions for modified versions,
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22 approved by the Free Software Foundation. -->
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23 <!-- Created by GNU Texinfo 6.3, http://www.gnu.org/software/texinfo/ -->
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24 <head>
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25 <title>FFTW 3.3.8: Real-data DFT Array Format</title>
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26
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27 <meta name="description" content="FFTW 3.3.8: Real-data DFT Array Format">
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28 <meta name="keywords" content="FFTW 3.3.8: Real-data DFT Array Format">
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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="Basic-Interface.html#Basic-Interface" rel="up" title="Basic Interface">
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37 <link href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" rel="next" title="Real-to-Real Transforms">
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38 <link href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" rel="prev" title="Real-data DFTs">
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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="Real_002ddata-DFT-Array-Format"></a>
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72 <div class="header">
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73 <p>
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74 Next: <a href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" accesskey="n" rel="next">Real-to-Real Transforms</a>, Previous: <a href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" accesskey="p" rel="prev">Real-data DFTs</a>, Up: <a href="Basic-Interface.html#Basic-Interface" accesskey="u" rel="up">Basic Interface</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="Real_002ddata-DFT-Array-Format-1"></a>
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78 <h4 class="subsection">4.3.4 Real-data DFT Array Format</h4>
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79 <a name="index-r2c_002fc2r-multi_002ddimensional-array-format-1"></a>
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80
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81 <p>The output of a DFT of real data (r2c) contains symmetries that, in
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82 principle, make half of the outputs redundant (see <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>). (Similarly for the input of an inverse c2r transform.) In
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83 practice, it is not possible to entirely realize these savings in an
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84 efficient and understandable format that generalizes to
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85 multi-dimensional transforms. Instead, the output of the r2c
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86 transforms is <em>slightly</em> over half of the output of the
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87 corresponding complex transform. We do not “pack” the data in any
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88 way, but store it as an ordinary array of <code>fftw_complex</code> values.
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89 In fact, this data is simply a subsection of what would be the array in
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90 the corresponding complex transform.
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91 </p>
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92 <p>Specifically, for a real transform of <em>d</em> (= <code>rank</code>)
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93 dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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94 , the complex data is an n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × (n<sub>d-1</sub>/2 + 1)
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95 array of
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96 <code>fftw_complex</code> values in row-major order (with the division rounded
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97 down). That is, we only store the <em>lower</em> half (non-negative
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98 frequencies), plus one element, of the last dimension of the data from
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99 the ordinary complex transform. (We could have instead taken half of
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100 any other dimension, but implementation turns out to be simpler if the
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101 last, contiguous, dimension is used.)
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102 </p>
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103 <a name="index-out_002dof_002dplace-1"></a>
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104 <p>For an out-of-place transform, the real data is simply an array with
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105 physical dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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106 in row-major order.
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107 </p>
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108 <a name="index-in_002dplace-4"></a>
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109 <a name="index-padding-3"></a>
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110 <p>For an in-place transform, some complications arise since the complex data
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111 is slightly larger than the real data. In this case, the final
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112 dimension of the real data must be <em>padded</em> with extra values to
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113 accommodate the size of the complex data—two extra if the last
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114 dimension is even and one if it is odd. That is, the last dimension of
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115 the real data must physically contain
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116 2 * (n<sub>d-1</sub>/2+1)
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117 <code>double</code> values (exactly enough to hold the complex data). This
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118 physical array size does not, however, change the <em>logical</em> array
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119 size—only
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120 n<sub>d-1</sub>
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121 values are actually stored in the last dimension, and
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122 n<sub>d-1</sub>
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123 is the last dimension passed to the planner.
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124 </p>
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125 <hr>
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126 <div class="header">
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127 <p>
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128 Next: <a href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" accesskey="n" rel="next">Real-to-Real Transforms</a>, Previous: <a href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" accesskey="p" rel="prev">Real-data DFTs</a>, Up: <a href="Basic-Interface.html#Basic-Interface" accesskey="u" rel="up">Basic Interface</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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129 </div>
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130
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131
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132
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133 </body>
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134 </html>
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