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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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21 except that this permission notice may be stated in a translation
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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: Multi-Dimensional DFTs of Real Data</title>
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26
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27 <meta name="description" content="FFTW 3.3.8: Multi-Dimensional DFTs of Real Data">
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28 <meta name="keywords" content="FFTW 3.3.8: Multi-Dimensional DFTs of Real Data">
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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="Tutorial.html#Tutorial" rel="up" title="Tutorial">
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37 <link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="next" title="More DFTs of Real Data">
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38 <link href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" rel="prev" title="One-Dimensional DFTs of Real Data">
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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-DFTs-of-Real-Data"></a>
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72 <div class="header">
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73 <p>
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74 Next: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="n" rel="next">More DFTs of Real Data</a>, Previous: <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" accesskey="p" rel="prev">One-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</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-DFTs-of-Real-Data-1"></a>
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78 <h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3>
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79
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80 <p>Multi-dimensional DFTs of real data use the following planner routines:
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81 </p>
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82 <div class="example">
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83 <pre class="example">fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
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84 double *in, fftw_complex *out,
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85 unsigned flags);
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86 fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
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87 double *in, fftw_complex *out,
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88 unsigned flags);
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89 fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
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90 double *in, fftw_complex *out,
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91 unsigned flags);
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92 </pre></div>
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93 <a name="index-fftw_005fplan_005fdft_005fr2c_005f2d"></a>
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94 <a name="index-fftw_005fplan_005fdft_005fr2c_005f3d"></a>
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95 <a name="index-fftw_005fplan_005fdft_005fr2c"></a>
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96
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97 <p>as well as the corresponding <code>c2r</code> routines with the input/output
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98 types swapped. These routines work similarly to their complex
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99 analogues, except for the fact that here the complex output array is cut
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100 roughly in half and the real array requires padding for in-place
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101 transforms (as in 1d, above).
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102 </p>
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103 <p>As before, <code>n</code> is the logical size of the array, and the
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104 consequences of this on the the format of the complex arrays deserve
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105 careful attention.
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106 <a name="index-r2c_002fc2r-multi_002ddimensional-array-format"></a>
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107 Suppose that the real data has dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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108 (in row-major order).
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109 Then, after an r2c transform, the output 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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110 array of
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111 <code>fftw_complex</code> values in row-major order, corresponding to slightly
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112 over half of the output of the corresponding complex DFT. (The division
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113 is rounded down.) The ordering of the data is otherwise exactly the
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114 same as in the complex-DFT case.
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115 </p>
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116 <p>For out-of-place transforms, this is the end of the story: the real
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117 data is stored as a row-major array of size n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
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118 and the complex
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119 data is stored as a row-major array of size n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × (n<sub>d-1</sub>/2 + 1)
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120 .
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121 </p>
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122 <p>For in-place transforms, however, extra padding of the real-data array
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123 is necessary because the complex array is larger than the real array,
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124 and the two arrays share the same memory locations. Thus, for
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125 in-place transforms, the final dimension of the real-data array must
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126 be padded with extra values to accommodate the size of the complex
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127 data—two values if the last dimension is even and one if it is odd.
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128 <a name="index-padding-1"></a>
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129 That is, the last dimension of the real data must physically contain
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130 2 * (n<sub>d-1</sub>/2+1)
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131 <code>double</code> values (exactly enough to hold the complex data).
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132 This physical array size does not, however, change the <em>logical</em>
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133 array size—only
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134 n<sub>d-1</sub>
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135 values are actually stored in the last dimension, and
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136 n<sub>d-1</sub>
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137 is the last dimension passed to the plan-creation routine.
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138 </p>
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139 <p>For example, consider the transform of a two-dimensional real array of
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140 size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a
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141 two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where
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142 the <code>y</code> dimension has been cut nearly in half because of
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143 redundancies in the output. Because <code>fftw_complex</code> is twice the
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144 size of <code>double</code>, the output array is slightly bigger than the
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145 input array. Thus, if we want to compute the transform in place, we
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146 must <em>pad</em> the input array so that it is of size <code>n0</code> by
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147 <code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding
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148 elements at the end of each row (which need not be initialized, as they
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149 are only used for output).
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150 </p>
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151 <p>The following illustration depicts the input and output arrays just
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152 described, for both the out-of-place and in-place transforms (with the
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153 arrows indicating consecutive memory locations):
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154 <img src="rfftwnd-for-html.png" alt="rfftwnd-for-html">
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155 </p>
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156 <p>These transforms are unnormalized, so an r2c followed by a c2r
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157 transform (or vice versa) will result in the original data scaled by
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158 the number of real data elements—that is, the product of the
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159 (logical) dimensions of the real data.
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160 <a name="index-normalization-1"></a>
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161 </p>
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162
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163 <p>(Because the last dimension is treated specially, if it is equal to
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164 <code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional
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165 r2c/c2r transform. In that case, the last complex dimension also has
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166 size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the
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167 complex transforms.)
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168 </p>
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169 <hr>
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170 <div class="header">
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171 <p>
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172 Next: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="n" rel="next">More DFTs of Real Data</a>, Previous: <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" accesskey="p" rel="prev">One-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</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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173 </div>
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174
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175
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176
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177 </body>
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178 </html>
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