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1 <html lang="en">
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2 <head>
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3 <title>Multi-Dimensional DFTs of Real Data - 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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6 <meta name="generator" content="makeinfo 4.13">
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7 <link title="Top" rel="start" href="index.html#Top">
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8 <link rel="up" href="Tutorial.html#Tutorial" title="Tutorial">
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9 <link rel="prev" href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" title="One-Dimensional DFTs of Real Data">
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10 <link rel="next" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" title="More DFTs of Real Data">
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11 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
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12 <!--
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13 This manual is for FFTW
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14 (version 3.3.3, 25 November 2012).
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15
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16 Copyright (C) 2003 Matteo Frigo.
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17
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18 Copyright (C) 2003 Massachusetts Institute of Technology.
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19
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20 Permission is granted to make and distribute verbatim copies of
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21 this manual provided the copyright notice and this permission
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22 notice are preserved on all copies.
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23
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24 Permission is granted to copy and distribute modified versions of
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25 this manual under the conditions for verbatim copying, provided
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26 that the entire resulting derived work is distributed under the
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27 terms of a permission notice identical to this one.
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28
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29 Permission is granted to copy and distribute translations of this
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30 manual into another language, under the above conditions for
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31 modified versions, except that this permission notice may be
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32 stated in a translation approved by the Free Software Foundation.
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33 -->
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34 <meta http-equiv="Content-Style-Type" content="text/css">
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35 <style type="text/css"><!--
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45 --></style>
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46 </head>
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47 <body>
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48 <div class="node">
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49 <a name="Multi-Dimensional-DFTs-of-Real-Data"></a>
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50 <a name="Multi_002dDimensional-DFTs-of-Real-Data"></a>
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51 <p>
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52 Next: <a rel="next" accesskey="n" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>,
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53 Previous: <a rel="previous" accesskey="p" href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data">One-Dimensional DFTs of Real Data</a>,
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54 Up: <a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a>
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55 <hr>
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56 </div>
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57
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58 <h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3>
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59
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60 <p>Multi-dimensional DFTs of real data use the following planner routines:
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61
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62 <pre class="example"> fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
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63 double *in, fftw_complex *out,
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64 unsigned flags);
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65 fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
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66 double *in, fftw_complex *out,
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67 unsigned flags);
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68 fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
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69 double *in, fftw_complex *out,
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70 unsigned flags);
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71 </pre>
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72 <p><a name="index-fftw_005fplan_005fdft_005fr2c_005f2d-59"></a><a name="index-fftw_005fplan_005fdft_005fr2c_005f3d-60"></a><a name="index-fftw_005fplan_005fdft_005fr2c-61"></a>
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73 as well as the corresponding <code>c2r</code> routines with the input/output
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74 types swapped. These routines work similarly to their complex
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75 analogues, except for the fact that here the complex output array is cut
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76 roughly in half and the real array requires padding for in-place
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77 transforms (as in 1d, above).
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78
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79 <p>As before, <code>n</code> is the logical size of the array, and the
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80 consequences of this on the the format of the complex arrays deserve
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81 careful attention.
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82 <a name="index-r2c_002fc2r-multi_002ddimensional-array-format-62"></a>Suppose that the real data has dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub> (in row-major order).
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83 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) array of
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84 <code>fftw_complex</code> values in row-major order, corresponding to slightly
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85 over half of the output of the corresponding complex DFT. (The division
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86 is rounded down.) The ordering of the data is otherwise exactly the
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87 same as in the complex-DFT case.
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88
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89 <p>For out-of-place transforms, this is the end of the story: the real
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90 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> and the complex
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91 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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92
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93 <p>For in-place transforms, however, extra padding of the real-data array
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94 is necessary because the complex array is larger than the real array,
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95 and the two arrays share the same memory locations. Thus, for
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96 in-place transforms, the final dimension of the real-data array must
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97 be padded with extra values to accommodate the size of the complex
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98 data—two values if the last dimension is even and one if it is odd.
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99 <a name="index-padding-63"></a>That is, the last dimension of the real data must physically contain
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100 2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data).
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101 This physical array size does not, however, change the <em>logical</em>
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102 array size—only
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103 n<sub>d-1</sub>values are actually stored in the last dimension, and
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104 n<sub>d-1</sub>is the last dimension passed to the plan-creation routine.
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105
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106 <p>For example, consider the transform of a two-dimensional real array of
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107 size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a
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108 two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where
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109 the <code>y</code> dimension has been cut nearly in half because of
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110 redundancies in the output. Because <code>fftw_complex</code> is twice the
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111 size of <code>double</code>, the output array is slightly bigger than the
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112 input array. Thus, if we want to compute the transform in place, we
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113 must <em>pad</em> the input array so that it is of size <code>n0</code> by
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114 <code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding
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115 elements at the end of each row (which need not be initialized, as they
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116 are only used for output).
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117
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118 <p>The following illustration depicts the input and output arrays just
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119 described, for both the out-of-place and in-place transforms (with the
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120 arrows indicating consecutive memory locations):
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121 <img src="rfftwnd-for-html.png" alt="rfftwnd-for-html.png">
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122
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123 <p>These transforms are unnormalized, so an r2c followed by a c2r
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124 transform (or vice versa) will result in the original data scaled by
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125 the number of real data elements—that is, the product of the
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126 (logical) dimensions of the real data.
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127 <a name="index-normalization-64"></a>
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128
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129 <p>(Because the last dimension is treated specially, if it is equal to
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130 <code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional
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131 r2c/c2r transform. In that case, the last complex dimension also has
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132 size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the
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133 complex transforms.)
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134
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135 <!-- -->
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136 </body></html>
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137
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