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2 <head>
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3 <title>Real-data DFT Array Format - FFTW 3.3.4</title>
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4 <meta http-equiv="Content-Type" content="text/html">
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5 <meta name="description" content="FFTW 3.3.4">
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7 <link title="Top" rel="start" href="index.html#Top">
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8 <link rel="up" href="Basic-Interface.html#Basic-Interface" title="Basic Interface">
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9 <link rel="prev" href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" title="Real-data DFTs">
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10 <link rel="next" href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" title="Real-to-Real Transforms">
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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 3.3.4, 20 September 2013).
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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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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="Real-data-DFT-Array-Format"></a>
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50 <a name="Real_002ddata-DFT-Array-Format"></a>
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51 <p>
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52 Next: <a rel="next" accesskey="n" href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms">Real-to-Real Transforms</a>,
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53 Previous: <a rel="previous" accesskey="p" href="Real_002ddata-DFTs.html#Real_002ddata-DFTs">Real-data DFTs</a>,
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54 Up: <a rel="up" accesskey="u" href="Basic-Interface.html#Basic-Interface">Basic Interface</a>
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55 <hr>
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56 </div>
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57
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58 <h4 class="subsection">4.3.4 Real-data DFT Array Format</h4>
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59
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60 <p><a name="index-r2c_002fc2r-multi_002ddimensional-array-format-201"></a>
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61 The output of a DFT of real data (r2c) contains symmetries that, in
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62 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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63 practice, it is not possible to entirely realize these savings in an
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64 efficient and understandable format that generalizes to
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65 multi-dimensional transforms. Instead, the output of the r2c
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66 transforms is <em>slightly</em> over half of the output of the
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67 corresponding complex transform. We do not “pack” the data in any
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68 way, but store it as an ordinary array of <code>fftw_complex</code> values.
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69 In fact, this data is simply a subsection of what would be the array in
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70 the corresponding complex transform.
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71
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72 <p>Specifically, for a real transform of d (= <code>rank</code>)
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73 dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>, the complex data 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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74 <code>fftw_complex</code> values in row-major order (with the division rounded
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75 down). That is, we only store the <em>lower</em> half (non-negative
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76 frequencies), plus one element, of the last dimension of the data from
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77 the ordinary complex transform. (We could have instead taken half of
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78 any other dimension, but implementation turns out to be simpler if the
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79 last, contiguous, dimension is used.)
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80
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81 <p><a name="index-out_002dof_002dplace-202"></a>For an out-of-place transform, the real data is simply an array with
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82 physical 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
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84 <p><a name="index-in_002dplace-203"></a><a name="index-padding-204"></a>For an in-place transform, some complications arise since the complex data
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85 is slightly larger than the real data. In this case, the final
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86 dimension of the real data must be <em>padded</em> with extra values to
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87 accommodate the size of the complex data—two extra if the last
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88 dimension is even and one if it is odd. That is, the last dimension of
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89 the real data must physically contain
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90 2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data). This
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91 physical array size does not, however, change the <em>logical</em> array
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92 size—only
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93 n<sub>d-1</sub>values are actually stored in the last dimension, and
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94 n<sub>d-1</sub>is the last dimension passed to the planner.
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95
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96 <!-- =========> -->
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97 </body></html>
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98
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