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comparison src/fftw-3.3.5/doc/html/Real_002ddata-DFT-Array-Format.html @ 127:7867fa7e1b6b
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| author | Chris Cannam <cannam@all-day-breakfast.com> |
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| date | Tue, 18 Oct 2016 13:40:26 +0100 |
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| 126:4a7071416412 | 127:7867fa7e1b6b |
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| 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> | |
| 2 <html> | |
| 3 <!-- This manual is for FFTW | |
| 4 (version 3.3.5, 30 July 2016). | |
| 5 | |
| 6 Copyright (C) 2003 Matteo Frigo. | |
| 7 | |
| 8 Copyright (C) 2003 Massachusetts Institute of Technology. | |
| 9 | |
| 10 Permission is granted to make and distribute verbatim copies of this | |
| 11 manual provided the copyright notice and this permission notice are | |
| 12 preserved on all copies. | |
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| 14 Permission is granted to copy and distribute modified versions of this | |
| 15 manual under the conditions for verbatim copying, provided that the | |
| 16 entire resulting derived work is distributed under the terms of a | |
| 17 permission notice identical to this one. | |
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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. --> | |
| 23 <!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ --> | |
| 24 <head> | |
| 25 <title>FFTW 3.3.5: Real-data DFT Array Format</title> | |
| 26 | |
| 27 <meta name="description" content="FFTW 3.3.5: Real-data DFT Array Format"> | |
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| 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> | |
| 36 <link href="Basic-Interface.html#Basic-Interface" rel="up" title="Basic Interface"> | |
| 37 <link href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" rel="next" title="Real-to-Real Transforms"> | |
| 38 <link href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" rel="prev" title="Real-data DFTs"> | |
| 39 <style type="text/css"> | |
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| 66 </style> | |
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| 69 </head> | |
| 70 | |
| 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000"> | |
| 72 <a name="Real_002ddata-DFT-Array-Format"></a> | |
| 73 <div class="header"> | |
| 74 <p> | |
| 75 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> | |
| 76 </div> | |
| 77 <hr> | |
| 78 <a name="Real_002ddata-DFT-Array-Format-1"></a> | |
| 79 <h4 class="subsection">4.3.4 Real-data DFT Array Format</h4> | |
| 80 <a name="index-r2c_002fc2r-multi_002ddimensional-array-format-1"></a> | |
| 81 | |
| 82 <p>The output of a DFT of real data (r2c) contains symmetries that, in | |
| 83 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 | |
| 84 practice, it is not possible to entirely realize these savings in an | |
| 85 efficient and understandable format that generalizes to | |
| 86 multi-dimensional transforms. Instead, the output of the r2c | |
| 87 transforms is <em>slightly</em> over half of the output of the | |
| 88 corresponding complex transform. We do not “pack” the data in any | |
| 89 way, but store it as an ordinary array of <code>fftw_complex</code> values. | |
| 90 In fact, this data is simply a subsection of what would be the array in | |
| 91 the corresponding complex transform. | |
| 92 </p> | |
| 93 <p>Specifically, for a real transform of <em>d</em> (= <code>rank</code>) | |
| 94 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 | |
| 95 <code>fftw_complex</code> values in row-major order (with the division rounded | |
| 96 down). That is, we only store the <em>lower</em> half (non-negative | |
| 97 frequencies), plus one element, of the last dimension of the data from | |
| 98 the ordinary complex transform. (We could have instead taken half of | |
| 99 any other dimension, but implementation turns out to be simpler if the | |
| 100 last, contiguous, dimension is used.) | |
| 101 </p> | |
| 102 <a name="index-out_002dof_002dplace-1"></a> | |
| 103 <p>For an out-of-place transform, the real data is simply an array with | |
| 104 physical dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub> in row-major order. | |
| 105 </p> | |
| 106 <a name="index-in_002dplace-4"></a> | |
| 107 <a name="index-padding-3"></a> | |
| 108 <p>For an in-place transform, some complications arise since the complex data | |
| 109 is slightly larger than the real data. In this case, the final | |
| 110 dimension of the real data must be <em>padded</em> with extra values to | |
| 111 accommodate the size of the complex data—two extra if the last | |
| 112 dimension is even and one if it is odd. That is, the last dimension of | |
| 113 the real data must physically contain | |
| 114 2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data). This | |
| 115 physical array size does not, however, change the <em>logical</em> array | |
| 116 size—only | |
| 117 n<sub>d-1</sub>values are actually stored in the last dimension, and | |
| 118 n<sub>d-1</sub>is the last dimension passed to the planner. | |
| 119 </p> | |
| 120 <hr> | |
| 121 <div class="header"> | |
| 122 <p> | |
| 123 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> | |
| 124 </div> | |
| 125 | |
| 126 | |
| 127 | |
| 128 </body> | |
| 129 </html> |