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| author | Chris Cannam |
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| date | Tue, 18 Oct 2016 13:40:26 +0100 |
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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 | |
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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 | |
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| 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. --> | |
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| 24 <head> | |
| 25 <title>FFTW 3.3.5: Multi-Dimensional DFTs of Real Data</title> | |
| 26 | |
| 27 <meta name="description" content="FFTW 3.3.5: Multi-Dimensional DFTs of Real Data"> | |
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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"> | |
| 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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| 69 </head> | |
| 70 | |
| 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000"> | |
| 72 <a name="Multi_002dDimensional-DFTs-of-Real-Data"></a> | |
| 73 <div class="header"> | |
| 74 <p> | |
| 75 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> | |
| 76 </div> | |
| 77 <hr> | |
| 78 <a name="Multi_002dDimensional-DFTs-of-Real-Data-1"></a> | |
| 79 <h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3> | |
| 80 | |
| 81 <p>Multi-dimensional DFTs of real data use the following planner routines: | |
| 82 </p> | |
| 83 <div class="example"> | |
| 84 <pre class="example">fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1, | |
| 85 double *in, fftw_complex *out, | |
| 86 unsigned flags); | |
| 87 fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2, | |
| 88 double *in, fftw_complex *out, | |
| 89 unsigned flags); | |
| 90 fftw_plan fftw_plan_dft_r2c(int rank, const int *n, | |
| 91 double *in, fftw_complex *out, | |
| 92 unsigned flags); | |
| 93 </pre></div> | |
| 94 <a name="index-fftw_005fplan_005fdft_005fr2c_005f2d"></a> | |
| 95 <a name="index-fftw_005fplan_005fdft_005fr2c_005f3d"></a> | |
| 96 <a name="index-fftw_005fplan_005fdft_005fr2c"></a> | |
| 97 | |
| 98 <p>as well as the corresponding <code>c2r</code> routines with the input/output | |
| 99 types swapped. These routines work similarly to their complex | |
| 100 analogues, except for the fact that here the complex output array is cut | |
| 101 roughly in half and the real array requires padding for in-place | |
| 102 transforms (as in 1d, above). | |
| 103 </p> | |
| 104 <p>As before, <code>n</code> is the logical size of the array, and the | |
| 105 consequences of this on the the format of the complex arrays deserve | |
| 106 careful attention. | |
| 107 <a name="index-r2c_002fc2r-multi_002ddimensional-array-format"></a> | |
| 108 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). | |
| 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) array of | |
| 110 <code>fftw_complex</code> values in row-major order, corresponding to slightly | |
| 111 over half of the output of the corresponding complex DFT. (The division | |
| 112 is rounded down.) The ordering of the data is otherwise exactly the | |
| 113 same as in the complex-DFT case. | |
| 114 </p> | |
| 115 <p>For out-of-place transforms, this is the end of the story: the real | |
| 116 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 | |
| 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>/2 + 1). | |
| 118 </p> | |
| 119 <p>For in-place transforms, however, extra padding of the real-data array | |
| 120 is necessary because the complex array is larger than the real array, | |
| 121 and the two arrays share the same memory locations. Thus, for | |
| 122 in-place transforms, the final dimension of the real-data array must | |
| 123 be padded with extra values to accommodate the size of the complex | |
| 124 data—two values if the last dimension is even and one if it is odd. | |
| 125 <a name="index-padding-1"></a> | |
| 126 That is, the last dimension of the real data must physically contain | |
| 127 2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data). | |
| 128 This physical array size does not, however, change the <em>logical</em> | |
| 129 array size—only | |
| 130 n<sub>d-1</sub>values are actually stored in the last dimension, and | |
| 131 n<sub>d-1</sub>is the last dimension passed to the plan-creation routine. | |
| 132 </p> | |
| 133 <p>For example, consider the transform of a two-dimensional real array of | |
| 134 size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a | |
| 135 two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where | |
| 136 the <code>y</code> dimension has been cut nearly in half because of | |
| 137 redundancies in the output. Because <code>fftw_complex</code> is twice the | |
| 138 size of <code>double</code>, the output array is slightly bigger than the | |
| 139 input array. Thus, if we want to compute the transform in place, we | |
| 140 must <em>pad</em> the input array so that it is of size <code>n0</code> by | |
| 141 <code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding | |
| 142 elements at the end of each row (which need not be initialized, as they | |
| 143 are only used for output). | |
| 144 </p> | |
| 145 <p>The following illustration depicts the input and output arrays just | |
| 146 described, for both the out-of-place and in-place transforms (with the | |
| 147 arrows indicating consecutive memory locations): | |
| 148 <img src="rfftwnd-for-html.png" alt="rfftwnd-for-html"> | |
| 149 </p> | |
| 150 <p>These transforms are unnormalized, so an r2c followed by a c2r | |
| 151 transform (or vice versa) will result in the original data scaled by | |
| 152 the number of real data elements—that is, the product of the | |
| 153 (logical) dimensions of the real data. | |
| 154 <a name="index-normalization-1"></a> | |
| 155 </p> | |
| 156 | |
| 157 <p>(Because the last dimension is treated specially, if it is equal to | |
| 158 <code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional | |
| 159 r2c/c2r transform. In that case, the last complex dimension also has | |
| 160 size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the | |
| 161 complex transforms.) | |
| 162 </p> | |
| 163 <hr> | |
| 164 <div class="header"> | |
| 165 <p> | |
| 166 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> | |
| 167 </div> | |
| 168 | |
| 169 | |
| 170 | |
| 171 </body> | |
| 172 </html> |