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| author | Chris Cannam |
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| date | Wed, 20 Mar 2013 15:35:50 +0000 |
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| 1 <html lang="en"> | |
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| 3 <title>Multi-Dimensional DFTs of Real Data - FFTW 3.3.3</title> | |
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| 12 <!-- | |
| 13 This manual is for FFTW | |
| 14 (version 3.3.3, 25 November 2012). | |
| 15 | |
| 16 Copyright (C) 2003 Matteo Frigo. | |
| 17 | |
| 18 Copyright (C) 2003 Massachusetts Institute of Technology. | |
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| 49 <a name="Multi-Dimensional-DFTs-of-Real-Data"></a> | |
| 50 <a name="Multi_002dDimensional-DFTs-of-Real-Data"></a> | |
| 51 <p> | |
| 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>, | |
| 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>, | |
| 54 Up: <a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a> | |
| 55 <hr> | |
| 56 </div> | |
| 57 | |
| 58 <h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3> | |
| 59 | |
| 60 <p>Multi-dimensional DFTs of real data use the following planner routines: | |
| 61 | |
| 62 <pre class="example"> fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1, | |
| 63 double *in, fftw_complex *out, | |
| 64 unsigned flags); | |
| 65 fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2, | |
| 66 double *in, fftw_complex *out, | |
| 67 unsigned flags); | |
| 68 fftw_plan fftw_plan_dft_r2c(int rank, const int *n, | |
| 69 double *in, fftw_complex *out, | |
| 70 unsigned flags); | |
| 71 </pre> | |
| 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> | |
| 73 as well as the corresponding <code>c2r</code> routines with the input/output | |
| 74 types swapped. These routines work similarly to their complex | |
| 75 analogues, except for the fact that here the complex output array is cut | |
| 76 roughly in half and the real array requires padding for in-place | |
| 77 transforms (as in 1d, above). | |
| 78 | |
| 79 <p>As before, <code>n</code> is the logical size of the array, and the | |
| 80 consequences of this on the the format of the complex arrays deserve | |
| 81 careful attention. | |
| 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). | |
| 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 | |
| 84 <code>fftw_complex</code> values in row-major order, corresponding to slightly | |
| 85 over half of the output of the corresponding complex DFT. (The division | |
| 86 is rounded down.) The ordering of the data is otherwise exactly the | |
| 87 same as in the complex-DFT case. | |
| 88 | |
| 89 <p>For out-of-place transforms, this is the end of the story: the real | |
| 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 | |
| 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). | |
| 92 | |
| 93 <p>For in-place transforms, however, extra padding of the real-data array | |
| 94 is necessary because the complex array is larger than the real array, | |
| 95 and the two arrays share the same memory locations. Thus, for | |
| 96 in-place transforms, the final dimension of the real-data array must | |
| 97 be padded with extra values to accommodate the size of the complex | |
| 98 data—two values if the last dimension is even and one if it is odd. | |
| 99 <a name="index-padding-63"></a>That is, the last dimension of the real data must physically contain | |
| 100 2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data). | |
| 101 This physical array size does not, however, change the <em>logical</em> | |
| 102 array size—only | |
| 103 n<sub>d-1</sub>values are actually stored in the last dimension, and | |
| 104 n<sub>d-1</sub>is the last dimension passed to the plan-creation routine. | |
| 105 | |
| 106 <p>For example, consider the transform of a two-dimensional real array of | |
| 107 size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a | |
| 108 two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where | |
| 109 the <code>y</code> dimension has been cut nearly in half because of | |
| 110 redundancies in the output. Because <code>fftw_complex</code> is twice the | |
| 111 size of <code>double</code>, the output array is slightly bigger than the | |
| 112 input array. Thus, if we want to compute the transform in place, we | |
| 113 must <em>pad</em> the input array so that it is of size <code>n0</code> by | |
| 114 <code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding | |
| 115 elements at the end of each row (which need not be initialized, as they | |
| 116 are only used for output). | |
| 117 | |
| 118 <p>The following illustration depicts the input and output arrays just | |
| 119 described, for both the out-of-place and in-place transforms (with the | |
| 120 arrows indicating consecutive memory locations): | |
| 121 <img src="rfftwnd-for-html.png" alt="rfftwnd-for-html.png"> | |
| 122 | |
| 123 <p>These transforms are unnormalized, so an r2c followed by a c2r | |
| 124 transform (or vice versa) will result in the original data scaled by | |
| 125 the number of real data elements—that is, the product of the | |
| 126 (logical) dimensions of the real data. | |
| 127 <a name="index-normalization-64"></a> | |
| 128 | |
| 129 <p>(Because the last dimension is treated specially, if it is equal to | |
| 130 <code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional | |
| 131 r2c/c2r transform. In that case, the last complex dimension also has | |
| 132 size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the | |
| 133 complex transforms.) | |
| 134 | |
| 135 <!-- --> | |
| 136 </body></html> | |
| 137 |