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author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
parents bd3cc4d1df30
children
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cannam@167 3 <!-- This manual is for FFTW
cannam@167 4 (version 3.3.8, 24 May 2018).
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cannam@167 6 Copyright (C) 2003 Matteo Frigo.
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cannam@167 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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cannam@167 24 <head>
cannam@167 25 <title>FFTW 3.3.8: Multi-Dimensional DFTs of Real Data</title>
cannam@167 26
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cannam@167 68 </head>
cannam@167 69
cannam@167 70 <body lang="en">
cannam@167 71 <a name="Multi_002dDimensional-DFTs-of-Real-Data"></a>
cannam@167 72 <div class="header">
cannam@167 73 <p>
cannam@167 74 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> &nbsp; [<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>
cannam@167 75 </div>
cannam@167 76 <hr>
cannam@167 77 <a name="Multi_002dDimensional-DFTs-of-Real-Data-1"></a>
cannam@167 78 <h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3>
cannam@167 79
cannam@167 80 <p>Multi-dimensional DFTs of real data use the following planner routines:
cannam@167 81 </p>
cannam@167 82 <div class="example">
cannam@167 83 <pre class="example">fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
cannam@167 84 double *in, fftw_complex *out,
cannam@167 85 unsigned flags);
cannam@167 86 fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
cannam@167 87 double *in, fftw_complex *out,
cannam@167 88 unsigned flags);
cannam@167 89 fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
cannam@167 90 double *in, fftw_complex *out,
cannam@167 91 unsigned flags);
cannam@167 92 </pre></div>
cannam@167 93 <a name="index-fftw_005fplan_005fdft_005fr2c_005f2d"></a>
cannam@167 94 <a name="index-fftw_005fplan_005fdft_005fr2c_005f3d"></a>
cannam@167 95 <a name="index-fftw_005fplan_005fdft_005fr2c"></a>
cannam@167 96
cannam@167 97 <p>as well as the corresponding <code>c2r</code> routines with the input/output
cannam@167 98 types swapped. These routines work similarly to their complex
cannam@167 99 analogues, except for the fact that here the complex output array is cut
cannam@167 100 roughly in half and the real array requires padding for in-place
cannam@167 101 transforms (as in 1d, above).
cannam@167 102 </p>
cannam@167 103 <p>As before, <code>n</code> is the logical size of the array, and the
cannam@167 104 consequences of this on the the format of the complex arrays deserve
cannam@167 105 careful attention.
cannam@167 106 <a name="index-r2c_002fc2r-multi_002ddimensional-array-format"></a>
cannam@167 107 Suppose that the real data has dimensions n<sub>0</sub>&nbsp;&times;&nbsp;n<sub>1</sub>&nbsp;&times;&nbsp;n<sub>2</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;n<sub>d-1</sub>
cannam@167 108 (in row-major order).
cannam@167 109 Then, after an r2c transform, the output is an n<sub>0</sub>&nbsp;&times;&nbsp;n<sub>1</sub>&nbsp;&times;&nbsp;n<sub>2</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;(n<sub>d-1</sub>/2 + 1)
cannam@167 110 array of
cannam@167 111 <code>fftw_complex</code> values in row-major order, corresponding to slightly
cannam@167 112 over half of the output of the corresponding complex DFT. (The division
cannam@167 113 is rounded down.) The ordering of the data is otherwise exactly the
cannam@167 114 same as in the complex-DFT case.
cannam@167 115 </p>
cannam@167 116 <p>For out-of-place transforms, this is the end of the story: the real
cannam@167 117 data is stored as a row-major array of size n<sub>0</sub>&nbsp;&times;&nbsp;n<sub>1</sub>&nbsp;&times;&nbsp;n<sub>2</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;n<sub>d-1</sub>
cannam@167 118 and the complex
cannam@167 119 data is stored as a row-major array of size n<sub>0</sub>&nbsp;&times;&nbsp;n<sub>1</sub>&nbsp;&times;&nbsp;n<sub>2</sub>&nbsp;&times;&nbsp;&hellip;&nbsp;&times;&nbsp;(n<sub>d-1</sub>/2 + 1)
cannam@167 120 .
cannam@167 121 </p>
cannam@167 122 <p>For in-place transforms, however, extra padding of the real-data array
cannam@167 123 is necessary because the complex array is larger than the real array,
cannam@167 124 and the two arrays share the same memory locations. Thus, for
cannam@167 125 in-place transforms, the final dimension of the real-data array must
cannam@167 126 be padded with extra values to accommodate the size of the complex
cannam@167 127 data&mdash;two values if the last dimension is even and one if it is odd.
cannam@167 128 <a name="index-padding-1"></a>
cannam@167 129 That is, the last dimension of the real data must physically contain
cannam@167 130 2 * (n<sub>d-1</sub>/2+1)
cannam@167 131 <code>double</code> values (exactly enough to hold the complex data).
cannam@167 132 This physical array size does not, however, change the <em>logical</em>
cannam@167 133 array size&mdash;only
cannam@167 134 n<sub>d-1</sub>
cannam@167 135 values are actually stored in the last dimension, and
cannam@167 136 n<sub>d-1</sub>
cannam@167 137 is the last dimension passed to the plan-creation routine.
cannam@167 138 </p>
cannam@167 139 <p>For example, consider the transform of a two-dimensional real array of
cannam@167 140 size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a
cannam@167 141 two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where
cannam@167 142 the <code>y</code> dimension has been cut nearly in half because of
cannam@167 143 redundancies in the output. Because <code>fftw_complex</code> is twice the
cannam@167 144 size of <code>double</code>, the output array is slightly bigger than the
cannam@167 145 input array. Thus, if we want to compute the transform in place, we
cannam@167 146 must <em>pad</em> the input array so that it is of size <code>n0</code> by
cannam@167 147 <code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding
cannam@167 148 elements at the end of each row (which need not be initialized, as they
cannam@167 149 are only used for output).
cannam@167 150 </p>
cannam@167 151 <p>The following illustration depicts the input and output arrays just
cannam@167 152 described, for both the out-of-place and in-place transforms (with the
cannam@167 153 arrows indicating consecutive memory locations):
cannam@167 154 <img src="rfftwnd-for-html.png" alt="rfftwnd-for-html">
cannam@167 155 </p>
cannam@167 156 <p>These transforms are unnormalized, so an r2c followed by a c2r
cannam@167 157 transform (or vice versa) will result in the original data scaled by
cannam@167 158 the number of real data elements&mdash;that is, the product of the
cannam@167 159 (logical) dimensions of the real data.
cannam@167 160 <a name="index-normalization-1"></a>
cannam@167 161 </p>
cannam@167 162
cannam@167 163 <p>(Because the last dimension is treated specially, if it is equal to
cannam@167 164 <code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional
cannam@167 165 r2c/c2r transform. In that case, the last complex dimension also has
cannam@167 166 size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the
cannam@167 167 complex transforms.)
cannam@167 168 </p>
cannam@167 169 <hr>
cannam@167 170 <div class="header">
cannam@167 171 <p>
cannam@167 172 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> &nbsp; [<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>
cannam@167 173 </div>
cannam@167 174
cannam@167 175
cannam@167 176
cannam@167 177 </body>
cannam@167 178 </html>