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author Geogaddi\David <d.m.ronan@qmul.ac.uk>
date Thu, 09 Jul 2015 01:12:16 +0100
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1 <html lang="en">
2 <head>
3 <title>Multi-Dimensional DFTs of Real Data - FFTW 3.2.1</title>
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50 <a name="Multi-Dimensional-DFTs-of-Real-Data"></a>
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52 Next:&nbsp;<a rel="next" accesskey="n" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>,
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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-58"></a><a name="index-fftw_005fplan_005fdft_005fr2c_005f3d-59"></a><a name="index-fftw_005fplan_005fdft_005fr2c-60"></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-61"></a>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> (in row-major order).
83 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) 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>Since the complex data is slightly larger than the real data, some
90 complications arise for in-place transforms. In this case, the final
91 dimension of the real data must be padded with extra values to
92 accommodate the size of the complex data&mdash;two values if the last
93 dimension is even and one if it is odd.
94 <a name="index-padding-62"></a>That is, the last dimension of the real data must physically contain
95 2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data).
96 This physical array size does not, however, change the <em>logical</em>
97 array size&mdash;only
98 n<sub>d-1</sub>values are actually stored in the last dimension, and
99 n<sub>d-1</sub>is the last dimension passed to the plan-creation routine.
100
101 <p>For example, consider the transform of a two-dimensional real array of
102 size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a
103 two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where
104 the <code>y</code> dimension has been cut nearly in half because of
105 redundancies in the output. Because <code>fftw_complex</code> is twice the
106 size of <code>double</code>, the output array is slightly bigger than the
107 input array. Thus, if we want to compute the transform in place, we
108 must <em>pad</em> the input array so that it is of size <code>n0</code> by
109 <code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding
110 elements at the end of each row (which need not be initialized, as they
111 are only used for output).
112
113 <p>The following illustration depicts the input and output arrays just
114 described, for both the out-of-place and in-place transforms (with the
115 arrows indicating consecutive memory locations):
116
117 <div class="block-image"><img src="rfftwnd.png" alt="rfftwnd.png"></div>
118
119 <p>These transforms are unnormalized, so an r2c followed by a c2r
120 transform (or vice versa) will result in the original data scaled by
121 the number of real data elements&mdash;that is, the product of the
122 (logical) dimensions of the real data.
123 <a name="index-normalization-63"></a>
124 (Because the last dimension is treated specially, if it is equal to
125 <code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional
126 r2c/c2r transform. In that case, the last complex dimension also has
127 size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the
128 complex transforms.)
129
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