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| author | Chris Cannam <cannam@all-day-breakfast.com> |
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| date | Wed, 20 Mar 2013 15:35:50 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.3/doc/html/Multi_002dDimensional-DFTs-of-Real-Data.html Wed Mar 20 15:35:50 2013 +0000 @@ -0,0 +1,137 @@ +<html lang="en"> +<head> +<title>Multi-Dimensional DFTs of Real Data - FFTW 3.3.3</title> +<meta http-equiv="Content-Type" content="text/html"> +<meta name="description" content="FFTW 3.3.3"> +<meta name="generator" content="makeinfo 4.13"> +<link title="Top" rel="start" href="index.html#Top"> +<link rel="up" href="Tutorial.html#Tutorial" title="Tutorial"> +<link rel="prev" href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" title="One-Dimensional DFTs of Real Data"> +<link rel="next" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" title="More DFTs of Real Data"> +<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> +<!-- +This manual is for FFTW +(version 3.3.3, 25 November 2012). + +Copyright (C) 2003 Matteo Frigo. + +Copyright (C) 2003 Massachusetts Institute of Technology. + + Permission is granted to make and distribute verbatim copies of + this manual provided the copyright notice and this permission + notice are preserved on all copies. + + Permission is granted to copy and distribute modified versions of + this manual under the conditions for verbatim copying, provided + that the entire resulting derived work is distributed under the + terms of a permission notice identical to this one. + + Permission is granted to copy and distribute translations of this + manual into another language, under the above conditions for + modified versions, except that this permission notice may be + stated in a translation approved by the Free Software Foundation. + --> +<meta http-equiv="Content-Style-Type" content="text/css"> +<style type="text/css"><!-- + pre.display { font-family:inherit } + pre.format { font-family:inherit } + pre.smalldisplay { font-family:inherit; font-size:smaller } + pre.smallformat { font-family:inherit; font-size:smaller } + pre.smallexample { font-size:smaller } + pre.smalllisp { font-size:smaller } + span.sc { font-variant:small-caps } + span.roman { font-family:serif; font-weight:normal; } + span.sansserif { font-family:sans-serif; font-weight:normal; } +--></style> +</head> +<body> +<div class="node"> +<a name="Multi-Dimensional-DFTs-of-Real-Data"></a> +<a name="Multi_002dDimensional-DFTs-of-Real-Data"></a> +<p> +Next: <a rel="next" accesskey="n" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>, +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>, +Up: <a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a> +<hr> +</div> + +<h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3> + +<p>Multi-dimensional DFTs of real data use the following planner routines: + +<pre class="example"> fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1, + double *in, fftw_complex *out, + unsigned flags); + fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2, + double *in, fftw_complex *out, + unsigned flags); + fftw_plan fftw_plan_dft_r2c(int rank, const int *n, + double *in, fftw_complex *out, + unsigned flags); +</pre> + <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> +as well as the corresponding <code>c2r</code> routines with the input/output +types swapped. These routines work similarly to their complex +analogues, except for the fact that here the complex output array is cut +roughly in half and the real array requires padding for in-place +transforms (as in 1d, above). + + <p>As before, <code>n</code> is the logical size of the array, and the +consequences of this on the the format of the complex arrays deserve +careful attention. +<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). +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 +<code>fftw_complex</code> values in row-major order, corresponding to slightly +over half of the output of the corresponding complex DFT. (The division +is rounded down.) The ordering of the data is otherwise exactly the +same as in the complex-DFT case. + + <p>For out-of-place transforms, this is the end of the story: the real +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 +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). + + <p>For in-place transforms, however, extra padding of the real-data array +is necessary because the complex array is larger than the real array, +and the two arrays share the same memory locations. Thus, for +in-place transforms, the final dimension of the real-data array must +be padded with extra values to accommodate the size of the complex +data—two values if the last dimension is even and one if it is odd. +<a name="index-padding-63"></a>That is, the last dimension of the real data must physically contain +2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data). +This physical array size does not, however, change the <em>logical</em> +array size—only +n<sub>d-1</sub>values are actually stored in the last dimension, and +n<sub>d-1</sub>is the last dimension passed to the plan-creation routine. + + <p>For example, consider the transform of a two-dimensional real array of +size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a +two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where +the <code>y</code> dimension has been cut nearly in half because of +redundancies in the output. Because <code>fftw_complex</code> is twice the +size of <code>double</code>, the output array is slightly bigger than the +input array. Thus, if we want to compute the transform in place, we +must <em>pad</em> the input array so that it is of size <code>n0</code> by +<code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding +elements at the end of each row (which need not be initialized, as they +are only used for output). + + <p>The following illustration depicts the input and output arrays just +described, for both the out-of-place and in-place transforms (with the +arrows indicating consecutive memory locations): +<img src="rfftwnd-for-html.png" alt="rfftwnd-for-html.png"> + + <p>These transforms are unnormalized, so an r2c followed by a c2r +transform (or vice versa) will result in the original data scaled by +the number of real data elements—that is, the product of the +(logical) dimensions of the real data. +<a name="index-normalization-64"></a> + + <p>(Because the last dimension is treated specially, if it is equal to +<code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional +r2c/c2r transform. In that case, the last complex dimension also has +size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the +complex transforms.) + +<!-- --> + </body></html> +