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| author | Chris Cannam |
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| date | Tue, 18 Oct 2016 13:40:26 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.5/doc/html/Multi_002dDimensional-DFTs-of-Real-Data.html Tue Oct 18 13:40:26 2016 +0100 @@ -0,0 +1,172 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> +<html> +<!-- This manual is for FFTW +(version 3.3.5, 30 July 2016). + +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. --> +<!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ --> +<head> +<title>FFTW 3.3.5: Multi-Dimensional DFTs of Real Data</title> + +<meta name="description" content="FFTW 3.3.5: Multi-Dimensional DFTs of Real Data"> +<meta name="keywords" content="FFTW 3.3.5: Multi-Dimensional DFTs of Real Data"> +<meta name="resource-type" content="document"> +<meta name="distribution" content="global"> +<meta name="Generator" content="makeinfo"> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8"> +<link href="index.html#Top" rel="start" title="Top"> +<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> +<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> +<link href="Tutorial.html#Tutorial" rel="up" title="Tutorial"> +<link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="next" title="More DFTs of Real Data"> +<link href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" rel="prev" title="One-Dimensional DFTs of Real Data"> +<style type="text/css"> +<!-- +a.summary-letter {text-decoration: none} +blockquote.smallquotation {font-size: smaller} +div.display {margin-left: 3.2em} +div.example {margin-left: 3.2em} +div.indentedblock {margin-left: 3.2em} +div.lisp {margin-left: 3.2em} +div.smalldisplay {margin-left: 3.2em} +div.smallexample {margin-left: 3.2em} +div.smallindentedblock {margin-left: 3.2em; font-size: smaller} +div.smalllisp {margin-left: 3.2em} +kbd {font-style:oblique} +pre.display {font-family: inherit} +pre.format {font-family: inherit} +pre.menu-comment {font-family: serif} +pre.menu-preformatted {font-family: serif} +pre.smalldisplay {font-family: inherit; font-size: smaller} +pre.smallexample {font-size: smaller} +pre.smallformat {font-family: inherit; font-size: smaller} +pre.smalllisp {font-size: smaller} +span.nocodebreak {white-space:nowrap} +span.nolinebreak {white-space:nowrap} +span.roman {font-family:serif; font-weight:normal} +span.sansserif {font-family:sans-serif; font-weight:normal} +ul.no-bullet {list-style: none} +--> +</style> + + +</head> + +<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000"> +<a name="Multi_002dDimensional-DFTs-of-Real-Data"></a> +<div class="header"> +<p> +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> +</div> +<hr> +<a name="Multi_002dDimensional-DFTs-of-Real-Data-1"></a> +<h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3> + +<p>Multi-dimensional DFTs of real data use the following planner routines: +</p> +<div class="example"> +<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></div> +<a name="index-fftw_005fplan_005fdft_005fr2c_005f2d"></a> +<a name="index-fftw_005fplan_005fdft_005fr2c_005f3d"></a> +<a name="index-fftw_005fplan_005fdft_005fr2c"></a> + +<p>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> +<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"></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> +<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> +<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-1"></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> +<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> +<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"> +</p> +<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-1"></a> +</p> + +<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.) +</p> +<hr> +<div class="header"> +<p> +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> +</div> + + + +</body> +</html>