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diff src/fftw-3.3.5/doc/html/Real_002ddata-DFT-Array-Format.html @ 42:2cd0e3b3e1fd
Current fftw source
| 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/Real_002ddata-DFT-Array-Format.html Tue Oct 18 13:40:26 2016 +0100 @@ -0,0 +1,129 @@ +<!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: Real-data DFT Array Format</title> + +<meta name="description" content="FFTW 3.3.5: Real-data DFT Array Format"> +<meta name="keywords" content="FFTW 3.3.5: Real-data DFT Array Format"> +<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="Basic-Interface.html#Basic-Interface" rel="up" title="Basic Interface"> +<link href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" rel="next" title="Real-to-Real Transforms"> +<link href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" rel="prev" title="Real-data DFTs"> +<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="Real_002ddata-DFT-Array-Format"></a> +<div class="header"> +<p> +Next: <a href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" accesskey="n" rel="next">Real-to-Real Transforms</a>, Previous: <a href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" accesskey="p" rel="prev">Real-data DFTs</a>, Up: <a href="Basic-Interface.html#Basic-Interface" accesskey="u" rel="up">Basic Interface</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="Real_002ddata-DFT-Array-Format-1"></a> +<h4 class="subsection">4.3.4 Real-data DFT Array Format</h4> +<a name="index-r2c_002fc2r-multi_002ddimensional-array-format-1"></a> + +<p>The output of a DFT of real data (r2c) contains symmetries that, in +principle, make half of the outputs redundant (see <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>). (Similarly for the input of an inverse c2r transform.) In +practice, it is not possible to entirely realize these savings in an +efficient and understandable format that generalizes to +multi-dimensional transforms. Instead, the output of the r2c +transforms is <em>slightly</em> over half of the output of the +corresponding complex transform. We do not “pack” the data in any +way, but store it as an ordinary array of <code>fftw_complex</code> values. +In fact, this data is simply a subsection of what would be the array in +the corresponding complex transform. +</p> +<p>Specifically, for a real transform of <em>d</em> (= <code>rank</code>) +dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>, the complex data 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 (with the division rounded +down). That is, we only store the <em>lower</em> half (non-negative +frequencies), plus one element, of the last dimension of the data from +the ordinary complex transform. (We could have instead taken half of +any other dimension, but implementation turns out to be simpler if the +last, contiguous, dimension is used.) +</p> +<a name="index-out_002dof_002dplace-1"></a> +<p>For an out-of-place transform, the real data is simply an array with +physical dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub> in row-major order. +</p> +<a name="index-in_002dplace-4"></a> +<a name="index-padding-3"></a> +<p>For an in-place transform, some complications arise since the complex data +is slightly larger than the real data. In this case, the final +dimension of the real data must be <em>padded</em> with extra values to +accommodate the size of the complex data—two extra if the last +dimension is even and one if it is odd. 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 planner. +</p> +<hr> +<div class="header"> +<p> +Next: <a href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" accesskey="n" rel="next">Real-to-Real Transforms</a>, Previous: <a href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" accesskey="p" rel="prev">Real-data DFTs</a>, Up: <a href="Basic-Interface.html#Basic-Interface" accesskey="u" rel="up">Basic Interface</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>