Mercurial > hg > sv-dependency-builds
diff src/fftw-3.3.3/doc/html/Real_002ddata-DFT-Array-Format.html @ 10:37bf6b4a2645
Add FFTW3
| author | Chris Cannam |
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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/Real_002ddata-DFT-Array-Format.html Wed Mar 20 15:35:50 2013 +0000 @@ -0,0 +1,98 @@ +<html lang="en"> +<head> +<title>Real-data DFT Array Format - 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="Basic-Interface.html#Basic-Interface" title="Basic Interface"> +<link rel="prev" href="Real_002ddata-DFTs.html#Real_002ddata-DFTs" title="Real-data DFTs"> +<link rel="next" href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms" title="Real-to-Real Transforms"> +<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="Real-data-DFT-Array-Format"></a> +<a name="Real_002ddata-DFT-Array-Format"></a> +<p> +Next: <a rel="next" accesskey="n" href="Real_002dto_002dReal-Transforms.html#Real_002dto_002dReal-Transforms">Real-to-Real Transforms</a>, +Previous: <a rel="previous" accesskey="p" href="Real_002ddata-DFTs.html#Real_002ddata-DFTs">Real-data DFTs</a>, +Up: <a rel="up" accesskey="u" href="Basic-Interface.html#Basic-Interface">Basic Interface</a> +<hr> +</div> + +<h4 class="subsection">4.3.4 Real-data DFT Array Format</h4> + +<p><a name="index-r2c_002fc2r-multi_002ddimensional-array-format-199"></a> +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>Specifically, for a real transform of d (= <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-200"></a>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-201"></a><a name="index-padding-202"></a>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. + +<!-- =========> --> + </body></html> +