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| 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/The-Halfcomplex_002dformat-DFT.html Wed Mar 20 15:35:50 2013 +0000 @@ -0,0 +1,111 @@ +<html lang="en"> +<head> +<title>The Halfcomplex-format DFT - 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="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" title="More DFTs of Real Data"> +<link rel="prev" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" title="More DFTs of Real Data"> +<link rel="next" href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" title="Real even/odd DFTs (cosine/sine 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="The-Halfcomplex-format-DFT"></a> +<a name="The-Halfcomplex_002dformat-DFT"></a> +<p> +Next: <a rel="next" accesskey="n" href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029">Real even/odd DFTs (cosine/sine transforms)</a>, +Previous: <a rel="previous" accesskey="p" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>, +Up: <a rel="up" accesskey="u" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a> +<hr> +</div> + +<h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4> + +<p>An r2r kind of <code>FFTW_R2HC</code> (<dfn>r2hc</dfn>) corresponds to an r2c DFT +<a name="index-FFTW_005fR2HC-72"></a><a name="index-r2c-73"></a><a name="index-r2hc-74"></a>(see <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data">One-Dimensional DFTs of Real Data</a>) but with “halfcomplex” +format output, and may sometimes be faster and/or more convenient than +the latter. +<a name="index-halfcomplex-format-75"></a>The inverse <dfn>hc2r</dfn> transform is of kind <code>FFTW_HC2R</code>. +<a name="index-FFTW_005fHC2R-76"></a><a name="index-hc2r-77"></a>This consists of the non-redundant half of the complex output for a 1d +real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real +numbers (<code>double</code>) in the format: + + <p><p align=center> +r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>n/2</sub>, i<sub>(n+1)/2-1</sub>, ..., i<sub>2</sub>, i<sub>1</sub> +</p> + + <p>Here, +r<sub>k</sub>is the real part of the kth output, and +i<sub>k</sub>is the imaginary part. (Division by 2 is rounded down.) For a +halfcomplex array <code>hc[n]</code>, the kth component thus has its +real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with +the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter +only if <code>n</code> is even)—in these two cases, the imaginary part is +zero due to symmetries of the real-input DFT, and is not stored. +Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of +length <code>n</code>, and vice versa for hc2r. +<a name="index-normalization-78"></a> + + <p>Aside from the differing format, the output of +<code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for +the corresponding 1d r2c/c2r transform +(i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively). +Recall that these transforms are unnormalized, so r2hc followed by hc2r +will result in the original data multiplied by <code>n</code>. Furthermore, +like the c2r transform, an out-of-place hc2r transform will +<em>destroy its input</em> array. + + <p>Although these halfcomplex transforms can be used with the +multi-dimensional r2r interface, the interpretation of such a separable +product of transforms along each dimension is problematic. For example, +consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc +transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC, +FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows +(of size <code>n1</code>) to produce halfcomplex rows, and then transforms the +columns (of size <code>n0</code>). Half of these column transforms, however, +are of imaginary parts, and should therefore be multiplied by i +and combined with the r2hc transforms of the real columns to produce the +2d DFT amplitudes; FFTW's r2r transform does <em>not</em> perform this +combination for you. Thus, if a multi-dimensional real-input/output DFT +is required, we recommend using the ordinary r2c/c2r +interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>). + +<!-- =========> --> + </body></html> +