Chris@42: <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> Chris@42: <html> Chris@42: <!-- This manual is for FFTW Chris@42: (version 3.3.5, 30 July 2016). Chris@42: Chris@42: Copyright (C) 2003 Matteo Frigo. Chris@42: Chris@42: Copyright (C) 2003 Massachusetts Institute of Technology. Chris@42: Chris@42: Permission is granted to make and distribute verbatim copies of this Chris@42: manual provided the copyright notice and this permission notice are Chris@42: preserved on all copies. Chris@42: Chris@42: Permission is granted to copy and distribute modified versions of this Chris@42: manual under the conditions for verbatim copying, provided that the Chris@42: entire resulting derived work is distributed under the terms of a Chris@42: permission notice identical to this one. Chris@42: Chris@42: Permission is granted to copy and distribute translations of this manual Chris@42: into another language, under the above conditions for modified versions, Chris@42: except that this permission notice may be stated in a translation Chris@42: approved by the Free Software Foundation. --> Chris@42: <!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ --> Chris@42: <head> Chris@42: <title>FFTW 3.3.5: The Halfcomplex-format DFT</title> Chris@42: Chris@42: <meta name="description" content="FFTW 3.3.5: The Halfcomplex-format DFT"> Chris@42: <meta name="keywords" content="FFTW 3.3.5: The Halfcomplex-format DFT"> Chris@42: <meta name="resource-type" content="document"> Chris@42: <meta name="distribution" content="global"> Chris@42: <meta name="Generator" content="makeinfo"> Chris@42: <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> Chris@42: <link href="index.html#Top" rel="start" title="Top"> Chris@42: <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> Chris@42: <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> Chris@42: <link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="up" title="More DFTs of Real Data"> Chris@42: <link href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" rel="next" title="Real even/odd DFTs (cosine/sine transforms)"> Chris@42: <link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="prev" title="More DFTs of Real Data"> Chris@42: <style type="text/css"> Chris@42: <!-- Chris@42: a.summary-letter {text-decoration: none} Chris@42: blockquote.smallquotation {font-size: smaller} Chris@42: div.display {margin-left: 3.2em} Chris@42: div.example {margin-left: 3.2em} Chris@42: div.indentedblock {margin-left: 3.2em} Chris@42: div.lisp {margin-left: 3.2em} Chris@42: div.smalldisplay {margin-left: 3.2em} Chris@42: div.smallexample {margin-left: 3.2em} Chris@42: div.smallindentedblock {margin-left: 3.2em; font-size: smaller} Chris@42: div.smalllisp {margin-left: 3.2em} Chris@42: kbd {font-style:oblique} Chris@42: pre.display {font-family: inherit} Chris@42: pre.format {font-family: inherit} Chris@42: pre.menu-comment {font-family: serif} Chris@42: pre.menu-preformatted {font-family: serif} Chris@42: pre.smalldisplay {font-family: inherit; font-size: smaller} Chris@42: pre.smallexample {font-size: smaller} Chris@42: pre.smallformat {font-family: inherit; font-size: smaller} Chris@42: pre.smalllisp {font-size: smaller} Chris@42: span.nocodebreak {white-space:nowrap} Chris@42: span.nolinebreak {white-space:nowrap} Chris@42: span.roman {font-family:serif; font-weight:normal} Chris@42: span.sansserif {font-family:sans-serif; font-weight:normal} Chris@42: ul.no-bullet {list-style: none} Chris@42: --> Chris@42: </style> Chris@42: Chris@42: Chris@42: </head> Chris@42: Chris@42: <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000"> Chris@42: <a name="The-Halfcomplex_002dformat-DFT"></a> Chris@42: <div class="header"> Chris@42: <p> Chris@42: Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</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> Chris@42: </div> Chris@42: <hr> Chris@42: <a name="The-Halfcomplex_002dformat-DFT-1"></a> Chris@42: <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4> Chris@42: Chris@42: <p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT Chris@42: <a name="index-FFTW_005fR2HC"></a> Chris@42: <a name="index-r2c-1"></a> Chris@42: <a name="index-r2hc"></a> Chris@42: (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” Chris@42: format output, and may sometimes be faster and/or more convenient than Chris@42: the latter. Chris@42: <a name="index-halfcomplex-format-1"></a> Chris@42: The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>. Chris@42: <a name="index-FFTW_005fHC2R"></a> Chris@42: <a name="index-hc2r"></a> Chris@42: This consists of the non-redundant half of the complex output for a 1d Chris@42: real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real Chris@42: numbers (<code>double</code>) in the format: Chris@42: </p> Chris@42: <p align=center> Chris@42: 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> Chris@42: </p> Chris@42: <p>Here, Chris@42: r<sub>k</sub>is the real part of the <em>k</em>th output, and Chris@42: i<sub>k</sub>is the imaginary part. (Division by 2 is rounded down.) For a Chris@42: halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its Chris@42: real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with Chris@42: the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter Chris@42: only if <code>n</code> is even)—in these two cases, the imaginary part is Chris@42: zero due to symmetries of the real-input DFT, and is not stored. Chris@42: Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of Chris@42: length <code>n</code>, and vice versa for hc2r. Chris@42: <a name="index-normalization-2"></a> Chris@42: </p> Chris@42: Chris@42: <p>Aside from the differing format, the output of Chris@42: <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for Chris@42: the corresponding 1d r2c/c2r transform Chris@42: (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively). Chris@42: Recall that these transforms are unnormalized, so r2hc followed by hc2r Chris@42: will result in the original data multiplied by <code>n</code>. Furthermore, Chris@42: like the c2r transform, an out-of-place hc2r transform will Chris@42: <em>destroy its input</em> array. Chris@42: </p> Chris@42: <p>Although these halfcomplex transforms can be used with the Chris@42: multi-dimensional r2r interface, the interpretation of such a separable Chris@42: product of transforms along each dimension is problematic. For example, Chris@42: consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc Chris@42: transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC, Chris@42: FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows Chris@42: (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the Chris@42: columns (of size <code>n0</code>). Half of these column transforms, however, Chris@42: are of imaginary parts, and should therefore be multiplied by <em>i</em> Chris@42: and combined with the r2hc transforms of the real columns to produce the Chris@42: 2d DFT amplitudes; FFTW’s r2r transform does <em>not</em> perform this Chris@42: combination for you. Thus, if a multi-dimensional real-input/output DFT Chris@42: is required, we recommend using the ordinary r2c/c2r Chris@42: interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>). Chris@42: </p> Chris@42: <hr> Chris@42: <div class="header"> Chris@42: <p> Chris@42: Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</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> Chris@42: </div> Chris@42: Chris@42: Chris@42: Chris@42: </body> Chris@42: </html>