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Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 19 Nov 2019 14:52:55 +0000
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+<!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 3.3.8, 24 May 2018).
+
+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 6.3, http://www.gnu.org/software/texinfo/ -->
+<head>
+<title>FFTW 3.3.8: The Halfcomplex-format DFT</title>
+
+<meta name="description" content="FFTW 3.3.8: The Halfcomplex-format DFT">
+<meta name="keywords" content="FFTW 3.3.8: The Halfcomplex-format DFT">
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+<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="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="up" title="More DFTs of Real Data">
+<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)">
+<link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="prev" title="More DFTs of Real Data">
+<style type="text/css">
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+</head>
+
+<body lang="en">
+<a name="The-Halfcomplex_002dformat-DFT"></a>
+<div class="header">
+<p>
+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> &nbsp; [<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="The-Halfcomplex_002dformat-DFT-1"></a>
+<h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
+
+<p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
+<a name="index-FFTW_005fR2HC"></a>
+<a name="index-r2c-1"></a>
+<a name="index-r2hc"></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 &ldquo;halfcomplex&rdquo;
+format output, and may sometimes be faster and/or more convenient than
+the latter.
+<a name="index-halfcomplex-format-1"></a>
+The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
+<a name="index-FFTW_005fHC2R"></a>
+<a name="index-hc2r"></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 <em>k</em>th 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 <em>k</em>th 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)&mdash;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-2"></a>
+</p>
+
+<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>
+<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 <em>i</em>
+and combined with the r2hc transforms of the real columns to produce the
+2d DFT amplitudes; FFTW&rsquo;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>).
+</p>
+<hr>
+<div class="header">
+<p>
+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> &nbsp; [<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>