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Current fftw source
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 18 Oct 2016 13:40:26 +0100
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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 3.3.5, 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: The Discrete Hartley Transform</title>
+
+<meta name="description" content="FFTW 3.3.5: The Discrete Hartley Transform">
+<meta name="keywords" content="FFTW 3.3.5: The Discrete Hartley Transform">
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+<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="Other-Important-Topics.html#Other-Important-Topics" rel="next" title="Other Important Topics">
+<link href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" rel="prev" title="Real even/odd DFTs (cosine/sine transforms)">
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+<!--
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+</head>
+
+<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
+<a name="The-Discrete-Hartley-Transform"></a>
+<div class="header">
+<p>
+Previous: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</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-Discrete-Hartley-Transform-1"></a>
+<h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
+
+<p>If you are planning to use the DHT because you&rsquo;ve heard that it is
+&ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>.  The DHT is not
+faster than the DFT.  That story is an old but enduring misconception
+that was debunked in 1987.
+</p>
+<p>The discrete Hartley transform (DHT) is an invertible linear transform
+closely related to the DFT.  In the DFT, one multiplies each input by
+<em>cos - i * sin</em> (a complex exponential), whereas in the DHT each
+input is multiplied by simply <em>cos + sin</em>.  Thus, the DHT
+transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
+convenient property of being its own inverse.  In FFTW, a DHT (of any
+positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
+<a name="index-FFTW_005fDHT"></a>
+<a name="index-discrete-Hartley-transform"></a>
+<a name="index-DHT"></a>
+</p>
+<p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
+size <code>n</code> followed by another DHT of the same size will result in
+the original array multiplied by <code>n</code>.
+<a name="index-normalization-4"></a>
+</p>
+<p>The DHT was originally proposed as a more efficient alternative to the
+DFT for real data, but it was subsequently shown that a specialized DFT
+(such as FFTW&rsquo;s r2hc or r2c transforms) could be just as fast.  In FFTW,
+the DHT is actually computed by post-processing an r2hc transform, so
+there is ordinarily no reason to prefer it from a performance
+perspective.<a name="DOCF5" href="#FOOT5"><sup>5</sup></a>
+However, we have heard rumors that the DHT might be the most appropriate
+transform in its own right for certain applications, and we would be
+very interested to hear from anyone who finds it useful.
+</p>
+<p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
+multi-dimensional transform, FFTW computes the separable product of 1d
+DHTs along each dimension.  Unfortunately, this is not quite the same
+thing as a true multi-dimensional DHT; you can compute the latter, if
+necessary, with at most <code>rank-1</code> post-processing passes
+[see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
+</p>
+<p>For the precise mathematical definition of the DHT as used by FFTW, see
+<a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
+</p>
+<div class="footnote">
+<hr>
+<h4 class="footnotes-heading">Footnotes</h4>
+
+<h3><a name="FOOT5" href="#DOCF5">(5)</a></h3>
+<p>We provide the DHT mainly as a byproduct of some
+internal algorithms. FFTW computes a real input/output DFT of
+<em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
+and then using Rader&rsquo;s prime-DFT algorithm adapted to the DHT.</p>
+</div>
+<hr>
+<div class="header">
+<p>
+Previous: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</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>