mirror of
https://github.com/tildearrow/furnace.git
synced 2024-11-23 21:15:11 +00:00
129 lines
6.3 KiB
HTML
129 lines
6.3 KiB
HTML
<!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 3.3.10, 10 December 2020).
|
|
|
|
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 6.7, http://www.gnu.org/software/texinfo/ -->
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
|
<title>The Discrete Hartley Transform (FFTW 3.3.10)</title>
|
|
|
|
<meta name="description" content="The Discrete Hartley Transform (FFTW 3.3.10)">
|
|
<meta name="keywords" content="The Discrete Hartley Transform (FFTW 3.3.10)">
|
|
<meta name="resource-type" content="document">
|
|
<meta name="distribution" content="global">
|
|
<meta name="Generator" content="makeinfo">
|
|
<link href="index.html" rel="start" title="Top">
|
|
<link href="Concept-Index.html" 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" rel="up" title="More DFTs of Real Data">
|
|
<link href="Other-Important-Topics.html" rel="next" title="Other Important Topics">
|
|
<link href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" rel="prev" title="Real even/odd DFTs (cosine/sine transforms)">
|
|
<style type="text/css">
|
|
<!--
|
|
a.summary-letter {text-decoration: none}
|
|
blockquote.indentedblock {margin-right: 0em}
|
|
div.display {margin-left: 3.2em}
|
|
div.example {margin-left: 3.2em}
|
|
div.lisp {margin-left: 3.2em}
|
|
kbd {font-style: oblique}
|
|
pre.display {font-family: inherit}
|
|
pre.format {font-family: inherit}
|
|
pre.menu-comment {font-family: serif}
|
|
pre.menu-preformatted {font-family: serif}
|
|
span.nolinebreak {white-space: nowrap}
|
|
span.roman {font-family: initial; font-weight: normal}
|
|
span.sansserif {font-family: sans-serif; font-weight: normal}
|
|
ul.no-bullet {list-style: none}
|
|
-->
|
|
</style>
|
|
|
|
|
|
</head>
|
|
|
|
<body lang="en">
|
|
<span id="The-Discrete-Hartley-Transform"></span><div class="header">
|
|
<p>
|
|
Previous: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</a>, Up: <a href="More-DFTs-of-Real-Data.html" 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" title="Index" rel="index">Index</a>]</p>
|
|
</div>
|
|
<hr>
|
|
<span id="The-Discrete-Hartley-Transform-1"></span><h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
|
|
|
|
<p>If you are planning to use the DHT because you’ve heard that it is
|
|
“faster” 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>.
|
|
<span id="index-FFTW_005fDHT"></span>
|
|
<span id="index-discrete-Hartley-transform"></span>
|
|
<span id="index-DHT"></span>
|
|
</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>.
|
|
<span id="index-normalization-4"></span>
|
|
</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’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 id="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>, 264–266 (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</a>.
|
|
</p>
|
|
<div class="footnote">
|
|
<hr>
|
|
<h4 class="footnotes-heading">Footnotes</h4>
|
|
|
|
<h5><a id="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’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" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</a>, Up: <a href="More-DFTs-of-Real-Data.html" 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" title="Index" rel="index">Index</a>]</p>
|
|
</div>
|
|
|
|
|
|
|
|
</body>
|
|
</html>
|