Mercurial > hg > sv-dependency-builds

annotate src/fftw-3.3.5/doc/html/The-Discrete-Hartley-Transform.html @ 42:2cd0e3b3e1fd

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Current fftw source
author Chris Cannam
date Tue, 18 Oct 2016 13:40:26 +0100
parents
children
rev   line source
Chris@42 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
Chris@42 2 <html>
Chris@42 3 <!-- This manual is for FFTW
Chris@42 4 (version 3.3.5, 30 July 2016).
Chris@42 5
Chris@42 6 Copyright (C) 2003 Matteo Frigo.
Chris@42 7
Chris@42 8 Copyright (C) 2003 Massachusetts Institute of Technology.
Chris@42 9
Chris@42 10 Permission is granted to make and distribute verbatim copies of this
Chris@42 11 manual provided the copyright notice and this permission notice are
Chris@42 12 preserved on all copies.
Chris@42 13
Chris@42 14 Permission is granted to copy and distribute modified versions of this
Chris@42 15 manual under the conditions for verbatim copying, provided that the
Chris@42 16 entire resulting derived work is distributed under the terms of a
Chris@42 17 permission notice identical to this one.
Chris@42 18
Chris@42 19 Permission is granted to copy and distribute translations of this manual
Chris@42 20 into another language, under the above conditions for modified versions,
Chris@42 21 except that this permission notice may be stated in a translation
Chris@42 22 approved by the Free Software Foundation. -->
Chris@42 23 <!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ -->
Chris@42 24 <head>
Chris@42 25 <title>FFTW 3.3.5: The Discrete Hartley Transform</title>
Chris@42 26
Chris@42 27 <meta name="description" content="FFTW 3.3.5: The Discrete Hartley Transform">
Chris@42 28 <meta name="keywords" content="FFTW 3.3.5: The Discrete Hartley Transform">
Chris@42 29 <meta name="resource-type" content="document">
Chris@42 30 <meta name="distribution" content="global">
Chris@42 31 <meta name="Generator" content="makeinfo">
Chris@42 32 <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
Chris@42 33 <link href="index.html#Top" rel="start" title="Top">
Chris@42 34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
Chris@42 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
Chris@42 36 <link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="up" title="More DFTs of Real Data">
Chris@42 37 <link href="Other-Important-Topics.html#Other-Important-Topics" rel="next" title="Other Important Topics">
Chris@42 38 <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)">
Chris@42 39 <style type="text/css">
Chris@42 40 <!--
Chris@42 41 a.summary-letter {text-decoration: none}
Chris@42 42 blockquote.smallquotation {font-size: smaller}
Chris@42 43 div.display {margin-left: 3.2em}
Chris@42 44 div.example {margin-left: 3.2em}
Chris@42 45 div.indentedblock {margin-left: 3.2em}
Chris@42 46 div.lisp {margin-left: 3.2em}
Chris@42 47 div.smalldisplay {margin-left: 3.2em}
Chris@42 48 div.smallexample {margin-left: 3.2em}
Chris@42 49 div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
Chris@42 50 div.smalllisp {margin-left: 3.2em}
Chris@42 51 kbd {font-style:oblique}
Chris@42 52 pre.display {font-family: inherit}
Chris@42 53 pre.format {font-family: inherit}
Chris@42 54 pre.menu-comment {font-family: serif}
Chris@42 55 pre.menu-preformatted {font-family: serif}
Chris@42 56 pre.smalldisplay {font-family: inherit; font-size: smaller}
Chris@42 57 pre.smallexample {font-size: smaller}
Chris@42 58 pre.smallformat {font-family: inherit; font-size: smaller}
Chris@42 59 pre.smalllisp {font-size: smaller}
Chris@42 60 span.nocodebreak {white-space:nowrap}
Chris@42 61 span.nolinebreak {white-space:nowrap}
Chris@42 62 span.roman {font-family:serif; font-weight:normal}
Chris@42 63 span.sansserif {font-family:sans-serif; font-weight:normal}
Chris@42 64 ul.no-bullet {list-style: none}
Chris@42 65 -->
Chris@42 66 </style>
Chris@42 67
Chris@42 68
Chris@42 69 </head>
Chris@42 70
Chris@42 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
Chris@42 72 <a name="The-Discrete-Hartley-Transform"></a>
Chris@42 73 <div class="header">
Chris@42 74 <p>
Chris@42 75 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>
Chris@42 76 </div>
Chris@42 77 <hr>
Chris@42 78 <a name="The-Discrete-Hartley-Transform-1"></a>
Chris@42 79 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
Chris@42 80
Chris@42 81 <p>If you are planning to use the DHT because you&rsquo;ve heard that it is
Chris@42 82 &ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>. The DHT is not
Chris@42 83 faster than the DFT. That story is an old but enduring misconception
Chris@42 84 that was debunked in 1987.
Chris@42 85 </p>
Chris@42 86 <p>The discrete Hartley transform (DHT) is an invertible linear transform
Chris@42 87 closely related to the DFT. In the DFT, one multiplies each input by
Chris@42 88 <em>cos - i * sin</em> (a complex exponential), whereas in the DHT each
Chris@42 89 input is multiplied by simply <em>cos + sin</em>. Thus, the DHT
Chris@42 90 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
Chris@42 91 convenient property of being its own inverse. In FFTW, a DHT (of any
Chris@42 92 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
Chris@42 93 <a name="index-FFTW_005fDHT"></a>
Chris@42 94 <a name="index-discrete-Hartley-transform"></a>
Chris@42 95 <a name="index-DHT"></a>
Chris@42 96 </p>
Chris@42 97 <p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
Chris@42 98 size <code>n</code> followed by another DHT of the same size will result in
Chris@42 99 the original array multiplied by <code>n</code>.
Chris@42 100 <a name="index-normalization-4"></a>
Chris@42 101 </p>
Chris@42 102 <p>The DHT was originally proposed as a more efficient alternative to the
Chris@42 103 DFT for real data, but it was subsequently shown that a specialized DFT
Chris@42 104 (such as FFTW&rsquo;s r2hc or r2c transforms) could be just as fast. In FFTW,
Chris@42 105 the DHT is actually computed by post-processing an r2hc transform, so
Chris@42 106 there is ordinarily no reason to prefer it from a performance
Chris@42 107 perspective.<a name="DOCF5" href="#FOOT5"><sup>5</sup></a>
Chris@42 108 However, we have heard rumors that the DHT might be the most appropriate
Chris@42 109 transform in its own right for certain applications, and we would be
Chris@42 110 very interested to hear from anyone who finds it useful.
Chris@42 111 </p>
Chris@42 112 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
Chris@42 113 multi-dimensional transform, FFTW computes the separable product of 1d
Chris@42 114 DHTs along each dimension. Unfortunately, this is not quite the same
Chris@42 115 thing as a true multi-dimensional DHT; you can compute the latter, if
Chris@42 116 necessary, with at most <code>rank-1</code> post-processing passes
Chris@42 117 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
Chris@42 118 </p>
Chris@42 119 <p>For the precise mathematical definition of the DHT as used by FFTW, see
Chris@42 120 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
Chris@42 121 </p>
Chris@42 122 <div class="footnote">
Chris@42 123 <hr>
Chris@42 124 <h4 class="footnotes-heading">Footnotes</h4>
Chris@42 125
Chris@42 126 <h3><a name="FOOT5" href="#DOCF5">(5)</a></h3>
Chris@42 127 <p>We provide the DHT mainly as a byproduct of some
Chris@42 128 internal algorithms. FFTW computes a real input/output DFT of
Chris@42 129 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
Chris@42 130 and then using Rader&rsquo;s prime-DFT algorithm adapted to the DHT.</p>
Chris@42 131 </div>
Chris@42 132 <hr>
Chris@42 133 <div class="header">
Chris@42 134 <p>
Chris@42 135 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>
Chris@42 136 </div>
Chris@42 137
Chris@42 138
Chris@42 139
Chris@42 140 </body>
Chris@42 141 </html>