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Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam
date Tue, 19 Nov 2019 14:52:55 +0000
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Chris@82 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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Chris@82 3 <!-- This manual is for FFTW
Chris@82 4 (version 3.3.8, 24 May 2018).
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Chris@82 6 Copyright (C) 2003 Matteo Frigo.
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Chris@82 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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Chris@82 24 <head>
Chris@82 25 <title>FFTW 3.3.8: The Discrete Hartley Transform</title>
Chris@82 26
Chris@82 27 <meta name="description" content="FFTW 3.3.8: The Discrete Hartley Transform">
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Chris@82 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
Chris@82 36 <link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="up" title="More DFTs of Real Data">
Chris@82 37 <link href="Other-Important-Topics.html#Other-Important-Topics" rel="next" title="Other Important Topics">
Chris@82 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)">
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Chris@82 66
Chris@82 67
Chris@82 68 </head>
Chris@82 69
Chris@82 70 <body lang="en">
Chris@82 71 <a name="The-Discrete-Hartley-Transform"></a>
Chris@82 72 <div class="header">
Chris@82 73 <p>
Chris@82 74 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@82 75 </div>
Chris@82 76 <hr>
Chris@82 77 <a name="The-Discrete-Hartley-Transform-1"></a>
Chris@82 78 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
Chris@82 79
Chris@82 80 <p>If you are planning to use the DHT because you&rsquo;ve heard that it is
Chris@82 81 &ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>. The DHT is not
Chris@82 82 faster than the DFT. That story is an old but enduring misconception
Chris@82 83 that was debunked in 1987.
Chris@82 84 </p>
Chris@82 85 <p>The discrete Hartley transform (DHT) is an invertible linear transform
Chris@82 86 closely related to the DFT. In the DFT, one multiplies each input by
Chris@82 87 <em>cos - i * sin</em> (a complex exponential), whereas in the DHT each
Chris@82 88 input is multiplied by simply <em>cos + sin</em>. Thus, the DHT
Chris@82 89 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
Chris@82 90 convenient property of being its own inverse. In FFTW, a DHT (of any
Chris@82 91 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
Chris@82 92 <a name="index-FFTW_005fDHT"></a>
Chris@82 93 <a name="index-discrete-Hartley-transform"></a>
Chris@82 94 <a name="index-DHT"></a>
Chris@82 95 </p>
Chris@82 96 <p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
Chris@82 97 size <code>n</code> followed by another DHT of the same size will result in
Chris@82 98 the original array multiplied by <code>n</code>.
Chris@82 99 <a name="index-normalization-4"></a>
Chris@82 100 </p>
Chris@82 101 <p>The DHT was originally proposed as a more efficient alternative to the
Chris@82 102 DFT for real data, but it was subsequently shown that a specialized DFT
Chris@82 103 (such as FFTW&rsquo;s r2hc or r2c transforms) could be just as fast. In FFTW,
Chris@82 104 the DHT is actually computed by post-processing an r2hc transform, so
Chris@82 105 there is ordinarily no reason to prefer it from a performance
Chris@82 106 perspective.<a name="DOCF5" href="#FOOT5"><sup>5</sup></a>
Chris@82 107 However, we have heard rumors that the DHT might be the most appropriate
Chris@82 108 transform in its own right for certain applications, and we would be
Chris@82 109 very interested to hear from anyone who finds it useful.
Chris@82 110 </p>
Chris@82 111 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
Chris@82 112 multi-dimensional transform, FFTW computes the separable product of 1d
Chris@82 113 DHTs along each dimension. Unfortunately, this is not quite the same
Chris@82 114 thing as a true multi-dimensional DHT; you can compute the latter, if
Chris@82 115 necessary, with at most <code>rank-1</code> post-processing passes
Chris@82 116 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
Chris@82 117 </p>
Chris@82 118 <p>For the precise mathematical definition of the DHT as used by FFTW, see
Chris@82 119 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
Chris@82 120 </p>
Chris@82 121 <div class="footnote">
Chris@82 122 <hr>
Chris@82 123 <h4 class="footnotes-heading">Footnotes</h4>
Chris@82 124
Chris@82 125 <h3><a name="FOOT5" href="#DOCF5">(5)</a></h3>
Chris@82 126 <p>We provide the DHT mainly as a byproduct of some
Chris@82 127 internal algorithms. FFTW computes a real input/output DFT of
Chris@82 128 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
Chris@82 129 and then using Rader&rsquo;s prime-DFT algorithm adapted to the DHT.</p>
Chris@82 130 </div>
Chris@82 131 <hr>
Chris@82 132 <div class="header">
Chris@82 133 <p>
Chris@82 134 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@82 135 </div>
Chris@82 136
Chris@82 137
Chris@82 138
Chris@82 139 </body>
Chris@82 140 </html>