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