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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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cannam@127 3 <!-- This manual is for FFTW
cannam@127 4 (version 3.3.5, 30 July 2016).
cannam@127 5
cannam@127 6 Copyright (C) 2003 Matteo Frigo.
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cannam@127 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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cannam@127 24 <head>
cannam@127 25 <title>FFTW 3.3.5: The Discrete Hartley Transform</title>
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cannam@127 70
cannam@127 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
cannam@127 72 <a name="The-Discrete-Hartley-Transform"></a>
cannam@127 73 <div class="header">
cannam@127 74 <p>
cannam@127 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>
cannam@127 76 </div>
cannam@127 77 <hr>
cannam@127 78 <a name="The-Discrete-Hartley-Transform-1"></a>
cannam@127 79 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
cannam@127 80
cannam@127 81 <p>If you are planning to use the DHT because you&rsquo;ve heard that it is
cannam@127 82 &ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>. The DHT is not
cannam@127 83 faster than the DFT. That story is an old but enduring misconception
cannam@127 84 that was debunked in 1987.
cannam@127 85 </p>
cannam@127 86 <p>The discrete Hartley transform (DHT) is an invertible linear transform
cannam@127 87 closely related to the DFT. In the DFT, one multiplies each input by
cannam@127 88 <em>cos - i * sin</em> (a complex exponential), whereas in the DHT each
cannam@127 89 input is multiplied by simply <em>cos + sin</em>. Thus, the DHT
cannam@127 90 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
cannam@127 91 convenient property of being its own inverse. In FFTW, a DHT (of any
cannam@127 92 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
cannam@127 93 <a name="index-FFTW_005fDHT"></a>
cannam@127 94 <a name="index-discrete-Hartley-transform"></a>
cannam@127 95 <a name="index-DHT"></a>
cannam@127 96 </p>
cannam@127 97 <p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
cannam@127 98 size <code>n</code> followed by another DHT of the same size will result in
cannam@127 99 the original array multiplied by <code>n</code>.
cannam@127 100 <a name="index-normalization-4"></a>
cannam@127 101 </p>
cannam@127 102 <p>The DHT was originally proposed as a more efficient alternative to the
cannam@127 103 DFT for real data, but it was subsequently shown that a specialized DFT
cannam@127 104 (such as FFTW&rsquo;s r2hc or r2c transforms) could be just as fast. In FFTW,
cannam@127 105 the DHT is actually computed by post-processing an r2hc transform, so
cannam@127 106 there is ordinarily no reason to prefer it from a performance
cannam@127 107 perspective.<a name="DOCF5" href="#FOOT5"><sup>5</sup></a>
cannam@127 108 However, we have heard rumors that the DHT might be the most appropriate
cannam@127 109 transform in its own right for certain applications, and we would be
cannam@127 110 very interested to hear from anyone who finds it useful.
cannam@127 111 </p>
cannam@127 112 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
cannam@127 113 multi-dimensional transform, FFTW computes the separable product of 1d
cannam@127 114 DHTs along each dimension. Unfortunately, this is not quite the same
cannam@127 115 thing as a true multi-dimensional DHT; you can compute the latter, if
cannam@127 116 necessary, with at most <code>rank-1</code> post-processing passes
cannam@127 117 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
cannam@127 118 </p>
cannam@127 119 <p>For the precise mathematical definition of the DHT as used by FFTW, see
cannam@127 120 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
cannam@127 121 </p>
cannam@127 122 <div class="footnote">
cannam@127 123 <hr>
cannam@127 124 <h4 class="footnotes-heading">Footnotes</h4>
cannam@127 125
cannam@127 126 <h3><a name="FOOT5" href="#DOCF5">(5)</a></h3>
cannam@127 127 <p>We provide the DHT mainly as a byproduct of some
cannam@127 128 internal algorithms. FFTW computes a real input/output DFT of
cannam@127 129 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
cannam@127 130 and then using Rader&rsquo;s prime-DFT algorithm adapted to the DHT.</p>
cannam@127 131 </div>
cannam@127 132 <hr>
cannam@127 133 <div class="header">
cannam@127 134 <p>
cannam@127 135 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>
cannam@127 136 </div>
cannam@127 137
cannam@127 138
cannam@127 139
cannam@127 140 </body>
cannam@127 141 </html>