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author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
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cannam@167 3 <!-- This manual is for FFTW
cannam@167 4 (version 3.3.8, 24 May 2018).
cannam@167 5
cannam@167 6 Copyright (C) 2003 Matteo Frigo.
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cannam@167 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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cannam@167 24 <head>
cannam@167 25 <title>FFTW 3.3.8: The Discrete Hartley Transform</title>
cannam@167 26
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cannam@167 68 </head>
cannam@167 69
cannam@167 70 <body lang="en">
cannam@167 71 <a name="The-Discrete-Hartley-Transform"></a>
cannam@167 72 <div class="header">
cannam@167 73 <p>
cannam@167 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>
cannam@167 75 </div>
cannam@167 76 <hr>
cannam@167 77 <a name="The-Discrete-Hartley-Transform-1"></a>
cannam@167 78 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
cannam@167 79
cannam@167 80 <p>If you are planning to use the DHT because you&rsquo;ve heard that it is
cannam@167 81 &ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>. The DHT is not
cannam@167 82 faster than the DFT. That story is an old but enduring misconception
cannam@167 83 that was debunked in 1987.
cannam@167 84 </p>
cannam@167 85 <p>The discrete Hartley transform (DHT) is an invertible linear transform
cannam@167 86 closely related to the DFT. In the DFT, one multiplies each input by
cannam@167 87 <em>cos - i * sin</em> (a complex exponential), whereas in the DHT each
cannam@167 88 input is multiplied by simply <em>cos + sin</em>. Thus, the DHT
cannam@167 89 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
cannam@167 90 convenient property of being its own inverse. In FFTW, a DHT (of any
cannam@167 91 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
cannam@167 92 <a name="index-FFTW_005fDHT"></a>
cannam@167 93 <a name="index-discrete-Hartley-transform"></a>
cannam@167 94 <a name="index-DHT"></a>
cannam@167 95 </p>
cannam@167 96 <p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
cannam@167 97 size <code>n</code> followed by another DHT of the same size will result in
cannam@167 98 the original array multiplied by <code>n</code>.
cannam@167 99 <a name="index-normalization-4"></a>
cannam@167 100 </p>
cannam@167 101 <p>The DHT was originally proposed as a more efficient alternative to the
cannam@167 102 DFT for real data, but it was subsequently shown that a specialized DFT
cannam@167 103 (such as FFTW&rsquo;s r2hc or r2c transforms) could be just as fast. In FFTW,
cannam@167 104 the DHT is actually computed by post-processing an r2hc transform, so
cannam@167 105 there is ordinarily no reason to prefer it from a performance
cannam@167 106 perspective.<a name="DOCF5" href="#FOOT5"><sup>5</sup></a>
cannam@167 107 However, we have heard rumors that the DHT might be the most appropriate
cannam@167 108 transform in its own right for certain applications, and we would be
cannam@167 109 very interested to hear from anyone who finds it useful.
cannam@167 110 </p>
cannam@167 111 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
cannam@167 112 multi-dimensional transform, FFTW computes the separable product of 1d
cannam@167 113 DHTs along each dimension. Unfortunately, this is not quite the same
cannam@167 114 thing as a true multi-dimensional DHT; you can compute the latter, if
cannam@167 115 necessary, with at most <code>rank-1</code> post-processing passes
cannam@167 116 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
cannam@167 117 </p>
cannam@167 118 <p>For the precise mathematical definition of the DHT as used by FFTW, see
cannam@167 119 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
cannam@167 120 </p>
cannam@167 121 <div class="footnote">
cannam@167 122 <hr>
cannam@167 123 <h4 class="footnotes-heading">Footnotes</h4>
cannam@167 124
cannam@167 125 <h3><a name="FOOT5" href="#DOCF5">(5)</a></h3>
cannam@167 126 <p>We provide the DHT mainly as a byproduct of some
cannam@167 127 internal algorithms. FFTW computes a real input/output DFT of
cannam@167 128 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
cannam@167 129 and then using Rader&rsquo;s prime-DFT algorithm adapted to the DHT.</p>
cannam@167 130 </div>
cannam@167 131 <hr>
cannam@167 132 <div class="header">
cannam@167 133 <p>
cannam@167 134 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>
cannam@167 135 </div>
cannam@167 136
cannam@167 137
cannam@167 138
cannam@167 139 </body>
cannam@167 140 </html>