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date Wed, 20 Mar 2013 15:35:50 +0000
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3 <title>The Discrete Hartley Transform - FFTW 3.3.3</title>
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48 <a name="The-Discrete-Hartley-Transform"></a>
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50 Previous:&nbsp;<a rel="previous" accesskey="p" href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029">Real even/odd DFTs (cosine/sine transforms)</a>,
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54
55 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
56
57 <p>If you are planning to use the DHT because you've heard that it is
58 &ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>. The DHT is not
59 faster than the DFT. That story is an old but enduring misconception
60 that was debunked in 1987.
61
62 <p>The discrete Hartley transform (DHT) is an invertible linear transform
63 closely related to the DFT. In the DFT, one multiplies each input by
64 cos - i * sin (a complex exponential), whereas in the DHT each
65 input is multiplied by simply cos + sin. Thus, the DHT
66 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
67 convenient property of being its own inverse. In FFTW, a DHT (of any
68 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
69 <a name="index-FFTW_005fDHT-98"></a><a name="index-discrete-Hartley-transform-99"></a><a name="index-DHT-100"></a>
70 Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
71 size <code>n</code> followed by another DHT of the same size will result in
72 the original array multiplied by <code>n</code>.
73 <a name="index-normalization-101"></a>
74 The DHT was originally proposed as a more efficient alternative to the
75 DFT for real data, but it was subsequently shown that a specialized DFT
76 (such as FFTW's r2hc or r2c transforms) could be just as fast. In FFTW,
77 the DHT is actually computed by post-processing an r2hc transform, so
78 there is ordinarily no reason to prefer it from a performance
79 perspective.<a rel="footnote" href="#fn-1" name="fnd-1"><sup>1</sup></a>
80 However, we have heard rumors that the DHT might be the most appropriate
81 transform in its own right for certain applications, and we would be
82 very interested to hear from anyone who finds it useful.
83
84 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
85 multi-dimensional transform, FFTW computes the separable product of 1d
86 DHTs along each dimension. Unfortunately, this is not quite the same
87 thing as a true multi-dimensional DHT; you can compute the latter, if
88 necessary, with at most <code>rank-1</code> post-processing passes
89 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
90
91 <p>For the precise mathematical definition of the DHT as used by FFTW, see
92 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
93
94 <div class="footnote">
95 <hr>
96 <h4>Footnotes</h4><p class="footnote"><small>[<a name="fn-1" href="#fnd-1">1</a>]</small> We provide the DHT mainly as a byproduct of some
97 internal algorithms. FFTW computes a real input/output DFT of
98 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
99 and then using Rader's prime-DFT algorithm adapted to the DHT.</p>
100
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102
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104