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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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Chris@10 3 <title>The Discrete Hartley Transform - FFTW 3.3.3</title>
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Chris@10 48 <a name="The-Discrete-Hartley-Transform"></a>
Chris@10 49 <p>
Chris@10 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>,
Chris@10 51 Up:&nbsp;<a rel="up" accesskey="u" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>
Chris@10 52 <hr>
Chris@10 53 </div>
Chris@10 54
Chris@10 55 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
Chris@10 56
Chris@10 57 <p>If you are planning to use the DHT because you've heard that it is
Chris@10 58 &ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>. The DHT is not
Chris@10 59 faster than the DFT. That story is an old but enduring misconception
Chris@10 60 that was debunked in 1987.
Chris@10 61
Chris@10 62 <p>The discrete Hartley transform (DHT) is an invertible linear transform
Chris@10 63 closely related to the DFT. In the DFT, one multiplies each input by
Chris@10 64 cos - i * sin (a complex exponential), whereas in the DHT each
Chris@10 65 input is multiplied by simply cos + sin. Thus, the DHT
Chris@10 66 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
Chris@10 67 convenient property of being its own inverse. In FFTW, a DHT (of any
Chris@10 68 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
Chris@10 69 <a name="index-FFTW_005fDHT-98"></a><a name="index-discrete-Hartley-transform-99"></a><a name="index-DHT-100"></a>
Chris@10 70 Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
Chris@10 71 size <code>n</code> followed by another DHT of the same size will result in
Chris@10 72 the original array multiplied by <code>n</code>.
Chris@10 73 <a name="index-normalization-101"></a>
Chris@10 74 The DHT was originally proposed as a more efficient alternative to the
Chris@10 75 DFT for real data, but it was subsequently shown that a specialized DFT
Chris@10 76 (such as FFTW's r2hc or r2c transforms) could be just as fast. In FFTW,
Chris@10 77 the DHT is actually computed by post-processing an r2hc transform, so
Chris@10 78 there is ordinarily no reason to prefer it from a performance
Chris@10 79 perspective.<a rel="footnote" href="#fn-1" name="fnd-1"><sup>1</sup></a>
Chris@10 80 However, we have heard rumors that the DHT might be the most appropriate
Chris@10 81 transform in its own right for certain applications, and we would be
Chris@10 82 very interested to hear from anyone who finds it useful.
Chris@10 83
Chris@10 84 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
Chris@10 85 multi-dimensional transform, FFTW computes the separable product of 1d
Chris@10 86 DHTs along each dimension. Unfortunately, this is not quite the same
Chris@10 87 thing as a true multi-dimensional DHT; you can compute the latter, if
Chris@10 88 necessary, with at most <code>rank-1</code> post-processing passes
Chris@10 89 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
Chris@10 90
Chris@10 91 <p>For the precise mathematical definition of the DHT as used by FFTW, see
Chris@10 92 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
Chris@10 93
Chris@10 94 <div class="footnote">
Chris@10 95 <hr>
Chris@10 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
Chris@10 97 internal algorithms. FFTW computes a real input/output DFT of
Chris@10 98 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
Chris@10 99 and then using Rader's prime-DFT algorithm adapted to the DHT.</p>
Chris@10 100
Chris@10 101 <hr></div>
Chris@10 102
Chris@10 103 </body></html>
Chris@10 104