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Vectorised everything and made use of unique_ptr so there should be no more memory leaks. Hurrah for RAII
author Geogaddi\David <d.m.ronan@qmul.ac.uk>
date Wed, 12 Aug 2015 22:25:06 +0100
parents 25bf17994ef1
children
rev   line source
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d@0 3 <title>The Discrete Hartley Transform - FFTW 3.2.1</title>
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d@0 49 <a name="The-Discrete-Hartley-Transform"></a>
d@0 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>,
d@0 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>
d@0 52 <hr>
d@0 53 </div>
d@0 54
d@0 55 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
d@0 56
d@0 57 <p>The discrete Hartley transform (DHT) is an invertible linear transform
d@0 58 closely related to the DFT. In the DFT, one multiplies each input by
d@0 59 cos - i * sin (a complex exponential), whereas in the DHT each
d@0 60 input is multiplied by simply cos + sin. Thus, the DHT
d@0 61 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
d@0 62 convenient property of being its own inverse. In FFTW, a DHT (of any
d@0 63 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
d@0 64 <a name="index-FFTW_005fDHT-97"></a><a name="index-discrete-Hartley-transform-98"></a><a name="index-DHT-99"></a>
d@0 65 If you are planning to use the DHT because you've heard that it is
d@0 66 &ldquo;faster&rdquo; than the DFT (FFT), <strong>stop here</strong>. That story is an old
d@0 67 but enduring misconception that was debunked in 1987: a properly
d@0 68 designed real-input FFT (such as FFTW's) has no more operations in
d@0 69 general than an FHT. Moreover, in FFTW, the DHT is ordinarily
d@0 70 <em>slower</em> than the DFT for composite sizes (see below).
d@0 71
d@0 72 <p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
d@0 73 size <code>n</code> followed by another DHT of the same size will result in
d@0 74 the original array multiplied by <code>n</code>.
d@0 75 <a name="index-normalization-100"></a>
d@0 76 The DHT was originally proposed as a more efficient alternative to the
d@0 77 DFT for real data, but it was subsequently shown that a specialized DFT
d@0 78 (such as FFTW's r2hc or r2c transforms) could be just as fast. In FFTW,
d@0 79 the DHT is actually computed by post-processing an r2hc transform, so
d@0 80 there is ordinarily no reason to prefer it from a performance
d@0 81 perspective.<a rel="footnote" href="#fn-1" name="fnd-1"><sup>1</sup></a>
d@0 82 However, we have heard rumors that the DHT might be the most appropriate
d@0 83 transform in its own right for certain applications, and we would be
d@0 84 very interested to hear from anyone who finds it useful.
d@0 85
d@0 86 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
d@0 87 multi-dimensional transform, FFTW computes the separable product of 1d
d@0 88 DHTs along each dimension. Unfortunately, this is not quite the same
d@0 89 thing as a true multi-dimensional DHT; you can compute the latter, if
d@0 90 necessary, with at most <code>rank-1</code> post-processing passes
d@0 91 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264&ndash;266 (1987)].
d@0 92
d@0 93 <p>For the precise mathematical definition of the DHT as used by FFTW, see
d@0 94 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
d@0 95
d@0 96 <!-- ************************************************************ -->
d@0 97 <div class="footnote">
d@0 98 <hr>
d@0 99 <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
d@0 100 internal algorithms. FFTW computes a real input/output DFT of
d@0 101 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
d@0 102 and then using Rader's prime-DFT algorithm adapted to the DHT.</p>
d@0 103
d@0 104 <p><hr></div>
d@0 105
d@0 106 </body></html>
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