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diff src/fftw-3.3.3/doc/html/The-Discrete-Hartley-Transform.html @ 95:89f5e221ed7b
Add FFTW3
| author | Chris Cannam <cannam@all-day-breakfast.com> |
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| date | Wed, 20 Mar 2013 15:35:50 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.3/doc/html/The-Discrete-Hartley-Transform.html Wed Mar 20 15:35:50 2013 +0000 @@ -0,0 +1,104 @@ +<html lang="en"> +<head> +<title>The Discrete Hartley Transform - FFTW 3.3.3</title> +<meta http-equiv="Content-Type" content="text/html"> +<meta name="description" content="FFTW 3.3.3"> +<meta name="generator" content="makeinfo 4.13"> +<link title="Top" rel="start" href="index.html#Top"> +<link rel="up" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" title="More DFTs of Real Data"> +<link rel="prev" href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" title="Real even/odd DFTs (cosine/sine transforms)"> +<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> +<!-- +This manual is for FFTW +(version 3.3.3, 25 November 2012). + +Copyright (C) 2003 Matteo Frigo. + +Copyright (C) 2003 Massachusetts Institute of Technology. + + Permission is granted to make and distribute verbatim copies of + this manual provided the copyright notice and this permission + notice are preserved on all copies. + + Permission is granted to copy and distribute modified versions of + this manual under the conditions for verbatim copying, provided + that the entire resulting derived work is distributed under the + terms of a permission notice identical to this one. + + Permission is granted to copy and distribute translations of this + manual into another language, under the above conditions for + modified versions, except that this permission notice may be + stated in a translation approved by the Free Software Foundation. + --> +<meta http-equiv="Content-Style-Type" content="text/css"> +<style type="text/css"><!-- + pre.display { font-family:inherit } + pre.format { font-family:inherit } + pre.smalldisplay { font-family:inherit; font-size:smaller } + pre.smallformat { font-family:inherit; font-size:smaller } + pre.smallexample { font-size:smaller } + pre.smalllisp { font-size:smaller } + span.sc { font-variant:small-caps } + span.roman { font-family:serif; font-weight:normal; } + span.sansserif { font-family:sans-serif; font-weight:normal; } +--></style> +</head> +<body> +<div class="node"> +<a name="The-Discrete-Hartley-Transform"></a> +<p> +Previous: <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>, +Up: <a rel="up" accesskey="u" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a> +<hr> +</div> + +<h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4> + +<p>If you are planning to use the DHT because you've heard that it is +“faster” than the DFT (FFT), <strong>stop here</strong>. The DHT is not +faster than the DFT. That story is an old but enduring misconception +that was debunked in 1987. + + <p>The discrete Hartley transform (DHT) is an invertible linear transform +closely related to the DFT. In the DFT, one multiplies each input by +cos - i * sin (a complex exponential), whereas in the DHT each +input is multiplied by simply cos + sin. Thus, the DHT +transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the +convenient property of being its own inverse. In FFTW, a DHT (of any +positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>. +<a name="index-FFTW_005fDHT-98"></a><a name="index-discrete-Hartley-transform-99"></a><a name="index-DHT-100"></a> +Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of +size <code>n</code> followed by another DHT of the same size will result in +the original array multiplied by <code>n</code>. +<a name="index-normalization-101"></a> +The DHT was originally proposed as a more efficient alternative to the +DFT for real data, but it was subsequently shown that a specialized DFT +(such as FFTW's r2hc or r2c transforms) could be just as fast. In FFTW, +the DHT is actually computed by post-processing an r2hc transform, so +there is ordinarily no reason to prefer it from a performance +perspective.<a rel="footnote" href="#fn-1" name="fnd-1"><sup>1</sup></a> +However, we have heard rumors that the DHT might be the most appropriate +transform in its own right for certain applications, and we would be +very interested to hear from anyone who finds it useful. + + <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a +multi-dimensional transform, FFTW computes the separable product of 1d +DHTs along each dimension. Unfortunately, this is not quite the same +thing as a true multi-dimensional DHT; you can compute the latter, if +necessary, with at most <code>rank-1</code> post-processing passes +[see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264–266 (1987)]. + + <p>For the precise mathematical definition of the DHT as used by FFTW, see +<a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>. + + <div class="footnote"> +<hr> +<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 +internal algorithms. FFTW computes a real input/output DFT of +<em>prime</em> size by re-expressing it as a DHT plus post/pre-processing +and then using Rader's prime-DFT algorithm adapted to the DHT.</p> + + <hr></div> + + </body></html> +