Mercurial > hg > batch-feature-extraction-tool
diff Lib/fftw-3.2.1/doc/html/.svn/text-base/The-Discrete-Hartley-Transform.html.svn-base @ 15:585caf503ef5 tip
Tidy up for ROLI
| author | Geogaddi\David <d.m.ronan@qmul.ac.uk> |
|---|---|
| date | Tue, 17 May 2016 18:50:19 +0100 |
| parents | 636c989477e7 |
| children |
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--- a/Lib/fftw-3.2.1/doc/html/.svn/text-base/The-Discrete-Hartley-Transform.html.svn-base Wed May 04 11:02:59 2016 +0100 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,107 +0,0 @@ -<html lang="en"> -<head> -<title>The Discrete Hartley Transform - FFTW 3.2.1</title> -<meta http-equiv="Content-Type" content="text/html"> -<meta name="description" content="FFTW 3.2.1"> -<meta name="generator" content="makeinfo 4.8"> -<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 3.2.1, 5 February 2009). - -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"> -<p> -<a name="The-Discrete-Hartley-Transform"></a> -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>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-97"></a><a name="index-discrete-Hartley-transform-98"></a><a name="index-DHT-99"></a> -If you are planning to use the DHT because you've heard that it is -“faster” than the DFT (FFT), <strong>stop here</strong>. That story is an old -but enduring misconception that was debunked in 1987: a properly -designed real-input FFT (such as FFTW's) has no more operations in -general than an FHT. Moreover, in FFTW, the DHT is ordinarily -<em>slower</em> than the DFT for composite sizes (see below). - - <p>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-100"></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> - - <p><hr></div> - - </body></html> -