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2 <head>
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3 <title>The Discrete Hartley Transform - FFTW 3.2.1</title>
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4 <meta http-equiv="Content-Type" content="text/html">
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5 <meta name="description" content="FFTW 3.2.1">
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7 <link title="Top" rel="start" href="index.html#Top">
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8 <link rel="up" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" title="More DFTs of Real Data">
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9 <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)">
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10 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
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11 <!--
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12 This manual is for FFTW
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13 (version 3.2.1, 5 February 2009).
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14
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15 Copyright (C) 2003 Matteo Frigo.
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16
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17 Copyright (C) 2003 Massachusetts Institute of Technology.
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18
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19 Permission is granted to make and distribute verbatim copies of
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20 this manual provided the copyright notice and this permission
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21 notice are preserved on all copies.
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22
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23 Permission is granted to copy and distribute modified versions of
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24 this manual under the conditions for verbatim copying, provided
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25 that the entire resulting derived work is distributed under the
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26 terms of a permission notice identical to this one.
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27
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28 Permission is granted to copy and distribute translations of this
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29 manual into another language, under the above conditions for
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30 modified versions, except that this permission notice may be
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31 stated in a translation approved by the Free Software Foundation.
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32 -->
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44 --></style>
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45 </head>
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46 <body>
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47 <div class="node">
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48 <p>
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49 <a name="The-Discrete-Hartley-Transform"></a>
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50 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>,
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51 Up: <a rel="up" accesskey="u" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>
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52 <hr>
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53 </div>
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54
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55 <h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
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56
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57 <p>The discrete Hartley transform (DHT) is an invertible linear transform
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58 closely related to the DFT. In the DFT, one multiplies each input by
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59 cos - i * sin (a complex exponential), whereas in the DHT each
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60 input is multiplied by simply cos + sin. Thus, the DHT
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61 transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
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62 convenient property of being its own inverse. In FFTW, a DHT (of any
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63 positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
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64 <a name="index-FFTW_005fDHT-97"></a><a name="index-discrete-Hartley-transform-98"></a><a name="index-DHT-99"></a>
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65 If you are planning to use the DHT because you've heard that it is
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66 “faster” than the DFT (FFT), <strong>stop here</strong>. That story is an old
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67 but enduring misconception that was debunked in 1987: a properly
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68 designed real-input FFT (such as FFTW's) has no more operations in
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69 general than an FHT. Moreover, in FFTW, the DHT is ordinarily
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70 <em>slower</em> than the DFT for composite sizes (see below).
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71
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72 <p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
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73 size <code>n</code> followed by another DHT of the same size will result in
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74 the original array multiplied by <code>n</code>.
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75 <a name="index-normalization-100"></a>
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76 The DHT was originally proposed as a more efficient alternative to the
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77 DFT for real data, but it was subsequently shown that a specialized DFT
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78 (such as FFTW's r2hc or r2c transforms) could be just as fast. In FFTW,
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79 the DHT is actually computed by post-processing an r2hc transform, so
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80 there is ordinarily no reason to prefer it from a performance
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81 perspective.<a rel="footnote" href="#fn-1" name="fnd-1"><sup>1</sup></a>
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82 However, we have heard rumors that the DHT might be the most appropriate
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83 transform in its own right for certain applications, and we would be
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84 very interested to hear from anyone who finds it useful.
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85
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86 <p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
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87 multi-dimensional transform, FFTW computes the separable product of 1d
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88 DHTs along each dimension. Unfortunately, this is not quite the same
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89 thing as a true multi-dimensional DHT; you can compute the latter, if
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90 necessary, with at most <code>rank-1</code> post-processing passes
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91 [see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264–266 (1987)].
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92
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93 <p>For the precise mathematical definition of the DHT as used by FFTW, see
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94 <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>.
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95
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96 <!-- ************************************************************ -->
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97 <div class="footnote">
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98 <hr>
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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
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100 internal algorithms. FFTW computes a real input/output DFT of
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101 <em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
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102 and then using Rader's prime-DFT algorithm adapted to the DHT.</p>
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103
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104 <p><hr></div>
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105
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106 </body></html>
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107
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