130 lines
6.3 KiB
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130 lines
6.3 KiB
HTML
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<!-- This manual is for FFTW
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(version 3.3.10, 10 December 2020).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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Permission is granted to make and distribute verbatim copies of this
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manual provided the copyright notice and this permission notice are
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Permission is granted to copy and distribute modified versions of this
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Permission is granted to copy and distribute translations of this manual
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<title>The Discrete Hartley Transform (FFTW 3.3.10)</title>
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<link href="Concept-Index.html" rel="index" title="Concept Index">
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<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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<link href="More-DFTs-of-Real-Data.html" rel="up" title="More DFTs of Real Data">
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<span id="The-Discrete-Hartley-Transform"></span><div class="header">
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<p>
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Previous: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</a>, Up: <a href="More-DFTs-of-Real-Data.html" accesskey="u" rel="up">More DFTs of Real Data</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<span id="The-Discrete-Hartley-Transform-1"></span><h4 class="subsection">2.5.3 The Discrete Hartley Transform</h4>
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<p>If you are planning to use the DHT because you’ve heard that it is
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“faster” than the DFT (FFT), <strong>stop here</strong>. The DHT is not
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faster than the DFT. That story is an old but enduring misconception
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that was debunked in 1987.
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</p>
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<p>The discrete Hartley transform (DHT) is an invertible linear transform
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closely related to the DFT. In the DFT, one multiplies each input by
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<em>cos - i * sin</em> (a complex exponential), whereas in the DHT each
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input is multiplied by simply <em>cos + sin</em>. Thus, the DHT
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transforms <code>n</code> real numbers to <code>n</code> real numbers, and has the
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convenient property of being its own inverse. In FFTW, a DHT (of any
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positive <code>n</code>) can be specified by an r2r kind of <code>FFTW_DHT</code>.
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<span id="index-FFTW_005fDHT"></span>
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<span id="index-discrete-Hartley-transform"></span>
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<span id="index-DHT"></span>
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</p>
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<p>Like the DFT, in FFTW the DHT is unnormalized, so computing a DHT of
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size <code>n</code> followed by another DHT of the same size will result in
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the original array multiplied by <code>n</code>.
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<span id="index-normalization-4"></span>
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</p>
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<p>The DHT was originally proposed as a more efficient alternative to the
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DFT for real data, but it was subsequently shown that a specialized DFT
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(such as FFTW’s r2hc or r2c transforms) could be just as fast. In FFTW,
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the DHT is actually computed by post-processing an r2hc transform, so
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there is ordinarily no reason to prefer it from a performance
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perspective.<a id="DOCF5" href="#FOOT5"><sup>5</sup></a>
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However, we have heard rumors that the DHT might be the most appropriate
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transform in its own right for certain applications, and we would be
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very interested to hear from anyone who finds it useful.
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</p>
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<p>If <code>FFTW_DHT</code> is specified for multiple dimensions of a
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multi-dimensional transform, FFTW computes the separable product of 1d
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DHTs along each dimension. Unfortunately, this is not quite the same
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thing as a true multi-dimensional DHT; you can compute the latter, if
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necessary, with at most <code>rank-1</code> post-processing passes
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[see e.g. H. Hao and R. N. Bracewell, <i>Proc. IEEE</i> <b>75</b>, 264–266 (1987)].
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</p>
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<p>For the precise mathematical definition of the DHT as used by FFTW, see
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<a href="What-FFTW-Really-Computes.html">What FFTW Really Computes</a>.
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</p>
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<div class="footnote">
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<hr>
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<h4 class="footnotes-heading">Footnotes</h4>
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<h5><a id="FOOT5" href="#DOCF5">(5)</a></h3>
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<p>We provide the DHT mainly as a byproduct of some
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internal algorithms. FFTW computes a real input/output DFT of
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<em>prime</em> size by re-expressing it as a DHT plus post/pre-processing
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and then using Rader’s prime-DFT algorithm adapted to the DHT.</p>
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</div>
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<hr>
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<div class="header">
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<p>
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Previous: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="p" rel="prev">Real even/odd DFTs (cosine/sine transforms)</a>, Up: <a href="More-DFTs-of-Real-Data.html" accesskey="u" rel="up">More DFTs of Real Data</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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