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6.9 KiB
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<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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preserved on all copies.
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Permission is granted to copy and distribute modified versions of this
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manual under the conditions for verbatim copying, provided that the
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entire resulting derived work is distributed under the terms of a
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permission notice identical to this one.
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Permission is granted to copy and distribute translations of this manual
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>The Halfcomplex-format DFT (FFTW 3.3.10)</title>
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<meta name="description" content="The Halfcomplex-format DFT (FFTW 3.3.10)">
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<link href="index.html" rel="start" title="Top">
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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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<link href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" rel="next" title="Real even/odd DFTs (cosine/sine transforms)">
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</head>
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<body lang="en">
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<span id="The-Halfcomplex_002dformat-DFT"></span><div class="header">
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<p>
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Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html" accesskey="p" rel="prev">More DFTs of Real Data</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-Halfcomplex_002dformat-DFT-1"></span><h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
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<p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
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<span id="index-FFTW_005fR2HC"></span>
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<span id="index-r2c-1"></span>
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<span id="index-r2hc"></span>
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(see <a href="One_002dDimensional-DFTs-of-Real-Data.html">One-Dimensional DFTs of Real Data</a>) but with “halfcomplex”
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format output, and may sometimes be faster and/or more convenient than
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the latter.
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<span id="index-halfcomplex-format-1"></span>
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The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
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<span id="index-FFTW_005fHC2R"></span>
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<span id="index-hc2r"></span>
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This consists of the non-redundant half of the complex output for a 1d
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real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
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numbers (<code>double</code>) in the format:
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</p>
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<p align=center>
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r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>n/2</sub>, i<sub>(n+1)/2-1</sub>, ..., i<sub>2</sub>, i<sub>1</sub>
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</p>
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<p>Here,
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r<sub>k</sub>
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is the real part of the <em>k</em>th output, and
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i<sub>k</sub>
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is the imaginary part. (Division by 2 is rounded down.) For a
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halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its
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real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
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the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
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only if <code>n</code> is even)—in these two cases, the imaginary part is
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zero due to symmetries of the real-input DFT, and is not stored.
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Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
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length <code>n</code>, and vice versa for hc2r.
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<span id="index-normalization-2"></span>
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</p>
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<p>Aside from the differing format, the output of
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<code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
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the corresponding 1d r2c/c2r transform
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(i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
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Recall that these transforms are unnormalized, so r2hc followed by hc2r
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will result in the original data multiplied by <code>n</code>. Furthermore,
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like the c2r transform, an out-of-place hc2r transform will
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<em>destroy its input</em> array.
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</p>
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<p>Although these halfcomplex transforms can be used with the
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multi-dimensional r2r interface, the interpretation of such a separable
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product of transforms along each dimension is problematic. For example,
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consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
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transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
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FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
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(of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
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columns (of size <code>n0</code>). Half of these column transforms, however,
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are of imaginary parts, and should therefore be multiplied by <em>i</em>
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and combined with the r2hc transforms of the real columns to produce the
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2d DFT amplitudes; FFTW’s r2r transform does <em>not</em> perform this
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combination for you. Thus, if a multi-dimensional real-input/output DFT
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is required, we recommend using the ordinary r2c/c2r
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interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html">Multi-Dimensional DFTs of Real Data</a>).
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</p>
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<hr>
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<div class="header">
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<p>
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Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html" accesskey="p" rel="prev">More DFTs of Real Data</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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</html>
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