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Replace these with versions built using an older toolset (so as to avoid ABI compatibilities when linking on Ubuntu 14.04 for packaging purposes)
author Chris Cannam
date Fri, 07 Feb 2020 11:51:13 +0000
parents 2cd0e3b3e1fd
children
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Chris@42 3 <!-- This manual is for FFTW
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Chris@42 24 <head>
Chris@42 25 <title>FFTW 3.3.5: The Halfcomplex-format DFT</title>
Chris@42 26
Chris@42 27 <meta name="description" content="FFTW 3.3.5: The Halfcomplex-format DFT">
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Chris@42 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
Chris@42 72 <a name="The-Halfcomplex_002dformat-DFT"></a>
Chris@42 73 <div class="header">
Chris@42 74 <p>
Chris@42 75 Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
Chris@42 76 </div>
Chris@42 77 <hr>
Chris@42 78 <a name="The-Halfcomplex_002dformat-DFT-1"></a>
Chris@42 79 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
Chris@42 80
Chris@42 81 <p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
Chris@42 82 <a name="index-FFTW_005fR2HC"></a>
Chris@42 83 <a name="index-r2c-1"></a>
Chris@42 84 <a name="index-r2hc"></a>
Chris@42 85 (see <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data">One-Dimensional DFTs of Real Data</a>) but with &ldquo;halfcomplex&rdquo;
Chris@42 86 format output, and may sometimes be faster and/or more convenient than
Chris@42 87 the latter.
Chris@42 88 <a name="index-halfcomplex-format-1"></a>
Chris@42 89 The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
Chris@42 90 <a name="index-FFTW_005fHC2R"></a>
Chris@42 91 <a name="index-hc2r"></a>
Chris@42 92 This consists of the non-redundant half of the complex output for a 1d
Chris@42 93 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
Chris@42 94 numbers (<code>double</code>) in the format:
Chris@42 95 </p>
Chris@42 96 <p align=center>
Chris@42 97 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>
Chris@42 98 </p>
Chris@42 99 <p>Here,
Chris@42 100 r<sub>k</sub>is the real part of the <em>k</em>th output, and
Chris@42 101 i<sub>k</sub>is the imaginary part. (Division by 2 is rounded down.) For a
Chris@42 102 halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its
Chris@42 103 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
Chris@42 104 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
Chris@42 105 only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
Chris@42 106 zero due to symmetries of the real-input DFT, and is not stored.
Chris@42 107 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
Chris@42 108 length <code>n</code>, and vice versa for hc2r.
Chris@42 109 <a name="index-normalization-2"></a>
Chris@42 110 </p>
Chris@42 111
Chris@42 112 <p>Aside from the differing format, the output of
Chris@42 113 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
Chris@42 114 the corresponding 1d r2c/c2r transform
Chris@42 115 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
Chris@42 116 Recall that these transforms are unnormalized, so r2hc followed by hc2r
Chris@42 117 will result in the original data multiplied by <code>n</code>. Furthermore,
Chris@42 118 like the c2r transform, an out-of-place hc2r transform will
Chris@42 119 <em>destroy its input</em> array.
Chris@42 120 </p>
Chris@42 121 <p>Although these halfcomplex transforms can be used with the
Chris@42 122 multi-dimensional r2r interface, the interpretation of such a separable
Chris@42 123 product of transforms along each dimension is problematic. For example,
Chris@42 124 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
Chris@42 125 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
Chris@42 126 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
Chris@42 127 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
Chris@42 128 columns (of size <code>n0</code>). Half of these column transforms, however,
Chris@42 129 are of imaginary parts, and should therefore be multiplied by <em>i</em>
Chris@42 130 and combined with the r2hc transforms of the real columns to produce the
Chris@42 131 2d DFT amplitudes; FFTW&rsquo;s r2r transform does <em>not</em> perform this
Chris@42 132 combination for you. Thus, if a multi-dimensional real-input/output DFT
Chris@42 133 is required, we recommend using the ordinary r2c/c2r
Chris@42 134 interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>).
Chris@42 135 </p>
Chris@42 136 <hr>
Chris@42 137 <div class="header">
Chris@42 138 <p>
Chris@42 139 Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
Chris@42 140 </div>
Chris@42 141
Chris@42 142
Chris@42 143
Chris@42 144 </body>
Chris@42 145 </html>