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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 d0c2a83c1364
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Chris@82 24 <head>
Chris@82 25 <title>FFTW 3.3.8: The Halfcomplex-format DFT</title>
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Chris@82 68 </head>
Chris@82 69
Chris@82 70 <body lang="en">
Chris@82 71 <a name="The-Halfcomplex_002dformat-DFT"></a>
Chris@82 72 <div class="header">
Chris@82 73 <p>
Chris@82 74 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@82 75 </div>
Chris@82 76 <hr>
Chris@82 77 <a name="The-Halfcomplex_002dformat-DFT-1"></a>
Chris@82 78 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
Chris@82 79
Chris@82 80 <p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
Chris@82 81 <a name="index-FFTW_005fR2HC"></a>
Chris@82 82 <a name="index-r2c-1"></a>
Chris@82 83 <a name="index-r2hc"></a>
Chris@82 84 (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@82 85 format output, and may sometimes be faster and/or more convenient than
Chris@82 86 the latter.
Chris@82 87 <a name="index-halfcomplex-format-1"></a>
Chris@82 88 The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
Chris@82 89 <a name="index-FFTW_005fHC2R"></a>
Chris@82 90 <a name="index-hc2r"></a>
Chris@82 91 This consists of the non-redundant half of the complex output for a 1d
Chris@82 92 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
Chris@82 93 numbers (<code>double</code>) in the format:
Chris@82 94 </p>
Chris@82 95 <p align=center>
Chris@82 96 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@82 97 </p>
Chris@82 98
Chris@82 99 <p>Here,
Chris@82 100 r<sub>k</sub>
Chris@82 101 is the real part of the <em>k</em>th output, and
Chris@82 102 i<sub>k</sub>
Chris@82 103 is the imaginary part. (Division by 2 is rounded down.) For a
Chris@82 104 halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its
Chris@82 105 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
Chris@82 106 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
Chris@82 107 only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
Chris@82 108 zero due to symmetries of the real-input DFT, and is not stored.
Chris@82 109 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
Chris@82 110 length <code>n</code>, and vice versa for hc2r.
Chris@82 111 <a name="index-normalization-2"></a>
Chris@82 112 </p>
Chris@82 113
Chris@82 114 <p>Aside from the differing format, the output of
Chris@82 115 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
Chris@82 116 the corresponding 1d r2c/c2r transform
Chris@82 117 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
Chris@82 118 Recall that these transforms are unnormalized, so r2hc followed by hc2r
Chris@82 119 will result in the original data multiplied by <code>n</code>. Furthermore,
Chris@82 120 like the c2r transform, an out-of-place hc2r transform will
Chris@82 121 <em>destroy its input</em> array.
Chris@82 122 </p>
Chris@82 123 <p>Although these halfcomplex transforms can be used with the
Chris@82 124 multi-dimensional r2r interface, the interpretation of such a separable
Chris@82 125 product of transforms along each dimension is problematic. For example,
Chris@82 126 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
Chris@82 127 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
Chris@82 128 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
Chris@82 129 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
Chris@82 130 columns (of size <code>n0</code>). Half of these column transforms, however,
Chris@82 131 are of imaginary parts, and should therefore be multiplied by <em>i</em>
Chris@82 132 and combined with the r2hc transforms of the real columns to produce the
Chris@82 133 2d DFT amplitudes; FFTW&rsquo;s r2r transform does <em>not</em> perform this
Chris@82 134 combination for you. Thus, if a multi-dimensional real-input/output DFT
Chris@82 135 is required, we recommend using the ordinary r2c/c2r
Chris@82 136 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@82 137 </p>
Chris@82 138 <hr>
Chris@82 139 <div class="header">
Chris@82 140 <p>
Chris@82 141 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@82 142 </div>
Chris@82 143
Chris@82 144
Chris@82 145
Chris@82 146 </body>
Chris@82 147 </html>