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Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 19 Nov 2019 14:52:55 +0000
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25 <title>FFTW 3.3.8: The Halfcomplex-format DFT</title>
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71 <a name="The-Halfcomplex_002dformat-DFT"></a>
72 <div class="header">
73 <p>
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>
75 </div>
76 <hr>
77 <a name="The-Halfcomplex_002dformat-DFT-1"></a>
78 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
79
80 <p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
81 <a name="index-FFTW_005fR2HC"></a>
82 <a name="index-r2c-1"></a>
83 <a name="index-r2hc"></a>
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;
85 format output, and may sometimes be faster and/or more convenient than
86 the latter.
87 <a name="index-halfcomplex-format-1"></a>
88 The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
89 <a name="index-FFTW_005fHC2R"></a>
90 <a name="index-hc2r"></a>
91 This consists of the non-redundant half of the complex output for a 1d
92 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
93 numbers (<code>double</code>) in the format:
94 </p>
95 <p align=center>
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>
97 </p>
98
99 <p>Here,
100 r<sub>k</sub>
101 is the real part of the <em>k</em>th output, and
102 i<sub>k</sub>
103 is the imaginary part. (Division by 2 is rounded down.) For a
104 halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its
105 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
106 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
107 only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
108 zero due to symmetries of the real-input DFT, and is not stored.
109 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
110 length <code>n</code>, and vice versa for hc2r.
111 <a name="index-normalization-2"></a>
112 </p>
113
114 <p>Aside from the differing format, the output of
115 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
116 the corresponding 1d r2c/c2r transform
117 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
118 Recall that these transforms are unnormalized, so r2hc followed by hc2r
119 will result in the original data multiplied by <code>n</code>. Furthermore,
120 like the c2r transform, an out-of-place hc2r transform will
121 <em>destroy its input</em> array.
122 </p>
123 <p>Although these halfcomplex transforms can be used with the
124 multi-dimensional r2r interface, the interpretation of such a separable
125 product of transforms along each dimension is problematic. For example,
126 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
127 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
128 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
129 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
130 columns (of size <code>n0</code>). Half of these column transforms, however,
131 are of imaginary parts, and should therefore be multiplied by <em>i</em>
132 and combined with the r2hc transforms of the real columns to produce the
133 2d DFT amplitudes; FFTW&rsquo;s r2r transform does <em>not</em> perform this
134 combination for you. Thus, if a multi-dimensional real-input/output DFT
135 is required, we recommend using the ordinary r2c/c2r
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>).
137 </p>
138 <hr>
139 <div class="header">
140 <p>
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>
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