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3 <title>The Halfcomplex-format DFT - FFTW 3.3.3</title>
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57
58 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
59
60 <p>An r2r kind of <code>FFTW_R2HC</code> (<dfn>r2hc</dfn>) corresponds to an r2c DFT
61 <a name="index-FFTW_005fR2HC-72"></a><a name="index-r2c-73"></a><a name="index-r2hc-74"></a>(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;
62 format output, and may sometimes be faster and/or more convenient than
63 the latter.
64 <a name="index-halfcomplex-format-75"></a>The inverse <dfn>hc2r</dfn> transform is of kind <code>FFTW_HC2R</code>.
65 <a name="index-FFTW_005fHC2R-76"></a><a name="index-hc2r-77"></a>This consists of the non-redundant half of the complex output for a 1d
66 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
67 numbers (<code>double</code>) in the format:
68
69 <p><p align=center>
70 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>
71 </p>
72
73 <p>Here,
74 r<sub>k</sub>is the real part of the kth output, and
75 i<sub>k</sub>is the imaginary part. (Division by 2 is rounded down.) For a
76 halfcomplex array <code>hc[n]</code>, the kth component thus has its
77 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
78 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
79 only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
80 zero due to symmetries of the real-input DFT, and is not stored.
81 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
82 length <code>n</code>, and vice versa for hc2r.
83 <a name="index-normalization-78"></a>
84
85 <p>Aside from the differing format, the output of
86 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
87 the corresponding 1d r2c/c2r transform
88 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
89 Recall that these transforms are unnormalized, so r2hc followed by hc2r
90 will result in the original data multiplied by <code>n</code>. Furthermore,
91 like the c2r transform, an out-of-place hc2r transform will
92 <em>destroy its input</em> array.
93
94 <p>Although these halfcomplex transforms can be used with the
95 multi-dimensional r2r interface, the interpretation of such a separable
96 product of transforms along each dimension is problematic. For example,
97 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
98 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
99 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
100 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
101 columns (of size <code>n0</code>). Half of these column transforms, however,
102 are of imaginary parts, and should therefore be multiplied by i
103 and combined with the r2hc transforms of the real columns to produce the
104 2d DFT amplitudes; FFTW's r2r transform does <em>not</em> perform this
105 combination for you. Thus, if a multi-dimensional real-input/output DFT
106 is required, we recommend using the ordinary r2c/c2r
107 interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>).
108
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