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Add null config files
author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
parents 89f5e221ed7b
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cannam@95 3 <title>The Halfcomplex-format DFT - FFTW 3.3.3</title>
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cannam@95 49 <a name="The-Halfcomplex-format-DFT"></a>
cannam@95 50 <a name="The-Halfcomplex_002dformat-DFT"></a>
cannam@95 51 <p>
cannam@95 52 Next:&nbsp;<a rel="next" accesskey="n" href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029">Real even/odd DFTs (cosine/sine transforms)</a>,
cannam@95 53 Previous:&nbsp;<a rel="previous" accesskey="p" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>,
cannam@95 54 Up:&nbsp;<a rel="up" accesskey="u" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>
cannam@95 55 <hr>
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cannam@95 57
cannam@95 58 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
cannam@95 59
cannam@95 60 <p>An r2r kind of <code>FFTW_R2HC</code> (<dfn>r2hc</dfn>) corresponds to an r2c DFT
cannam@95 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;
cannam@95 62 format output, and may sometimes be faster and/or more convenient than
cannam@95 63 the latter.
cannam@95 64 <a name="index-halfcomplex-format-75"></a>The inverse <dfn>hc2r</dfn> transform is of kind <code>FFTW_HC2R</code>.
cannam@95 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
cannam@95 66 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
cannam@95 67 numbers (<code>double</code>) in the format:
cannam@95 68
cannam@95 69 <p><p align=center>
cannam@95 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>
cannam@95 71 </p>
cannam@95 72
cannam@95 73 <p>Here,
cannam@95 74 r<sub>k</sub>is the real part of the kth output, and
cannam@95 75 i<sub>k</sub>is the imaginary part. (Division by 2 is rounded down.) For a
cannam@95 76 halfcomplex array <code>hc[n]</code>, the kth component thus has its
cannam@95 77 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
cannam@95 78 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
cannam@95 79 only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
cannam@95 80 zero due to symmetries of the real-input DFT, and is not stored.
cannam@95 81 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
cannam@95 82 length <code>n</code>, and vice versa for hc2r.
cannam@95 83 <a name="index-normalization-78"></a>
cannam@95 84
cannam@95 85 <p>Aside from the differing format, the output of
cannam@95 86 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
cannam@95 87 the corresponding 1d r2c/c2r transform
cannam@95 88 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
cannam@95 89 Recall that these transforms are unnormalized, so r2hc followed by hc2r
cannam@95 90 will result in the original data multiplied by <code>n</code>. Furthermore,
cannam@95 91 like the c2r transform, an out-of-place hc2r transform will
cannam@95 92 <em>destroy its input</em> array.
cannam@95 93
cannam@95 94 <p>Although these halfcomplex transforms can be used with the
cannam@95 95 multi-dimensional r2r interface, the interpretation of such a separable
cannam@95 96 product of transforms along each dimension is problematic. For example,
cannam@95 97 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
cannam@95 98 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
cannam@95 99 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
cannam@95 100 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
cannam@95 101 columns (of size <code>n0</code>). Half of these column transforms, however,
cannam@95 102 are of imaginary parts, and should therefore be multiplied by i
cannam@95 103 and combined with the r2hc transforms of the real columns to produce the
cannam@95 104 2d DFT amplitudes; FFTW's r2r transform does <em>not</em> perform this
cannam@95 105 combination for you. Thus, if a multi-dimensional real-input/output DFT
cannam@95 106 is required, we recommend using the ordinary r2c/c2r
cannam@95 107 interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>).
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