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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 37bf6b4a2645
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Chris@10 3 <title>The Halfcomplex-format DFT - FFTW 3.3.3</title>
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Chris@10 48 <div class="node">
Chris@10 49 <a name="The-Halfcomplex-format-DFT"></a>
Chris@10 50 <a name="The-Halfcomplex_002dformat-DFT"></a>
Chris@10 51 <p>
Chris@10 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>,
Chris@10 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>,
Chris@10 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>
Chris@10 55 <hr>
Chris@10 56 </div>
Chris@10 57
Chris@10 58 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
Chris@10 59
Chris@10 60 <p>An r2r kind of <code>FFTW_R2HC</code> (<dfn>r2hc</dfn>) corresponds to an r2c DFT
Chris@10 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;
Chris@10 62 format output, and may sometimes be faster and/or more convenient than
Chris@10 63 the latter.
Chris@10 64 <a name="index-halfcomplex-format-75"></a>The inverse <dfn>hc2r</dfn> transform is of kind <code>FFTW_HC2R</code>.
Chris@10 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
Chris@10 66 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
Chris@10 67 numbers (<code>double</code>) in the format:
Chris@10 68
Chris@10 69 <p><p align=center>
Chris@10 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>
Chris@10 71 </p>
Chris@10 72
Chris@10 73 <p>Here,
Chris@10 74 r<sub>k</sub>is the real part of the kth output, and
Chris@10 75 i<sub>k</sub>is the imaginary part. (Division by 2 is rounded down.) For a
Chris@10 76 halfcomplex array <code>hc[n]</code>, the kth component thus has its
Chris@10 77 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
Chris@10 78 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
Chris@10 79 only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
Chris@10 80 zero due to symmetries of the real-input DFT, and is not stored.
Chris@10 81 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
Chris@10 82 length <code>n</code>, and vice versa for hc2r.
Chris@10 83 <a name="index-normalization-78"></a>
Chris@10 84
Chris@10 85 <p>Aside from the differing format, the output of
Chris@10 86 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
Chris@10 87 the corresponding 1d r2c/c2r transform
Chris@10 88 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
Chris@10 89 Recall that these transforms are unnormalized, so r2hc followed by hc2r
Chris@10 90 will result in the original data multiplied by <code>n</code>. Furthermore,
Chris@10 91 like the c2r transform, an out-of-place hc2r transform will
Chris@10 92 <em>destroy its input</em> array.
Chris@10 93
Chris@10 94 <p>Although these halfcomplex transforms can be used with the
Chris@10 95 multi-dimensional r2r interface, the interpretation of such a separable
Chris@10 96 product of transforms along each dimension is problematic. For example,
Chris@10 97 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
Chris@10 98 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
Chris@10 99 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
Chris@10 100 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
Chris@10 101 columns (of size <code>n0</code>). Half of these column transforms, however,
Chris@10 102 are of imaginary parts, and should therefore be multiplied by i
Chris@10 103 and combined with the r2hc transforms of the real columns to produce the
Chris@10 104 2d DFT amplitudes; FFTW's r2r transform does <em>not</em> perform this
Chris@10 105 combination for you. Thus, if a multi-dimensional real-input/output DFT
Chris@10 106 is required, we recommend using the ordinary r2c/c2r
Chris@10 107 interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>).
Chris@10 108
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