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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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Chris@10 3 <title>The Halfcomplex-format DFT - FFTW 3.3.3</title>
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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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