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3 <title>The Halfcomplex-format DFT - FFTW 3.2.1</title>
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12 <!--
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13 This manual is for FFTW
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14 (version 3.2.1, 5 February 2009).
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15
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16 Copyright (C) 2003 Matteo Frigo.
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17
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18 Copyright (C) 2003 Massachusetts Institute of Technology.
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24 Permission is granted to copy and distribute modified versions of
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25 this manual under the conditions for verbatim copying, provided
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26 that the entire resulting derived work is distributed under the
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27 terms of a permission notice identical to this one.
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31 modified versions, except that this permission notice may be
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47 <body>
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48 <div class="node">
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49 <p>
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50 <a name="The-Halfcomplex-format-DFT"></a>
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51 <a name="The-Halfcomplex_002dformat-DFT"></a>
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52 Next: <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>,
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53 Previous: <a rel="previous" accesskey="p" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>,
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54 Up: <a rel="up" accesskey="u" href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data">More DFTs of Real Data</a>
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55 <hr>
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56 </div>
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57
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58 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
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59
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60 <p>An r2r kind of <code>FFTW_R2HC</code> (<dfn>r2hc</dfn>) corresponds to an r2c DFT
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61 <a name="index-FFTW_005fR2HC-71"></a><a name="index-r2c-72"></a><a name="index-r2hc-73"></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 “halfcomplex”
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62 format output, and may sometimes be faster and/or more convenient than
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63 the latter.
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64 <a name="index-halfcomplex-format-74"></a>The inverse <dfn>hc2r</dfn> transform is of kind <code>FFTW_HC2R</code>.
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65 <a name="index-FFTW_005fHC2R-75"></a><a name="index-hc2r-76"></a>This consists of the non-redundant half of the complex output for a 1d
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66 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
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67 numbers (<code>double</code>) in the format:
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68
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69 <p><p align=center>
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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>
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71 </p>
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72
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73 <p>Here,
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74 r<sub>k</sub>is the real part of the kth output, and
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75 i<sub>k</sub>is the imaginary part. (Division by 2 is rounded down.) For a
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76 halfcomplex array <code>hc[n]</code>, the kth component thus has its
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77 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
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78 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
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79 only if <code>n</code> is even)—in these two cases, the imaginary part is
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80 zero due to symmetries of the real-input DFT, and is not stored.
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81 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
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82 length <code>n</code>, and vice versa for hc2r.
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83 <a name="index-normalization-77"></a>
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84 Aside from the differing format, the output of
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85 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
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86 the corresponding 1d r2c/c2r transform
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87 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
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88 Recall that these transforms are unnormalized, so r2hc followed by hc2r
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89 will result in the original data multiplied by <code>n</code>. Furthermore,
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90 like the c2r transform, an out-of-place hc2r transform will
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91 <em>destroy its input</em> array.
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92
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93 <p>Although these halfcomplex transforms can be used with the
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94 multi-dimensional r2r interface, the interpretation of such a separable
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95 product of transforms along each dimension is problematic. For example,
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96 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
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97 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
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98 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
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99 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
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100 columns (of size <code>n0</code>). Half of these column transforms, however,
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101 are of imaginary parts, and should therefore be multiplied by i
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102 and combined with the r2hc transforms of the real columns to produce the
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103 2d DFT amplitudes; FFTW's r2r transform does <em>not</em> perform this
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104 combination for you. Thus, if a multi-dimensional real-input/output DFT
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105 is required, we recommend using the ordinary r2c/c2r
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106 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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107
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108 <!-- =========> -->
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109 </body></html>
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110
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