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1 <html lang="en">
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2 <head>
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3 <title>The 1d Real-data DFT - FFTW 3.3.3</title>
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4 <meta http-equiv="Content-Type" content="text/html">
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5 <meta name="description" content="FFTW 3.3.3">
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6 <meta name="generator" content="makeinfo 4.13">
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7 <link title="Top" rel="start" href="index.html#Top">
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8 <link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
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9 <link rel="prev" href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029" title="The 1d Discrete Fourier Transform (DFT)">
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10 <link rel="next" href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" title="1d Real-even DFTs (DCTs)">
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11 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
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12 <!--
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13 This manual is for FFTW
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14 (version 3.3.3, 25 November 2012).
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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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19
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20 Permission is granted to make and distribute verbatim copies of
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21 this manual provided the copyright notice and this permission
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22 notice are preserved on all copies.
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23
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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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28
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29 Permission is granted to copy and distribute translations of this
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30 manual into another language, under the above conditions for
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31 modified versions, except that this permission notice may be
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32 stated in a translation approved by the Free Software Foundation.
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33 -->
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35 <style type="text/css"><!--
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45 --></style>
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46 </head>
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47 <body>
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48 <div class="node">
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49 <a name="The-1d-Real-data-DFT"></a>
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50 <a name="The-1d-Real_002ddata-DFT"></a>
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51 <p>
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52 Next: <a rel="next" accesskey="n" href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029">1d Real-even DFTs (DCTs)</a>,
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53 Previous: <a rel="previous" accesskey="p" href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029">The 1d Discrete Fourier Transform (DFT)</a>,
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54 Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</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">4.8.2 The 1d Real-data DFT</h4>
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59
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60 <p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
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61 Y of the size <code>n</code> real array X, exactly as defined
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62 above, i.e.
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63 <center><img src="equation-dft.png" align="top">.</center>This output array Y can easily be shown to possess the
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64 “Hermitian” symmetry
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65 <a name="index-Hermitian-296"></a><i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,where we take Y to be periodic so that
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66 <i>Y<sub>n</sub> = Y</i><sub>0</sub>.
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67
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68 <p>As a result of this symmetry, half of the output Y is redundant
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69 (being the complex conjugate of the other half), and so the 1d r2c
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70 transforms only output elements 0<small class="dots">...</small>n/2 of Y
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71 (n/2+1 complex numbers), where the division by 2 is
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72 rounded down.
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73
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74 <p>Moreover, the Hermitian symmetry implies that
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75 <i>Y</i><sub>0</sub>and, if n is even, the
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76 <i>Y</i><sub><i>n</i>/2</sub>element, are purely real. So, for the <code>R2HC</code> r2r transform, these
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77 elements are not stored in the halfcomplex output format.
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78 <a name="index-r2r-297"></a><a name="index-R2HC-298"></a><a name="index-halfcomplex-format-299"></a>
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79
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80 <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
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81 <em>complex</em> array X with Hermitian symmetry, stored in the
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82 r2c/<code>R2HC</code> output formats, respectively, where the backward
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83 transform is defined exactly as for the complex case:
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84 <center><img src="equation-idft.png" align="top">.</center>The outputs <code>Y</code> of this transform can easily be seen to be purely
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85 real, and are stored as an array of real numbers.
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86
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87 <p><a name="index-normalization-300"></a>Like FFTW's complex DFT, these transforms are unnormalized. In other
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88 words, applying the real-to-complex (forward) and then the
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89 complex-to-real (backward) transform will multiply the input by
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90 n.
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91
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92 <!-- =========> -->
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93 </body></html>
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94
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