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3 <title>1d Real-odd DFTs (DSTs) - FFTW 3.3.3</title>
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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="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" title="1d Real-even DFTs (DCTs)">
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10 <link rel="next" href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" title="1d Discrete Hartley Transforms (DHTs)">
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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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22 notice are preserved on all copies.
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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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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="1d-Real-odd-DFTs-(DSTs)"></a>
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50 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></a>
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51 <p>
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52 Next: <a rel="next" accesskey="n" href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029">1d Discrete Hartley Transforms (DHTs)</a>,
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53 Previous: <a rel="previous" accesskey="p" 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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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">4.8.4 1d Real-odd DFTs (DSTs)</h4>
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59
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60 <p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
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61 forward (and backward) DFTs as defined above, where the input array
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62 X of length N is purely real and is also <dfn>odd</dfn> symmetry. In
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63 this case, the output is odd symmetry and purely imaginary.
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64 <a name="index-real_002dodd-DFT-312"></a><a name="index-RODFT-313"></a>
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65
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66 <p><a name="index-RODFT00-314"></a>For the case of <code>RODFT00</code>, this odd symmetry means that
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67 <i>X<sub>j</sub> = -X<sub>N-j</sub></i>,where we take X to be periodic so that
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68 <i>X<sub>N</sub> = X</i><sub>0</sub>. Because of this redundancy, only the first n real numbers
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69 starting at j=1 are actually stored (the j=0 element is
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70 zero), where N = 2(n+1).
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71
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72 <p>The proper definition of odd symmetry for <code>RODFT10</code>,
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73 <code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
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74 because of the shifts by 1/2 of the input and/or output, although
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75 the corresponding boundary conditions are given in <a 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>. Because of the odd symmetry, however,
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76 the cosine terms in the DFT all cancel and the remaining sine terms are
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77 written explicitly below. This formulation often leads people to call
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78 such a transform a <dfn>discrete sine transform</dfn> (DST), although it is
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79 really just a special case of the DFT.
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80 <a name="index-discrete-sine-transform-315"></a><a name="index-DST-316"></a>
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81
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82 <p>In each of the definitions below, we transform a real array X of
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83 length n to a real array Y of length n:
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84
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85 <h5 class="subsubheading">RODFT00 (DST-I)</h5>
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86
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87 <p><a name="index-RODFT00-317"></a>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
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88 <center><img src="equation-rodft00.png" align="top">.</center>
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89
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90 <h5 class="subsubheading">RODFT10 (DST-II)</h5>
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91
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92 <p><a name="index-RODFT10-318"></a>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
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93 <center><img src="equation-rodft10.png" align="top">.</center>
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94
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95 <h5 class="subsubheading">RODFT01 (DST-III)</h5>
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96
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97 <p><a name="index-RODFT01-319"></a>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
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98 <center><img src="equation-rodft01.png" align="top">.</center>In the case of n=1, this reduces to
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99 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
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100
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101 <h5 class="subsubheading">RODFT11 (DST-IV)</h5>
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102
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103 <p><a name="index-RODFT11-320"></a>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
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104 <center><img src="equation-rodft11.png" align="top">.</center>
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105
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106 <h5 class="subsubheading">Inverses and Normalization</h5>
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107
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108 <p>These definitions correspond directly to the unnormalized DFTs used
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109 elsewhere in FFTW (hence the factors of 2 in front of the
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110 summations). The unnormalized inverse of <code>RODFT00</code> is
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111 <code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
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112 of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
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113 in the original array multiplied by N, where N is the
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114 <em>logical</em> DFT size. For <code>RODFT00</code>, N=2(n+1);
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115 otherwise, N=2n.
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116 <a name="index-normalization-321"></a>
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117
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118 <p>In defining the discrete sine transform, some authors also include
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119 additional factors of
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120 √2(or its inverse) multiplying selected inputs and/or outputs. This is a
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121 mostly cosmetic change that makes the transform orthogonal, but
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122 sacrifices the direct equivalence to an antisymmetric DFT.
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123
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124 <!-- =========> -->
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125 </body></html>
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126
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