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3 <title>One-Dimensional DFTs of Real Data - FFTW 3.2.1</title>
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9 <link rel="prev" href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" title="Complex Multi-Dimensional DFTs">
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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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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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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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46 </head>
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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="One-Dimensional-DFTs-of-Real-Data"></a>
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51 <a name="One_002dDimensional-DFTs-of-Real-Data"></a>
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52 Next: <a rel="next" accesskey="n" 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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53 Previous: <a rel="previous" accesskey="p" href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs">Complex Multi-Dimensional DFTs</a>,
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54 Up: <a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a>
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55 <hr>
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56 </div>
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57
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58 <h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3>
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59
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60 <p>In many practical applications, the input data <code>in[i]</code> are purely
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61 real numbers, in which case the DFT output satisfies the “Hermitian”
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62 <a name="index-Hermitian-45"></a>redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
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63 possible to take advantage of these circumstances in order to achieve
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64 roughly a factor of two improvement in both speed and memory usage.
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65
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66 <p>In exchange for these speed and space advantages, the user sacrifices
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67 some of the simplicity of FFTW's complex transforms. First of all, the
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68 input and output arrays are of <em>different sizes and types</em>: the
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69 input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
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70 complex numbers (the non-redundant outputs); this also requires slight
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71 “padding” of the input array for
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72 <a name="index-padding-46"></a>in-place transforms. Second, the inverse transform (complex to real)
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73 has the side-effect of <em>destroying its input array</em>, by default.
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74 Neither of these inconveniences should pose a serious problem for
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75 users, but it is important to be aware of them.
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76
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77 <p>The routines to perform real-data transforms are almost the same as
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78 those for complex transforms: you allocate arrays of <code>double</code>
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79 and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code>),
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80 create an <code>fftw_plan</code>, execute it as many times as you want with
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81 <code>fftw_execute(plan)</code>, and clean up with
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82 <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
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83 differences are that the input (or output) is of type <code>double</code>
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84 and there are new routines to create the plan. In one dimension:
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85
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86 <pre class="example"> fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
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87 unsigned flags);
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88 fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
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89 unsigned flags);
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90 </pre>
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91 <p><a name="index-fftw_005fplan_005fdft_005fr2c_005f1d-47"></a><a name="index-fftw_005fplan_005fdft_005fc2r_005f1d-48"></a>
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92 for the real input to complex-Hermitian output (<dfn>r2c</dfn>) and
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93 complex-Hermitian input to real output (<dfn>c2r</dfn>) transforms.
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94 <a name="index-r2c-49"></a><a name="index-c2r-50"></a>Unlike the complex DFT planner, there is no <code>sign</code> argument.
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95 Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
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96 always <code>FFTW_BACKWARD</code>.
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97 <a name="index-FFTW_005fFORWARD-51"></a><a name="index-FFTW_005fBACKWARD-52"></a>(For single/long-double precision
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98 <code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
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99 <code>float</code> and <code>long double</code>, respectively.)
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100 <a name="index-precision-53"></a>
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101 Here, <code>n</code> is the “logical” size of the DFT, not necessarily the
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102 physical size of the array. In particular, the real (<code>double</code>)
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103 array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
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104 array has <code>n/2+1</code> elements (where the division is rounded down).
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105 For an in-place transform,
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106 <a name="index-in_002dplace-54"></a><code>in</code> and <code>out</code> are aliased to the same array, which must be
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107 big enough to hold both; so, the real array would actually have
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108 <code>2*(n/2+1)</code> elements, where the elements beyond the first <code>n</code>
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109 are unused padding. The kth element of the complex array is
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110 exactly the same as the kth element of the corresponding complex
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111 DFT. All positive <code>n</code> are supported; products of small factors are
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112 most efficient, but an <i>O</i>(<i>n</i> log <i>n</i>) algorithm is used even for prime
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113 sizes.
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114
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115 <p>As noted above, the c2r transform destroys its input array even for
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116 out-of-place transforms. This can be prevented, if necessary, by
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117 including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
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118 unfortunately some sacrifice in performance.
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119 <a name="index-flags-55"></a><a name="index-FFTW_005fPRESERVE_005fINPUT-56"></a>This flag is also not currently supported for multi-dimensional real
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120 DFTs (next section).
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121
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122 <p>Readers familiar with DFTs of real data will recall that the 0th (the
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123 “DC”) and <code>n/2</code>-th (the “Nyquist” frequency, when <code>n</code> is
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124 even) elements of the complex output are purely real. Some
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125 implementations therefore store the Nyquist element where the DC
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126 imaginary part would go, in order to make the input and output arrays
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127 the same size. Such packing, however, does not generalize well to
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128 multi-dimensional transforms, and the space savings are miniscule in
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129 any case; FFTW does not support it.
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130
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131 <p>An alternative interface for one-dimensional r2c and c2r DFTs can be
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132 found in the `<samp><span class="samp">r2r</span></samp>' interface (see <a href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>), with “halfcomplex”-format output that <em>is</em> the same size
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133 (and type) as the input array.
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134 <a name="index-halfcomplex-format-57"></a>That interface, although it is not very useful for multi-dimensional
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135 transforms, may sometimes yield better performance.
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136
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137 <!-- -->
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138 </body></html>
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139
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