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1 <html lang="en">
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2 <head>
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3 <title>The 1d Discrete Fourier Transform (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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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="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
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10 <link rel="next" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" title="The 1d Real-data DFT">
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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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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-Discrete-Fourier-Transform-(DFT)"></a>
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50 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
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51 <p>
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52 Next: <a rel="next" accesskey="n" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT">The 1d Real-data DFT</a>,
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53 Previous: <a rel="previous" accesskey="p" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</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">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
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59
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60 <p><a name="index-discrete-Fourier-transform-292"></a><a name="index-DFT-293"></a>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
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61 1d complex array X of size n computes an array Y,
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62 where:
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63 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
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64 <center><img src="equation-idft.png" align="top">.</center>
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65
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66 <p><a name="index-normalization-294"></a>FFTW computes an unnormalized transform, in that there is no coefficient
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67 in front of the summation in the DFT. In other words, applying the
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68 forward and then the backward transform will multiply the input by
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69 n.
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70
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71 <p><a name="index-frequency-295"></a>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
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72 -1 in the exponent of the DFT. Note also that we use the
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73 standard “in-order” output ordering—the k-th output
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74 corresponds to the frequency k/n (or k/T, where T
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75 is your total sampling period). For those who like to think in terms of
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76 positive and negative frequencies, this means that the positive
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77 frequencies are stored in the first half of the output and the negative
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78 frequencies are stored in backwards order in the second half of the
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79 output. (The frequency -k/n is the same as the frequency
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80 (n-k)/n.)
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81
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82 <!-- =========> -->
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83 </body></html>
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84
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