|
d@0
|
1 <html lang="en">
|
|
d@0
|
2 <head>
|
|
d@0
|
3 <title>The 1d Discrete Fourier Transform (DFT) - FFTW 3.2.1</title>
|
|
d@0
|
4 <meta http-equiv="Content-Type" content="text/html">
|
|
d@0
|
5 <meta name="description" content="FFTW 3.2.1">
|
|
d@0
|
6 <meta name="generator" content="makeinfo 4.8">
|
|
d@0
|
7 <link title="Top" rel="start" href="index.html#Top">
|
|
d@0
|
8 <link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
|
|
d@0
|
9 <link rel="prev" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
|
|
d@0
|
10 <link rel="next" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" title="The 1d Real-data DFT">
|
|
d@0
|
11 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
|
|
d@0
|
12 <!--
|
|
d@0
|
13 This manual is for FFTW
|
|
d@0
|
14 (version 3.2.1, 5 February 2009).
|
|
d@0
|
15
|
|
d@0
|
16 Copyright (C) 2003 Matteo Frigo.
|
|
d@0
|
17
|
|
d@0
|
18 Copyright (C) 2003 Massachusetts Institute of Technology.
|
|
d@0
|
19
|
|
d@0
|
20 Permission is granted to make and distribute verbatim copies of
|
|
d@0
|
21 this manual provided the copyright notice and this permission
|
|
d@0
|
22 notice are preserved on all copies.
|
|
d@0
|
23
|
|
d@0
|
24 Permission is granted to copy and distribute modified versions of
|
|
d@0
|
25 this manual under the conditions for verbatim copying, provided
|
|
d@0
|
26 that the entire resulting derived work is distributed under the
|
|
d@0
|
27 terms of a permission notice identical to this one.
|
|
d@0
|
28
|
|
d@0
|
29 Permission is granted to copy and distribute translations of this
|
|
d@0
|
30 manual into another language, under the above conditions for
|
|
d@0
|
31 modified versions, except that this permission notice may be
|
|
d@0
|
32 stated in a translation approved by the Free Software Foundation.
|
|
d@0
|
33 -->
|
|
d@0
|
34 <meta http-equiv="Content-Style-Type" content="text/css">
|
|
d@0
|
35 <style type="text/css"><!--
|
|
d@0
|
36 pre.display { font-family:inherit }
|
|
d@0
|
37 pre.format { font-family:inherit }
|
|
d@0
|
38 pre.smalldisplay { font-family:inherit; font-size:smaller }
|
|
d@0
|
39 pre.smallformat { font-family:inherit; font-size:smaller }
|
|
d@0
|
40 pre.smallexample { font-size:smaller }
|
|
d@0
|
41 pre.smalllisp { font-size:smaller }
|
|
d@0
|
42 span.sc { font-variant:small-caps }
|
|
d@0
|
43 span.roman { font-family:serif; font-weight:normal; }
|
|
d@0
|
44 span.sansserif { font-family:sans-serif; font-weight:normal; }
|
|
d@0
|
45 --></style>
|
|
d@0
|
46 </head>
|
|
d@0
|
47 <body>
|
|
d@0
|
48 <div class="node">
|
|
d@0
|
49 <p>
|
|
d@0
|
50 <a name="The-1d-Discrete-Fourier-Transform-(DFT)"></a>
|
|
d@0
|
51 <a name="The-1d-Discrete-Fourier-Transform-_0028DFT_0029"></a>
|
|
d@0
|
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>,
|
|
d@0
|
53 Previous: <a rel="previous" accesskey="p" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>,
|
|
d@0
|
54 Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
|
|
d@0
|
55 <hr>
|
|
d@0
|
56 </div>
|
|
d@0
|
57
|
|
d@0
|
58 <h4 class="subsection">4.8.1 The 1d Discrete Fourier Transform (DFT)</h4>
|
|
d@0
|
59
|
|
d@0
|
60 <p><a name="index-discrete-Fourier-transform-282"></a><a name="index-DFT-283"></a>The forward (<code>FFTW_FORWARD</code>) discrete Fourier transform (DFT) of a
|
|
d@0
|
61 1d complex array X of size n computes an array Y,
|
|
d@0
|
62 where:
|
|
d@0
|
63 <center><img src="equation-dft.png" align="top">.</center>The backward (<code>FFTW_BACKWARD</code>) DFT computes:
|
|
d@0
|
64 <center><img src="equation-idft.png" align="top">.</center>
|
|
d@0
|
65
|
|
d@0
|
66 <p><a name="index-normalization-284"></a>FFTW computes an unnormalized transform, in that there is no coefficient
|
|
d@0
|
67 in front of the summation in the DFT. In other words, applying the
|
|
d@0
|
68 forward and then the backward transform will multiply the input by
|
|
d@0
|
69 n.
|
|
d@0
|
70
|
|
d@0
|
71 <p><a name="index-frequency-285"></a>From above, an <code>FFTW_FORWARD</code> transform corresponds to a sign of
|
|
d@0
|
72 -1 in the exponent of the DFT. Note also that we use the
|
|
d@0
|
73 standard “in-order” output ordering—the k-th output
|
|
d@0
|
74 corresponds to the frequency k/n (or k/T, where T
|
|
d@0
|
75 is your total sampling period). For those who like to think in terms of
|
|
d@0
|
76 positive and negative frequencies, this means that the positive
|
|
d@0
|
77 frequencies are stored in the first half of the output and the negative
|
|
d@0
|
78 frequencies are stored in backwards order in the second half of the
|
|
d@0
|
79 output. (The frequency -k/n is the same as the frequency
|
|
d@0
|
80 (n-k)/n.)
|
|
d@0
|
81
|
|
d@0
|
82 <!-- =========> -->
|
|
d@0
|
83 </body></html>
|
|
d@0
|
84
|
