1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
|
<html lang="en">
<head>
<title>Overview of real data FFTs - GNU Scientific Library -- Reference Manual</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="GNU Scientific Library -- Reference Manual">
<meta name="generator" content="makeinfo 4.11">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Fast-Fourier-Transforms.html" title="Fast Fourier Transforms">
<link rel="prev" href="Mixed_002dradix-FFT-routines-for-complex-data.html" title="Mixed-radix FFT routines for complex data">
<link rel="next" href="Radix_002d2-FFT-routines-for-real-data.html" title="Radix-2 FFT routines for real data">
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
<!--
Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being ``GNU General Public License'' and ``Free Software
Needs Free Documentation'', the Front-Cover text being ``A GNU Manual'',
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled ``GNU Free Documentation
License''.
(a) The Back-Cover Text is: ``You have the freedom to copy and modify this
GNU Manual.''-->
<meta http-equiv="Content-Style-Type" content="text/css">
<style type="text/css"><!--
pre.display { font-family:inherit }
pre.format { font-family:inherit }
pre.smalldisplay { font-family:inherit; font-size:smaller }
pre.smallformat { font-family:inherit; font-size:smaller }
pre.smallexample { font-size:smaller }
pre.smalllisp { font-size:smaller }
span.sc { font-variant:small-caps }
span.roman { font-family:serif; font-weight:normal; }
span.sansserif { font-family:sans-serif; font-weight:normal; }
--></style>
</head>
<body>
<div class="node">
<p>
<a name="Overview-of-real-data-FFTs"></a>
Next: <a rel="next" accesskey="n" href="Radix_002d2-FFT-routines-for-real-data.html">Radix-2 FFT routines for real data</a>,
Previous: <a rel="previous" accesskey="p" href="Mixed_002dradix-FFT-routines-for-complex-data.html">Mixed-radix FFT routines for complex data</a>,
Up: <a rel="up" accesskey="u" href="Fast-Fourier-Transforms.html">Fast Fourier Transforms</a>
<hr>
</div>
<h3 class="section">16.5 Overview of real data FFTs</h3>
<p><a name="index-FFT-of-real-data-1493"></a>The functions for real data are similar to those for complex data.
However, there is an important difference between forward and inverse
transforms. The fourier transform of a real sequence is not real. It is
a complex sequence with a special symmetry:
<pre class="example"> z_k = z_{n-k}^*
</pre>
<p class="noindent">A sequence with this symmetry is called <dfn>conjugate-complex</dfn> or
<dfn>half-complex</dfn>. This different structure requires different
storage layouts for the forward transform (from real to half-complex)
and inverse transform (from half-complex back to real). As a
consequence the routines are divided into two sets: functions in
<code>gsl_fft_real</code> which operate on real sequences and functions in
<code>gsl_fft_halfcomplex</code> which operate on half-complex sequences.
<p>Functions in <code>gsl_fft_real</code> compute the frequency coefficients of a
real sequence. The half-complex coefficients c of a real sequence
x are given by fourier analysis,
<pre class="example"> c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)
</pre>
<p class="noindent">Functions in <code>gsl_fft_halfcomplex</code> compute inverse or backwards
transforms. They reconstruct real sequences by fourier synthesis from
their half-complex frequency coefficients, c,
<pre class="example"> x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)
</pre>
<p class="noindent">The symmetry of the half-complex sequence implies that only half of the
complex numbers in the output need to be stored. The remaining half can
be reconstructed using the half-complex symmetry condition. This works
for all lengths, even and odd—when the length is even the middle value
where k=n/2 is also real. Thus only <var>n</var> real numbers are
required to store the half-complex sequence, and the transform of a real
sequence can be stored in the same size array as the original data.
<p>The precise storage arrangements depend on the algorithm, and are
different for radix-2 and mixed-radix routines. The radix-2 function
operates in-place, which constrains the locations where each element can
be stored. The restriction forces real and imaginary parts to be stored
far apart. The mixed-radix algorithm does not have this restriction, and
it stores the real and imaginary parts of a given term in neighboring
locations (which is desirable for better locality of memory accesses).
<hr>The GNU Scientific Library - a free numerical library licensed under the GNU GPL<br>Back to the <a href="/software/gsl/">GNU Scientific Library Homepage</a></body></html>
|
