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Discrete Fourier Transforms
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Fourier
Fourier
[
list
]
finds the discrete Fourier transform of a list of complex numbers.
MORE INFORMATION
The discrete Fourier transform
v
s
of a list
u
r
of length
n
is by default defined to be
.
»
Note that the zero frequency term appears at position 1 in the resulting list.
Other definitions are used in some scientific and technical fields.
Different choices of definitions can be specified using the option
FourierParameters
.
With the setting
FourierParameters
->{
a
,
b
}
the discrete Fourier transform computed by
Fourier
is
.
»
Some common choices for
{
a
,
b
}
are
{0, 1}
(default),
{-1, 1}
(data analysis),
{1, -1}
(signal processing).
The setting
effectively corresponds to conjugating both input and output lists.
To ensure a unique inverse discrete Fourier transform,
b
must be relatively prime to
n
.
»
The list of data supplied to
Fourier
need not have a length equal to a power of two.
The
list
given in
Fourier
[
list
]
can be nested to represent an array of data in any number of dimensions.
The array of data must be rectangular.
If the elements of
list
are exact numbers,
Fourier
begins by applying
N
to them.
Fourier
can be used on
SparseArray
objects.
EXAMPLES
CLOSE ALL
Basic Examples
(2)
Find a discrete Fourier transform:
In[1]:=
Out[1]=
Find a power spectrum:
In[1]:=
Out[1]=
Scope
(4)
Options
(2)
Applications
(9)
Properties & Relations
(6)
Possible Issues
(2)
SEE ALSO
InverseFourier
FourierDCT
FourierDST
FourierTransform
FourierSinTransform
Fit
TUTORIALS
Manipulating Numerical Data
Discrete Fourier Transforms
RELATED LINKS
Implementation notes: Numerical and Related Functions
NKS|Online
(
A New Kind of Science
)
MORE ABOUT
Data Transforms and Smoothing
Image Processing
Integral Transforms
Numerical Data
Signal Processing
New in 1 | Last modified in 4
© 2008 Wolfram Research, Inc.
. »
. »
effectively corresponds to conjugating both input and output lists. 
b must be relatively prime to n. »
EXAMPLES
Basic Examples 
Scope 