# Fourier

Fourier[list]

finds the discrete Fourier transform of a list of complex numbers.

Fourier[list,{p1,p2,}]

returns the specified positions of the discrete Fourier transform.

# Details and Options • The discrete Fourier transform vs of a list ur of length n is by default defined to be  ure2π i(r-1)(s-1)/n. »
• 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  ure2π i b(r-1)(s-1)/n. »
• 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[list,{p1,p2,}] is typically equivalent to Extract[Fourier[list],{p1,p2,}]. Cases with just a few positions p are computed using an algorithm that takes less time and memory but is more subject to numerical error, particularly when the length of list is long.
• Fourier can be used on SparseArray objects.

# Examples

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## Basic Examples(2)

Find a discrete Fourier transform:

 In:= Out= Find a power spectrum:

 In:= Out= ## Possible Issues(4)

Introduced in 1988
(1.0)
|
Updated in 2012
(9.0)