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


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.


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Basic Examples  (2)

Find a discrete Fourier transform:

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Find a power spectrum:

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Scope  (3)

Options  (2)

Applications  (10)

Properties & Relations  (6)

Possible Issues  (4)

Introduced in 1988
Updated in 2012