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 OptionsDetails and Options

  • The discrete Fourier transform of a list 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 are (default), (data analysis), (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 equivalent to Extract[Fourier[list, {p1, p2, ...}] but may require less time and memory. »
  • Fourier can be used on SparseArray objects.
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