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


returns the specified positions of the discrete inverse Fourier transform.

Details and Options

  • The inverse Fourier transform of a list of length is defined to be . »
  • Note that the zero frequency term must appear at position 1 in the input 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 InverseFourier is .
  • Some common choices for {a,b} are {0,1} (default), {-1,1} (data analysis), {1,-1} (signal processing).
  • The setting b=-1 effectively corresponds to conjugating both input and output lists.
  • To ensure a unique discrete Fourier transform, Abs[b] must be relatively prime to .
  • The list of data need not have a length equal to a power of two.
  • The list given in InverseFourier[list] can be nested to represent an array of data in any number of dimensions.
  • The array of data must be rectangular.
  • InverseFourier[list,{p1,p2,}] is typically equivalent to Extract[InverseFourier[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.
  • If the elements of list are exact numbers, InverseFourier begins by applying N to them.


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Basic Examples  (2)

Inverse Fourier transform of a real list:

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Inverse Fourier transform of a complex list:

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Scope  (3)

Options  (3)

Applications  (1)

Properties & Relations  (2)

Possible Issues  (1)

See Also

Fourier  FourierDCT  FourierDST  FourierTransform  InverseFourierTransform


Introduced in 1988
| Updated in 2012