This is documentation for Mathematica 4, which was
based on an earlier version of the Wolfram Language.

Fourier

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

The discrete Fourier transform of a list of length 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, must be relatively prime to .

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.

See The Mathematica Book: Section 1.6.6 and Section 3.8.3.

Implementation Notes: see section A.9.4.