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

FilledSmallSquare The discrete Fourier transform of a list of length is by default defined to be .

FilledSmallSquare Note that the zero frequency term appears at position 1 in the resulting list.

FilledSmallSquare Other definitions are used in some scientific and technical fields.

FilledSmallSquare Different choices of definitions can be specified using the option FourierParameters.

FilledSmallSquare With the setting FourierParameters -> a, b the discrete Fourier transform computed by Fourier is .

FilledSmallSquare Some common choices for a, b are {0, 1} (default), {-1, 1} (data analysis), {1, -1} (signal processing).

FilledSmallSquare The setting effectively corresponds to conjugating both input and output lists.

FilledSmallSquare To ensure a unique inverse discrete Fourier transform, must be relatively prime to .

FilledSmallSquare The list of data supplied to Fourier need not have a length equal to a power of two.

FilledSmallSquare The list given in Fourier[list] can be nested to represent an array of data in any number of dimensions.

FilledSmallSquare The array of data must be rectangular.

FilledSmallSquare If the elements of list are exact numbers, Fourier begins by applying N to them.

FilledSmallSquare See The Mathematica Book: Section 1.6.6 and Section 3.8.3.

FilledSmallSquare Implementation Notes: see section A.9.4.

FilledSmallSquare See also: InverseFourier, FourierTransform, Fit.

Further Examples