BUILT-IN MATHEMATICA SYMBOL

# Fourier

Fourier[list]
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.

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

Find a discrete Fourier transform:

 Out[1]=

Find a power spectrum:

 Out[1]=