FourierMatrix

FourierMatrix[n]

returns an n×n Fourier matrix.

Details and Options

  • FourierMatrix of order n returns a list of the length-n discrete Fourier transform's basis sequences.
  • Each entry Frs of the Fourier matrix is by default defined as .
  • Rows of the FourierMatrix are basis sequences of the discrete Fourier transform.
  • The result F of FourierMatrix[n] is symmetric and unitary, meaning that F-1 is Conjugate[F].
  • The result of FourierMatrix[n].list is equivalent to Fourier[list] when list has length n. However, the computation of Fourier[list] is much faster and has less numerical error. »
  • With the setting FourierParameters->{a,b}, entry Frs of the Fourier matrix is defined as . The default setting is FourierParameters->{0,1}.
  • FourierMatrix[,WorkingPrecision->p] gives a matrix with entries of precision p.

Examples

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Basic Examples  (1)

A Fourier matrix:

Scope  (1)

The real and imaginary parts of the Fourier's basis sequences of length 128:

Options  (1)

WorkingPrecision  (1)

Use machine precision:

Use arbitrary precision:

Properties & Relations  (1)

The Fourier transform of a vector is equivalent to the vector multiplied by a Fourier matrix:

Fourier is much faster than the matrix-based computation:

Introduced in 2012
 (9.0)