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 complex 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:

Applications  (1)

The efficiency of the fast Fourier transform (FFT) relies on being able to form a larger Fourier matrix from two smaller ones. Generate two small Fourier matrices of sizes p and q:

The Fourier matrix of size p q can be expressed as a product of four simpler matrices:

Show that the resulting matrix is equivalent to the result of FourierMatrix:

The discrete Fourier transform of a vector can be computed by successively multiplying the factors of the Fourier matrix to the vector:

The result is equivalent to applying Fourier to the vector:

Properties & Relations  (2)

FourierMatrix can be represented as a VandermondeMatrix:

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:

Wolfram Research (2012), FourierMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierMatrix.html.

Text

Wolfram Research (2012), FourierMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierMatrix.html.

CMS

Wolfram Language. 2012. "FourierMatrix." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FourierMatrix.html.

APA

Wolfram Language. (2012). FourierMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierMatrix.html

BibTeX

@misc{reference.wolfram_2022_fouriermatrix, author="Wolfram Research", title="{FourierMatrix}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/FourierMatrix.html}", note=[Accessed: 09-June-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_fouriermatrix, organization={Wolfram Research}, title={FourierMatrix}, year={2012}, url={https://reference.wolfram.com/language/ref/FourierMatrix.html}, note=[Accessed: 09-June-2023 ]}