FourierMatrix
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
open allclose allApplications (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:
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