# 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 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}.
• gives a matrix with entries of precision p.

# Examples

open allclose all

## 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(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: