dftmtx

Signal Processing Toolbox

Discrete Fourier transform matrix.

A = dftmtx(n)

Description

A discrete Fourier transform matrix is a complex matrix of values around the unit circle, whose matrix product with a vector computes the discrete Fourier transform of the vector.

A = dftmtx(n)

returns the n-by-n complex matrix A that, when multiplied into a length n column vector x:

y = A*x

computes the discrete Fourier transform of x.

The inverse discrete Fourier transform matrix is

Ai = conj(dftmtx(n))/n

Example

In practice, the discrete Fourier transform is computed more efficiently and uses less memory with an FFT algorithm

x = 1:256;
y1 = fft(x);

than by using the Fourier transform matrix

n = length(x);
y2 = x*dftmtx(n);
norm(y1-y2)

ans = 

    2.0016e-09

Algorithm

dftmtx uses an outer product to generate the transform matrix.

`convmtx`	Convolution matrix.
`fft`	One-dimensional fast Fourier transform.