DFT matrix

The N-point Discrete Fourier Transform (DFT) can be expressed as a matrix multiply. with an N by N matrix, as follows:

X = W x

where x is the original input signal, and X is the DFT of the signal.

Examples of DFT matrices

The two point DFT is a simple case, in which the first entry is the DC (sum) and the second entry is the AC (difference).

${\begin{bmatrix}1&1\\1&-1\end{bmatrix}}/{\sqrt {2}}$

[1 1

0 -1]/sqrt(2).

The three point DFT has a special significance, e.g. as Symmetrical Components Transform (SCT) of Fortescue's 1918 paper, that defines three phase balance, i.e. the 3-DFT breaks a signal up into a DC component, as well as two AC components, one going clockwise, and the other going counter clockwise.

The four point DFT matrix is as follows: [1 1 1 1

1 -i -1  i
1 -1  1 -1
1  i -1 -i]/2.

Unitary transform

The DFT is (or can be, through appropriate selection of scaling) a unitary transform, i.e. one that preserves energy. The appropriate choice of scaling is 1/sqrt(N), so that the energy in the physical domain will be the same as the energy in the Fourier domain, i.e. to satisfy Parseval's theorem.