The autocorrelation matrix is used in various digital signal processing algorithms. It consists of elements of the discrete autocorrelation function, arranged in the following manner:

This is clearly a Hermitian matrix and a Toeplitz matrix

Toeplitz matrix

In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant...

. Furthermore, if is a real valued function, then it is a circulant matrix

Circulant matrix

In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector. In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence...

 since . Finally if is wide-sense stationary then its autocorrelation matrix will be nonnegative definite.

The autocovariance matrix is related to the autocorrelation matrix as follows:


Where is a vector giving the mean of signal at each index of time.