The autocorrelation of a WSS process as a linear operator

If I'm given a autocorrelation matrix of a WSS process what interpretation should I put on the resulting vector.

More concretely the matrix takes the form

$\begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ x_2 & x_1 & \ldots & x_{n-1} \\ \ldots & \\ x_n & x_2 & \ldots & x_1 \end{bmatrix}$

and I multiply it with a vector defined on the same space, what interpretation should I give $y = Mv$?

Note, any matrix which has constant diagonals is ok for my purposes. So circular matrices are ok. I.e the matrix could describe the circular autocorrelation function.

For example, the rvs $\{X_i\}$ could be independent Gaussian on $\mathbb{R}^n$, with $v$ being some vector of length n in $\mathbb{R}^n$ also.

• Are you sure your matrix is correct? I think you've switched second and last rows. – Phonon Jun 19 '14 at 16:47
• The matrix is supposed to be circulant, so shifting the 1st row right one spot (mod n) produces the second. Doing this n times puts $x_1$ at the end and $x_2$ at the beginning. – Tom Kealy Jun 19 '14 at 16:52
• No. That's circular autocorrelation matrix, assuming you're computing it using circular convolution. In general, it is Hermitian and Toeplitz. – Phonon Jun 19 '14 at 16:54
• For the purposes of what I'm trying to answer a Toeplitz matrix is fine (any matrix with constant diagonals). – Tom Kealy Jun 19 '14 at 16:57