I am studying the role of an auto-correlation matrix for random signals and the difference of energy between a lag 0 and lag 1 matrix.
Consider a complex input signal $x(k)=[x1,x2]^T$ and $x(k-1)=[x0,x1]^T$, as column vectors with auto-correlation matrix of lag 0, $R(0)$, with diagonal $r(0)$ and auto-correlation matrix of lag 1 $R(1)$, with diagonal $r(1)$.
where $R(0)=E[x(k)x^H(k)]$ and $R(1)=E[x(k)x^H(k-1)]$
- what is the relationship between the energy resulting from $R(0)$ and $R(1)$? can we thus say that eigenvalue max of $R(0)$ is greater then that of $R(1)$
- and the difference in diagonal entries $r(0)$ and $r(1)$?
given that $r(0)$ is always positive real and $r(1)$ can be complex.
- What can we comment of the real and imaginary part of the $r(1)$ entry?
and where does the energy resides i.e. real portion or imaginary?
Additionally to a wide sense stationary signal $x(k)$ can we assume that $r(1)$ is approximately equal to $r(0)$?
Thus if we have $B = R(0) + R(1)$, we know that $R(0)$ is positive definite, symmetric and hermitian however $R(1)$ is not. Can we say that eigenvalue max of $B$ is eigenvalue max of $R(0)$ + the norm of that of $R(1)$ assuming its complex?
My apologies for the successive questions however I am trying to understand the differences between the auto-correlation matrix at lag 0 and lag 1 in terms of eigenvalues and energy because I have to work with a matrix $B = R(0) + R(1)$.