Appendix A: Computation of $R_{y,M}(\theta,Q)$

The autocovariance y_n, r_y(m), equals the sum of the cross-covariances of the modes: $r_y(m)=\sum_{ij=1}^{L}\mbox{$E$} [x^{i}_nx^{j}_{n+m}]+\sigma^2\delta(m)=\sum_{ij=1}^{L}r_{ij}(m)+\sigma^2\delta(m)$.If i=j, r_ij(m) is the autocovariance of an autoregressive process and can be computed inverting the Yule Walker equation [Stoica & Moses 1997]:

$$\begin{eqnarray} &&\left(\begin{array} {ccc} 1 & a_1^i & a_2^i \\ a_1^i & 1+a_2^... ...\forall q\geq3,\quad r_{ii}(q)=-a_1^ir_{ii}(q-1)-a_2^ir_{ii}(q-2).\end{eqnarray}$$ (A1) (A2)

If $i\not=j$, multiplying Eq. (2) for k=i by x^j(n) and x^j(n+1) and multiplying Eq. (2) for k=j by xⁱ(n-q), $q\geq 2$,and taking the expectation, gives:

$$\begin{eqnarray} &&\left(\begin{array} {cccc} 1 & a_1^i & a_2^i&0 \\ 0& 1& a_1^i... ...forall q\geq 4,\quad r_{ij}(q)=-a_1^jr_{ij}(q-1)-a_2^jr_{ij}(q-2).\end{eqnarray}$$ (A3) (A4)

These equations show that Q_ij can be factored out in the cross-covariance matrix between xⁱ(n) and x^j(n) and then $R_{y,M}(\theta,Q)$ is a linear combination of the Q_ij.

This result allows a simple computation of the Cramér-Rao bound for the matrix Q using the simplified expression of the Fisher information matrix for a Gaussian distribution [Stoica & Moses 1997].