I am currently busy with least cost system identification developed by Michel Gevers. For this purpose, I started with the following paper: "Design of least costly identification experiments - The main philosophy accompanied by illustrative examples.". In this paper, I found an expression for the covariance matrix:
$P_\theta = \frac{\sigma_e}{N} (E(\psi(t,\theta_0) \psi(t,\theta_0)^T ))^{-1}$
This was applied to an example system:
$\mathcal{S}:y(t)=\frac{3.6z^{-1}}{1-0.7z^{-1}}u(t) + (1-0.9z^{-1}) e(t)$
In this system, e(t) is zero mean white noise with variance 1. It is now easy to find that the expression $E(\psi(t,\theta_0) \psi(t,\theta_0)^T )$ becomes:
$E(\psi(t,\theta_0) \psi(t,\theta_0)^T ) = E[(\frac{z}{(z-0.9)(z-0.7)}u)^2]+E[(\frac{z}{(z-0.9)^2}y)^2]+E[(\frac{3.6z}{(z-0.9)(z-0.7)^2}u)^2]$
My question is now on how to find the expected value of filter. Let's take the first expression for example $E[(\frac{z}{(z-0.9)(z-0.7)}u)^2]$. Let's suppose that the input is a zero-mean white noise signal with variance one. Then in this case, I think that the earlier expression should boil down to the following:
$E[(\frac{z}{(z-0.9)(z-0.7)}u)^2] = \mu_F \cdot E[u^2]= \mu_F \cdot 1 = \mu_F$, where $\mu_F$ is the mean of the filter. Is this thinking correct? Does this also apply when I have a single sinusoid for example?