Gaussian Random Process

Question

I have a random process, $$Y_i=Bf_i+W_i,\quad i=0,\ldots,N$$

The RVs $B,W_1,\ldots,W_n$ are i.i.d., with $B\thicksim\mathcal{N}(0,\sigma_b^2)$ and $W_i\thicksim\mathcal{N}(0,\sigma_w^2)$. The function $f_i$ is a known deterministic signal.

I am trying to determine the joint Gaussian for $Y_i$. I can see that $\mu_{\mathbf{y}}$ is $\mathbf{0}$, but I cannot get straight in my mind what the covariance matrix, $\mathbf{\Sigma}$ is....

... I am anticipating covariance components in the non-diagonal elements of $\mathbf{\Sigma}$ because if $B$ is slightly higher, then $Y_i$ will shift higher and $\textit{vice versa}$.

Answer 1

Since $E(Y_i) =0$ the covariance matrix $\Sigma_{ij} = E(Y_i Y_j)$. Writing explicitly: $E(Y_i Y_j) = E(B^2 f_i f_j + W_i W_j +B W_i f_i +B W_j f_j)$.

Note that $B,W_j$ are uncorrelated, leaving us with $E(B^2 f_i f_j + W_i W_j)$. Now consider the case $i=j$ versus $i\neq j$ and also recall that $f$ are deterministic. Can you take it from here?