I am looking at a basic radar problem. To skirt around sampling for the moment, I am "transmitting" a discrete signal, $$\mathbf{s}=\begin{bmatrix}f_1&f_2\end{bmatrix}^T$$
Assuming no attenuation, the signal is reflected from the target and picks up some noise so that, $$\mathbf{x}=\begin{bmatrix}f_2+W_2&f_1+W_1\end{bmatrix}^T,\qquad W_i\thicksim\mathcal{N}(0,\sigma_w^2)$$
The vectors $\mathbf{x}$ and $\mathbf{s}$ are correlated to yield, $$\mathbf{Y}=\begin{bmatrix}f_1^2+f_1W_1&2f_1f_2+f_2W_1+f_1W_2&f_2^2+f_2W_2\end{bmatrix}^T$$
The expected value of $Y_1$ for example is, \begin{align*} \mathrm{E}[Y_1]&=\mathrm{E}[f_1^2+f_1W_1]\\ &=f_1^2+f_1\,\mathrm{E}[W_1]\\ &=f_1^2 \end{align*}
The expected value for $\mathbf{Y}$ is, $$\mu_{\mathbf{Y}}=\begin{bmatrix}f_1^2&2f_1f_2&f_2^2\end{bmatrix}^T$$
The variance for $Y_i$ is solved according to, \begin{equation*} \mathrm{var}(Y_i)=\mathrm{E}\left[(Y_i-\mathrm{E}[Y_i])^2\right] \end{equation*}
I calculate the variance of $\mathbf{Y}$ and get, \begin{equation*} \mathrm{var}(\mathbf{Y})=\begin{bmatrix}f_1^2\sigma_w^2&(f_1^2+f_2^2)\sigma_w^2&f_2^2\sigma_w^2\end{bmatrix}^T \end{equation*}
I simulate this by creating several thousand realizations of $\mathbf{Y}=R_{\mathbf{SX}}(\tau)$ and then calculate the means and variances across all the $Y_1$, $Y_2$ and $Y_3$.
It all checks out nicely. But then I figured if I looked at, say, $\mathrm{covar}(Y_1,Y_2)$, I should get zero. But instead I get, $$\mathrm{covar}(Y_1,Y_2)=\mathrm{E}\left[(Y_1-\mathrm{E}[Y_1])(Y_2-\mathrm{E}[Y_2])\right]=f_1f_2\sigma_w^2$$
Could someone please give me a tip as how I should think about this?