Assuming all “real” number, consider the detection problem of $x$ given $y_1$ and $y_2$ below
$y_1= x + n_1$
$y_2= n_1 + n_2$
where $x$ is either $+A$ or $-A$ with equal probability, and $n_1$ and $n_2$ are i.i.d Gaussian random variables with zero mean and variance of $\sigma^2$.
Is observation of $y_2$ helpful for detection of $x$? whether $x = A$ or $x = - A$?
From $y_1$ one can see that it’s a simple BPSK signal detection. However, In my opinion the new observation in $y_2$ does not add any new information, since $n_2$ and $n_1$ are i.i.d Normal gaussian distributions. One idea is for example to write the received signals as $y=y_1 + \alpha y_2=x+(1+\alpha)n_1+n_2$. But from my understanding, this linear combination is the same as detecting $x$ from $y_1$. Is my understanding correct?