How do you "split apart" the output of a linear estimator? Let's say I have a true signal $x$ which is corrupted by noise $n$ such that:
$y = x + n$
I am attempting to use a Weiner filter to estimate the true signal $x$. This estimator is of the form:
$\hat{x} = Gy$
Now let's say I keep the original equation and variables and add a known input bias $c$
$y + c = x + c + n$
The Weiner filter estimator would now take on the form
$\widehat{x+c} = G(y + c)$
The issue with this form is that I am now estimating $x + c$ instead of just $x$. Assuming I know the input bias value $c$, is it &quot;okay&quot; to simply break up the estimated output like the following?
$\widehat{x} = G(y + c) - c$
My confusion primarily comes about breaking up the output of an estimator. I assume because the Weiner filter is linear, this is okay.
There is nothing wrong about subtracting the bias. For the Weiner filter, the operation is a matrix multiply, there are no non-linear operations taking place which is reason everything works out. There are two possible ways to do it:

Subtract the bias before the filter. In this case you'll have a Weiner filter, $G_1$, which will produce $\hat{x}$ given $x+n$.

\begin{align}
\hat{x}&amp;=G_1(y-c)
\end{align}

Subtract the bias after the filter. In this case you'll have a Weiner filter, $G_2$, which will produce $\widehat{x+c}$ given $x+c+n$ (this is the case you presented).

\begin{align}
\widehat{x+c}&amp;=G_2(y+c)\\
\hat{x}&amp;=\widehat{x+c}-c
\end{align}
Note: Be careful with the notation of $G$. The Weiner filter is constructed using the data so the filter constructed using the $x+n$ data will be different from the filter constructed using the $x+c+n$ data.