Suppose I have two time series $A:=[0,0,0,4,5,6]$ and $B:=[1,2,3,4,5,6]$. I implement the following filter (with initial value at the first element):
$$ f(t) = f(t-1) + \alpha*(f(t)-f(t-1))$$
with $\alpha \in [0,1]$.
Is there any way I can use filtered values of $B$ to recover the filtered values of $A$? For example suppose I can store all historical filtered values of $B$ for any choice of $\alpha$ in memory, is it possible to reconstruct what the filtered values for $A$ would have been for some choice of $\alpha$?
Intuitively this would involve subtracting the some filtered value(s) of $B$ at time step $3$ from the filtered value(s) of $B$ at time step $6$, to recover an approximation of $A$.