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Suppose there are two vector signals $x$, $z$. The observer 1 receives a linear version of $x$ plus Gaussian noise. Observer 2 receives a linear sum of both $x$ and $z$ plus Gaussian noise as shown below; $$y_1(t)=A_1(t)x(t)+n_1(t),$$ $$y_2(t)=A_2(t)x(t)+B_2(t)z(t)+n_2(t).$$ But observer 2 is only interested in estimating $z$ so ideally he would like to cancel the whole term $A_2x$. Furthermore $A_2$ is changing slowly over time. Slowly in the sense the changes are correlated over time. Suppose observer 1 can provide his estimates of $x$ say $x_1$ to observer 2. Is there any way for observer 2 to use that information to better estimate $z$? The matrices $A_1,A_2,B_2$ are independent.

Edit: Suppose $A_2(t+1)= \sum_{n=0}^{n=t}\alpha_nA_2(n) + z_t(t+1)$ where $z_(t)$ are independent Gaussian noise.

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    $\begingroup$ (1) One would think that the answer would depend on how Observer 1 is estimating $x(t)$ and how good those estimates are. (2) It seems you must assume more about $A_1, A_2$, and $B_2$, for otherwise the $x(t)$ and $z(t)$ will not be identifiable. In particular, you need some specific quantitative assumptions; "changing slowly over time" is not really amenable to useful analysis. $\endgroup$ – whuber Jan 17 '14 at 15:42
  • $\begingroup$ @whuber I created this problem out of a practical situation I have in my experiment. Could you please suggest such assumptions? I am not sure how to put "changing slowly" into sold equation. What I mean is $A_2(t)$ and $A_2(t+1)$ are different but are correlated. I will include my idea in a edit of the question. Thanks. $\endgroup$ – triomphe Jan 17 '14 at 16:31
  • $\begingroup$ Probably not, at least not using the formulas given. $\endgroup$ – Carl Nov 12 '18 at 20:18