Upper and Lower Bounds and Asymptotics of Optimal Filtering Error of a Stationary Process with a Small Information Rate

Upper and lower bounds are obtained for the mean-square error of the optimal (nonlinear) filtering of a discrete-time stationary process $X=\{X_j\}$ from the observations $Y=\{Y_j\}$, where $Y=X+Z$ and $Z=\{Z_j\}$ is a sequence of i.i.d. random variables. These bounds are linear functions of the information rate $\overline I(X;Y)$. It is shown that the lower bound is asymptotically tight in the case where both $\overline I(X;Y)$ and the peak power of the signal $X$ tend to zero. The situations where $X_n$ is estimated from either the observations $\{Y_j, j\leq n-1\}$ or the observations $\{Y_j, j\leq n\}$ are both considered.