Optimal Filtering of a Gaussian Signal Against a Background of Almost Gaussian Noise

An asymptotic expression as $\varepsilon\to 0$ is derived for the mean-square error of the optimal nonlinear filtering of a discrete-time stationary Gaussian process $X=\{X_j\}$ from the observations $Y=\{Y_j\}$ under the assumption that the observed process $Y$ is the sum $Y_j=X_j+N_j+\varepsilon Z_j, j=0,\pm 1,ј$, where the stationary processes $X=\{X_j\}$, $N=\{N_j\}$, and $Z=\{Z_j\}$ are mutually independent and, moreover, $N$ and $X$ are Gaussian processes having spectral densities and $Z$ is an entropy-regular second-order process. It is also shown that the optimal linear filter reconstructing the signal $X$ from the observations $X+N$ (i.e., when the weak additional noise $\varepsilon Z$ is missing ) is asymptotically optimal. If $\varepsilon Z$ is an entropy-singular process, then the mean-square error of the optimal filtering does not depend on $Z$ ($\{Z_j\}$ can be correctly reconstructed from the observations $\{Y_j\}$).