We consider the nonlinear filtering model with Ornstein-Uhlenbeck
process as noise and obtain an analogue of the Bayes' formula for
the filter. For this we need to consider a modified model, where the
instaneteneous effect $h(X_t)$ of the signal in the usual model is
replaced by $\xi_t^{\alpha} = \alpha \int _{(t- \frac{1}{\alpha})
\vee 0}^t h (X_u) \,du$, (where $\alpha$ is a large parameter).
This means that there is a lingering effect of the signal for a time
period $ \frac{1}{\alpha} $.
Further, we also show the filter with Ornstein-Uhlenbeck converges
to the usual filter in probability.