Assume we have a linear state-space model: $$ z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1}+ w_{k}. $$
We are interested in filtering, i.e. we aim to estimate $E[x_{n}|z_{0}, \dots, z_{n}]$. If the measurement noise goes to zero, i.e. $var(v_{k}) \to 0$, then filtering does not give much to us and the filtered data is almost the same as original.
What is the opposite situation, when $var(w_{k}) \approx 0$ and $var(v_{k})$ is fixed? What would be the outcome of Kalman filter?