# Small noise of state process and filtering

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?

• I think you have a mistake in your question. v_k is measurement noise. w_k is process noise. – Anton Jan 23 '20 at 13:14
• Dear @Anton, you are right! Thank you! I have edited. – ABK Jan 23 '20 at 19:27