1. The agenda: 1. The Kalman theory 2. Break for 20 minuts 3. More theory 4. Simulation of the filter. 5. Further discussion and exercises The Scalar Kalman filter

2. Defenitions: S(n) = a(n)S(n-1) + W(n) a(n) is the projection of the last value of the sequence S to the next value of the sequence S. a(n) is not considered further, and is therefore equal one. X(n)=S(n) + V(n) W(n) and V(n) are both white gausian noise and they are orthorgonal.

3. The Scalar Kalman filter Since a(n) = 1 for all n

4. The Scalar Kalman filter Basic Algorithm:

5. The Scalar Kalman filter

6. The P(n) is defined: To obtain the optimal estimator, the P(n) must be minimized. The recursive estimation of P(n) can be found to be:

7. Solve following to obtain optimal estimator:

8. Break for 20 minutes

9. Solve following to obtain optimal estimator:

10. The Scalar Kalman filter Algorithm: Initialize n = 1 P(1) = some value (0) (1) = some value (0) 1.start loop Get n = n +1 go back to 1.

11. The Scalar Kalman filter Simulations - Normal (002)

12. The Scalar Kalman filter When to use a Kalman filter, compared to a LP filter? What can we do if the frequency that the kalman filter has to work with increses?