Kalman equivalent lowpass filter

Equivalent lowpass filter of the Kalman in steady-state

A kalman filter based on a constant velocity model and constant process noise reaches is steady-state after few samples. In steady-state the actual position estimate of the kalman gives the same performances of a zero+two-pole lowpass filter applied on the measured position, while the velocity estimate is the result of a derivative plus lowpass filter of the measured position. The process noise value changes the bandwidth of the filters. By applying the Z-transform to the constant velocity kalman filter equations you can find the equivalent filters transfer functions.
This sample shows the equivalent filters equations and their Bode diagrams. It also shows that after few seconds the results of the kalman filter and the equivalent filter are the same.
You can reach similar results for a kalman filter based on a constant acceleration model.