I am trying to use the Kalman filter (the scalar version) to estimate the steady state of a set measurements which is a random process. I have used a constant dynamic model as the state equation,
$$ x(t+1) = x(t) + w(t) $$
Where $ w(t) \sim N(0, Q) $.
The measurement equation
$$ y(t) = x(t) + v(t) $$
Where $ v(t) \sim N (0, R) $.
The measurement noise $ v(t) $ is taken as the uncertainty of the instrument and R is a known value with a assumed Gaussain distribution. The process error, $w(t) = x(t+1) - x(t)$ has high values as a constant variance (Calculated using the total error variance) or a time varying (Taking last 2, 5 readings as the set f sample). However, the state reaches the steady state when $ Q $ is very small or $ Q = 0 $.
I have seen several other theoretical examples uses $ Q = 0 $ and reaches the steady state. But I am working with actual data. Also I have seen some other posts explaining the similar behavior. Can some explain this behavior with reasons, please?