What is the difference between Kalman filter algorithm and stationary Kalman filter algorithm? I want to compute the stationary Kalman filter algorithm but I haven't found any information about that algorithm ( not even the pseudo code ) so, I wonder what is the difference between the Kalman filter algorithm and the stationary Kalman filter algorithm and how I can derive it from the Kalman filter algorithm. Could someone point me out resources I can read about it?

I just found this post:

How to derive the stationary Kalman filter predictor?

and this document:

http://www.uh.edu/~bsorense/kalman.pdf on page 5.

Basically what I understand is that you calculate beforehand ( before the recursive process ) the Kalman gain and the state covariance matrix when the Kalman gain goes to infinite using these two equations:

$$\bar{P} = A\bar{P}A^T - A\bar{P}C^T ( C\bar{P}C^T + R )^{-1}C\bar{P}A^T + Q$$
$$\bar{K} = \bar{P}C^T(C\bar{P}C^T+R)^{-1}$$

and in the recursive function, this equation:

$$\hat{x}(t+1|t) = (A-A\bar{K}C)\hat{x}(t|t-1) + A\bar{K}y(t)$$

Is that correct?

Notes:

Yes, it is a question from a DSP book ( https://users.aalto.fi/~ssarkka/pub/cup_book_online_20131111.pdf excercise 4.6 on page 83 ) but there is no information in that book about the stationary Kalman filter algorithm.
I’m not sure what you mean by the stationary Kalman filter, but it seems to be what I would call the steady-state Kalman filter.

If that is the same thing, then you just solve for the Kalman gain at $t=\infty$ and apply the normal Kalman filter equations.

I’m not sure you mean “the Kalman gain goes to infinity” in a well-posed problem.