Kalman Filter - Deriving state transition function
Hi Matthias La: I hate to be critical but the first example at the link you provided is actually quite poor because they end up using exponential smoothing (their update for $\hat{x}$ can re-written as an exponential smoothing update) for the update of the state, $x_t$, without deriving it. The way to derive the exponential smoothing update for the state (They also come up with a heuristic formula for the exponential smoothing factor, $\alpha_{n}$, of $\frac{1}{n}$ which is not optimal. It actually ends up being a function of the SNR) is to write the whole system in the following way:
$y_t = x_t + \epsilon_t $        # Measurement Equation
$x_t = x_{t-1} + \gamma_t$         # State Equation
where $\epsilon_t$ and $\gamma_t$ are independent normal error terms.
With the framework above, if you derive the updating equations for the kalman filter, you will obtain the exponential smoothing update for the state ( and $y_t = \hat{x_t}$ as update for the measurement). In the statistics-econometrics literature, the KF setup above is called a &quot;random walk + noise&quot; model.  I'm not claiming that the  link is completely useless but, as far as an intro to KF, that is not a good first example because they pull things out of the sky and don't explain where they come from. That can be confusing when going through the later examples.
I hope the setup above helps to show how  to obtain the ES update. Oh, I recommend below for a dense look at the KF. Harvey has a terse style which can be difficult at first (it was for me) but, in the end ( after a few reads:) ), its hits you that he actually provides a very nice explanation. I've read various parts of it more than a few times.
https://www.amazon.com/Forecasting-Structural-Time-Harvey/dp/0521405734/ref=sr_1_1?dchild=1&amp;keywords=Andrew+Harvey+Kalman+Filter&amp;qid=1615252088&amp;sr=8-1
EDIT: I didn't mention it above but I remembered later that you'll need a method for estimating the measurement variance and the state variance if you want to run the updating equations.  Harvey uses prediction error decomposition but my impression is that dsp people have a different way of estimating them. Prediction error decomp is the only method I'm familiar with but you may know of a better one. If you do and can explain it to me, it's appreciated.