Author here. Yes, if the system has a high-pass behaviour, then the corresponding Kalman filter will converge to a high-pass filter, and so on. Non-low-pass-like behaviour can emerge from the prediction step, if the model is not unitary. If there's interest, I might write a followup article where a Kalman filter takes on a bandpass nature.

I read Gaussian and Cauchy distributions as inputs, but can you add arbitrary distributions other than those two examples, and expect the filter to yield artifacts from those distributions? Naively, I'm thinking of different coloured noise as inputs, and what kind of artifacts a Kalman filter would produce over them.

With multi modal distribution you can end up stuck in any of the modes or even stuck in a low likelihood region in between the modes.

I have seen it derived that it is equivalent to a matched filter, but I could not repeat said derivation.

I'd expect BPF behavior when estimating the rate of a noisy input (e.g. state: speed, observation: position), but I can't think of a process model that would be truly high-pass.

Ambient temperature effects on a photodiode.

I've used it to compensate for directional measurement bias.

If you pass a moving probe through a magnetic field your meaured results vary by direction (with or perpendicular to flux) and velocity.

If you want to measure a geomagnetic field using an aircraft, the heading matters.

To normalise in post processing you calibrate a Kalman filter by flying precessing butterfly wing patterns in a known relatively level flux area and then use that to remove the magnetic signature of the aircraft and heading from data collected over multiple headings and days.

( There are a few other twists - diurnal flux and induced field from the Earths GMF interactions needs to be isolated, etc ).

Conceptually, every step of applying a Kalman filter in the simplest case is just updating a Gaussian prior representing position with a new datapoint sampled with (assumed) Gaussian error. The result is equivalent to convolving a Gaussian representing position with a Gaussian representing noise.

In the frequency domain, that's equivalent to multiplying one Gaussian by another one with zero mean, which will always put higher weight on low frequencies. No matter what the gain is in the Kalman filter, that'll still be true as far as I can tell. As the gain varies, the cutoff within the low-pass filter will change though.

It gets harder to analyze when you start using non-linear models to update position though. Generally, I think the same logic applies.

Yes, it can easily correspond to a BPF, for example when trying to track a sinusoidal signal, say the AC line frequency. Then you can be reasonably certain that it will be near 50 Hz (or 60 Hz), and the resulting Kalman filter will be a sharp band-pass filter centered around that (when neglecting higher harmonics).

it's a noise filter, so it wouldn't be high pass or band path. it's approximately low pass filter when it hasn't converged precisely to a low pass filter.