I'm wondering if in some situations, a Kalman filter simplifies to equations that are easy to implement? What do degenerate forms of it look like?

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That's a really great question and, in fact, I'm writing another post about this!

Generally the Kalman filter is simplest with 1-dimensional systems, because the linear algebra just becomes... ordinary algebra. The more dimensions in the system, the more covariance terms and so on that you have to think about, so it gets more complicated.

But -- and this is a sneak preview for my next article -- even for higher dimensional systems there's a simpler version of the Kalman filter that you can usually get away with.

As long as the various matrices (state transition F, measurement noise R, process noise Q, etc) are constant -- which they very often are -- then the Kalman gain, K, gradually converges to a steady-state value as the filter runs. Using a steady-state Kalman gain makes the filter much simpler to implement, especially on limited hardware, because it has constant coefficients and you can skip all the matrix inversions. And that filter it converges to is actually the optimal discrete-time Wiener filter for that system, which is pretty neat!

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