# Kalman Filter with single input?

• Is a Kalman Filter still a Kalman filter if there are no inputs that explicitly predict the next state of the system?

• For instance, if I have a signal where I know it has characteristics of a first order dynamic system and I have a filter configuration that can in effect discount or ignore changes it considers unlikely is this essentially a Kalman filter without explicit state estimation?

I have tried to follow the derivation for the Kalman filter and I understand that you use some prior knowledge or knowledge of how other system elements may effect your output and so you predict the next state.

• Is a sufficient prior to be able to say

this signal is from a 1st order dynamic system with a time constant of $\tau_x$?

## 2 Answers

The Kalman filter is the least-squares optimal solution for the states of a set of linear dynamic equations under a Gaussian additive noise assumption. Hence, if you have an unforced linear first-order differential equation you can you use the Kalman filter to estimate the state of this differential equation. I'm not sure it will be very useful though, since the convergence rate of the observer has to be much faster than the dynamics of the first-order system in order to capture most of the transient response, which is the only response in an unforced system, and to converge if the system is unstable (unless you know the initial state, which is rare). High gain amplifies noise, so you might end up with a poor estimate. For such a simple system you might want to use bandwidth-limited numerical differentiation instead to find the inverse of the filter/system to find the state.

A first order dynamic system has a transfer function similar to $$T(s)=\frac{b}{s+a}$$ which can be converted to a discrete transfer function $T(z)$ easily by the bilinear transform, for example. From $T(z)$, it is straight forward to get the difference equation you need to use in the Kalman filter. The coefficients $a$ and $b$ can even be estimated in the same filter.

That means, your assumption from the question, that the behavior is first order, gives you directly the equation you need to do a good filter.