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 $\tau_x$?

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.