Optimal gains of FIR estimators for a class of discrete-time state-space models

The gain of an optimal finite impulse response (FIR) filter of the system k-state is derived and analyzed for discrete real-time state-space models with jointly independent zero-mean noises of arbitrary distributions and covariances. Several special cases are considered, and the gain is modified accordingly. It is stated that if the model is deterministic or slowly changing with time and such that an averaging horizon is large, N >> 1, then a relatively simple unbiased FIR filter produced by the deadbeat constraint yields virtually an optimal estimate. An application is given to the crystal clock time error estimation via GPS-based measurements with temporary uncertainty in the presence of sawtooth noise.