Studies of discrete-time unbiased FIR filters of polynomial state-space models

We study an unbiased finite impulse response (FIR) filter in applications to discrete-time state space models with polynomial representation of the states. The unique l-degree polynomial FIR filter gain and the estimate variance are found for a general case. The noise power gain (NG) is derived for white Gaussian noises in the model and in the measurement. The filter does not involve any knowledge about noise in the algorithm. It is unstable at short horizons, 2 ≤ N ≤ l, and inefficient (NG exceeds unity) in the narrow range l <; N ≤ N_b, where N_b is ascertained by the cross-components in the measurement matrix C. With N ≫ N_b, the filter NG poorly depends on C and fits the asymptotic function (l+1)²/N. With very large N ≫> 1, the estimate noise becomes negligible and the filter thus optimal in the sense of zero bias and zero noise. An example is given for a two-state system.