Unbiased FIR Filtering of Discrete-Time Polynomial State-Space Models

We address an unbiased finite impulse response (FIR) filter for 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 les N les l, and inefficient (NG exceeds unity) in the narrow range l < N les N
_b
, where N
_b
is ascertained by the cross-components in the measurement matrix C. With N GtN
_b
, the filter NG poorly depends on C and fits the asymptotic function (l +1)
²
/N . With very large N Gt>1, the estimate noise becomes negligible and the filter thus optimal in the sense of zero bias and zero noise. Having such properties, the proposed unbiased FIR filter fits well slowly changing with time models. An example is given for a two-state system.