Linear Optimal FIR Estimation of Discrete Time-Invariant State-Space Models

This paper addresses a general p-shift linear optimal finite impulse response (FIR) estimator intended for solving universally the problems of filtering (p = 0), smoothing (p < 0), and prediction (p > 0) of discrete time-invariant models in state space. An optimal solution is found in the batch form with the initial mean square state function self-determined by solving the discrete algebraic Riccati equation. An unbiased solution represented both in the batch and recursive forms does not involve any knowledge about noise and initial state. The mean square errors in both the optimal and unbiased estimates are found via the noise power gain (NPG) and a recursive algorithm for fast computation of the NPG is supplied. Applications are given for FIR filtering with fixed, receding, and full averaging horizons.