Optimal and unbiased FIR filtering in discrete time state space with smoothing and predictive properties

We address p-shift finite impulse response optimal (OFIR) and unbiased (UFIR) algorithms for predictive filtering (p > 0), filtering (p = 0), and smoothing filtering (p < 0) at a discrete point n over N neighboring points. The algorithms were designed for linear time-invariant state-space signal models with white Gaussian noise. The OFIR filter self-determines the initial mean square state function by solving the discrete algebraic Riccati equation. The UFIR one represented both in the batch and iterative Kalman-like forms does not require the noise covariances and initial errors. An example of applications is given for smoothing and predictive filtering of a two-state polynomial model. Based upon this example, we show that exact optimality is redundant when N ≫ 1 and still a nice suboptimal estimate can fairly be provided with a UFIR filter at a much lower cost.