Backward optimal FIR filtering and recursive forms for discrete LTV processes

Abstract Backward filtering is a useful tool to compute initial values in finite horizon signal processing. To make it optimally, a backward a posteriori optimal finite impulse response (B-OFIR) state estimator (filter) is proposed for discrete time-varying linear processes. The B-OFIR filter is derived in a discrete convolution-based batch form and represented with a fast iterative algorithm using recursions. The performance of the B-OFIR filter is compared to the forward OFIR filter (F-OFIR), KF, and unbiased FIR filter (UFIR). Simulations conducted based on a two-state tracking model and one degree-of-freedom torsion load system show that the B-OFIR filter is statistically equivalent to the F-OFIR filter, but operates with noise samples ordered back in time and therefore produces a bit different estimates.