FIR filters and recursive forms for continuous time-invariant state-space models

An FIR (finite impulse response) filter and an FIR smoother are introduced for continuous time-invariant state-space models. It is shown in this note that finite impulse responses of the FIR filter and smoother can be easily determined by solving a simple Riccati-type matrix differential equation on a finite interval. Especially for systems with stationary processes, finite impulse responses of the FIR filter and smoother become time-invariant and can be computed from simpler equations. For fast computational purposes, recursive forms of the FIR filter and smoother are derived by using adjoint variables. In this case all gains for recursive forms are shown to be constant.