Unified forms for Kalman and finite impulse response filtering and smoothing

The Kalman filter and smoother are optimal state estimators under certain conditions. The Kalman filter is typically presented in a predictor/corrector format, but the Kalman smoother has never been derived in that format. We derive the Kalman smoother in a predictor/corrector format, thus providing a unified form for the Kalman filter and smoother. We also discuss unbiased finite impulse response (UFIR) filters and smoothers, which can provide a suboptimal but robust alternative to Kalman estimators. We derive two unified forms for UFIR filters and smoothers, and we derive lower and upper bounds for their estimation error covariances.