Discrete Time $q$-Lag Maximum Likelihood FIR Smoothing and Iterative Recursive Algorithm

The finite impulse response (FIR) approach is known to be more robust than the Kalman approach. In this paper, we derive a batch
$q$
-lag maximum likelihood (ML) FIR smoother for full covariance matrices and represent it with an iterative algorithm using recursions for diagonal covariance matrices. It is shown that, under ideal conditions of fully known model, the ML FIR smoother occupies an intermediate place between the more accurate Rauch-Tung-Striebel (RTS) smoother and the less accurate unbiased FIR smoother. With uncertainties and errors in noise covariances, ML FIR smoothing is significantly superior to RTS smoothing. It is also shown experimentally that ML FIR smoothing is more robust than RTS smoothing against measurement outliers.