Least-squares finite impulse response fixed-lag smoother and filter in linear discrete-time stochastic systems

摘要

This paper proposes the least-squares (LS) finite impulse response (FIR) fixed-lag smoother and filter in linear discrete-time wide-sense stationary stochastic systems. The FIR fixed-lag smoothing estimate is given as a linear convolution sum of the impulse response function and the observed values. It is assumed that the signal is observed with additional white noise, which is uncorrelated with the signal process. By solving the simultaneous linear equations transformed from the Wiener–Hopf equation, the optimal impulse response function is obtained. The necessary information of the LS FIR fixed-lag smoothing algorithm is the auto-covariance function of the signal process and the variance of the observation noise process. In particular, this paper proposes the Levinson–Durbin algorithm, which needs less amount of arithmetic operations than the Gauss–Jordan elimination method in the inverse of the Toeplitz matrix, for the optimal impulse response function. From the numerical simulation example, the proposed LS FIR fixed-lag smoother and filter are superior in estimation accuracy to the RLS Wiener FIR estimators.