Least-Squares Finite Impulse Response Smoother Estimating Signal at Start Time of Fixed Interval in Linear Discrete-Time Stochastic Systems

Abstract

This paper proposes the least-squares (LS) finite impulse response (FIR) smoother estimating the signal at the start time of the fixed interval in the FIR smoother and filter in linear discrete-time wide-sense stationary stochastic systems. It is assumed that the signal is observed with additional white noise and is uncorrelated with the observation noise. The LS FIR smoothing estimate is given as a linear convolution of the impulse response function data and the observed values. By solving the simultaneous linear equations transformed from the Wiener-Hopf equation, we can calculate the finite number of impulse response function data. The necessary information in the LS FIR smoothing algorithm is the auto-covariance function data of the signal process, K(i), 1<=i<=L, for the finite interval  the variance of the signal process and the variance of the observation noise process. The auto-covariance function data of the signal process, K(i), 1<=i<=L, are equal to the auto-covariance function data of the observation process, Ky(i), 1<=i<=L. This paper also proposes the Levinson-Durbin algorithm, which needs less arithmetic operations than the Gauss-Jordan elimination method used in the matrix inversion, for the impulse response function data in the LS FIR smoothing problem. The proposed LS FIR smoother estimating the signal at the start time of the fixed interval and filter are numerically compared.