Application of Spectral Factorization to the Linear Discretetime Fixed-Point Smoothing Problem

An explicit expression for the minimum variance steady state fixed-point smoothing estimate of the output of linear, discretetime invariant system is obtained in terms of the measurement spectral factor. The filtered estimate of the states of the system is first derived by finding the spectral factor of the power density matrix of the measurement signal. The z-transform of the time-varying gain matrix, which produces the optimal smoothing estimate by multiplying the innovations process of the Kalman filter, is also obtained in terms of this factor. The results are easily extended to cases with colored measurement and driving noise signals and they are particularly simple to apply in the single-input single-output case.