RLS Wiener Smoother for Colored Observation Noise with Relation to Innovation Theory in Linear Discrete-Time Stochastic Systems

Almost estimators are designed for the white observation noise. In the estimation problems, rather than the white observation noise, there might be actual cases where the observation noise is colored. This paper, from the viewpoint of the innovation theory, based on the recursive least-squares (RLS) Wiener fixed-point smoother and filter for the colored observation noise, newly proposes the RLS Wiener fixed-interval smoothing algorithm in linear discrete-time wide-sense stationary stochastic systems. The observation y(k) is given as the sum of the signal z(k)=Hx(k) and the colored observation noise (v_c)(k). The RLS Wiener fixed-interval smoother uses the following information: (a) the system matrix for the state vector x(k); (b) the observation matrix H; (c) the variance of the state vector; (d) the system matrix for the colored observation noise (v_c)(k); (e) the variance of the colored observation noise; (f) the input noise variance in the state equation for the colored observation noise.