In the estimation problems, rather than the white observation noise, there are cases where the observation noise is modeled by the colored noise process. In the observation equation, the observed value y(k) is given as a sum of the signal z(k)=Hx(k) and the colored observation noise v_c(k). In this paper, the observation equation is converted to the new observation equation for the white observation noise. In accordance with the observation equation for the white observation noise, this paper proposes new RLS Wiener estimation algorithms for the fixed-point smoothing and filtering estimates in linear discrete-time wide-sense stationary stochastic systems. The RLS Wiener estimators require 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 x(k); (d) the system matrix for the colored observation noise v_c(k); (e) the variance of the colored observation noise.

This paper presents the new algorithm of the recursive least-squares (RLS) Wiener fixed-point smoother and filter based on the randomly delayed observed values by one sampling time in linear discrete-time wide-sense stationary stochastic systems. The observed value y(k) consists of the observed value y¯(k-1) with the probability p(k) and of y¯(k) with the probability 1-p(k). It is assumed that the delayed measurements are characterized by Bernoulli random variables. The observation y¯(k) is given as the sum of the signal z(k)=Hx(k) and the white observation noise v(k). The RLS Wiener estimators use 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 x(k); (d) the delayed probability p(k); (e) the variance of white observation noise v(k); (f) the input noise variance of the state equation for the augmented vector v¯(k) related with the observation noise.

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.