Kalman filtering in triplet Markov chains

Let x = {x(n)}(n is an element of IN) be a hidden process, y = {y(n))(n is an element of IN) an observed process, and r = {rn(}n is an element of IN) some additional process. We assume that t = (x, r, y) is a (so-called "Triplet") vector Markov chain (TMC). We first show that the linear TMC model encompasses and generalizes, among other models, the classical state-space systems with colored process and/or measurement noise(s). We next propose restoration Kalman-like filters for arbitrary linear Gaussian (LG) TMC.