Recursive Filtering and Smoothing for Discrete Index Gaussian Reciprocal Processes

We study minimum mean square error (mmse) estimation problems for discrete index Gaussian reciprocal processes (Grp's) (or boundary valued processes) with Dirichlet boundary conditions. Our contributions are: 1) deriving first order white noise driven representations from second order correlated noise driven representations; 2) deriving Kalman like recursive filtering equations for discrete index Grp's; and 3) deriving recursive smoothing equations for discrete index Grp's. Unlike previous work, our approach uses forward and backwards recursive representations for the Grp and leads to lower dimensional recursive filters and smoothers.