Recursive filtering and smoothing for reciprocal Gaussian processes with Dirichlet boundary conditions

The minimum mean square error (MMSE) estimation problem for a discrete-index reciprocal Gaussian process impaired by additive white Gaussian noise is completely solved in the general case of noisily observed Dirichlet random boundary conditions. Finite sets of recursive equations are obtained for the computation of the filtered sequence and of the fixed-point, fixed-interval, and fixed-lag smoothed sequences. Recursive expressions are also given for the MMSE performance of the presented estimators. Simulation results confirm the validity of the performance analysis.