Signal estimation for second-order vector difference equations

This note considers a linear estimation problem for a stochastic process viewed as the output signal of a linear second-order vector difference equation (VDE) driven by a white-noise input. An innovations approach is applied directly to develop the one-stage prediction estimator and associated error covariances. It is shown that the estimator can be expressed as a second-order recursion that preserves the mathematical structure of the given signal model with innovations feedback loops. It is also shown that the innovations can be computed through a first-order recursion in terms of one-stage prediction estimates and the measurements.