A New Approach to Linear Filtering and Prediction Based on High-Order Signal Models

—This research paper addresses the issue of filtering for signal models described by high-order vector difference equations (VDEs). The study is divided into two parts. The first part focuses on the filtering problem for a linear second-order VDE driven by white noise, while the second part extends these findings to high-order models of the same structure. The study develops a recursive equation for the filtered estimate based on the linear second-order model. The innovations approach is directly applied to the second-order model to derive a recursion for the filtered estimate. The resulting filter is defined as a second-order recursion that preserves the mathematical structure of the given model with innovations feedback loops. The study shows that the innovations satisfy a first-order recursion in terms of the filtered estimates and the measurements. The study formulates equations for the estimation of the filtered values and also determines the covariance matrices for the associated errors, based on the respective error values. This study presents the generalization of filtering results for high-order models of the same structure in the second part of the paper. The research considers a p th -order vector difference equation (VDE) model with additive white noise and a linear combination of the signal process with additive white noise as the observation process. The study develops a one-stage prediction estimator for the p th -order VDE signal model and presents the characterization of the innovations sequence in terms of the one-stage prediction estimates. The study also derives formulas for the estimator gains and demonstrates that the resulting estimator is a p th -order system that preserves the form of the given model with innovations feedback loops. The study further shows that the well-known Kalman filter is a special case of these findings.