Comparing frequency domain, optimal, and asymptotic filtering: A tutorial

This paper studies some connections between the main results of the Kalman-Bucy stochastic approach to filtering problems based mainly on linear stochastic estimation theory and emphasizing the optimality aspects of the achieved results and the classical deterministic frequency domain linear filters (such as Chebyshev, Butterworth, Bessel, etc.). A new non-stochastic but not necessarily deterministic (possibly nonlinear) alternative approach to signal filtering based mainly on concepts of signal power, signal energy, and an equivalence relation plays a dominant role in the presentation. Causality, error invariance, and especially error convergence properties are the most important and fundamental features of resulting filters. Therefore, it is natural to call them asymptotic filters. Although error convergence aspects are emphasized in the approach, it is shown that introducing the signal power as the quantitative measure of signal energy dissipation makes it possible to achieve reasonable results from the optimality point of view as well. The authors show that the notion of the asymptotic filter can be used as a proper tool for unifying stochastic and non-stochastic, linear and nonlinear approaches to signal filtering.