When do nonlinear filters achieve maximal accuracy?

TL;DR

This paper characterizes when nonlinear filters for ergodic signals in white noise reach maximal accuracy, with explicit conditions for finite states and connections to classical Gaussian results.

Contribution

It provides a general systems theoretic framework for understanding maximal accuracy in nonlinear filtering, including explicit conditions for finite state spaces.

Findings

Necessary and sufficient conditions for maximal accuracy in finite state spaces

Connection of results to classical Gaussian filtering theory

General characterization using systems theoretic notions

Abstract

The nonlinear filter for an ergodic signal observed in white noise is said to achieve maximal accuracy if the stationary filtering error vanishes as the signal to noise ratio diverges. We give a general characterization of the maximal accuracy property in terms of various systems theoretic notions. When the signal state space is a finite set explicit necessary and sufficient conditions are obtained, while the linear Gaussian case reduces to a classic result of Kwakernaak and Sivan (1972).