Bayesian robust optimal linear filters

Qualitatively, a robust filter maintains acceptable performance for signals statistically close to those for which it has been designed. For a parameterized family of signal models, we measure the robustness as the increase in error from applying the optimal filter for one state to the model for a different state. A Bayesian approach results from assuming that the state space possesses a probability distribution. In this case, the mean robustness for a state is the expected value of the error increase when the optimal filter for the given state is applied over all states of nature. A maximally robust state is one whose mean robustness is minimal. This paper treats robustness for optimal linear filters, in which case optimality depends on second-order statistics. Formulation of robustness is achieved by placing the matter into the context of canonical representation of random functions. Specifically, we use the representation of the optimal linear filter in terms of the cross-correlation between the signal to be estimated and the white-noise expansion of the observed signal. The general robustness formulation is reduced in particular cases, such as for linear degradation models and wide-sense stationary processes. For wide-sense stationary processes, robustness becomes a function of the power spectral densities. A maximally robust state depends on both statistical characteristics of the model and the distribution of the state vector. By incorporating the characteristics and state distribution into filter design, one can define a global filter that has good performance across all states.