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Chapter 4:
Optimal Robust Filtering
It is often unrealistic to assume that the design model is known with certainty, so it is prudent to assume that the true model belongs to an uncertainty class {(Xu, Yu)}u∈U of models parameterized by a vector u belonging to a parameter set U, also called the uncertainty class, that is in one-to-one correspondence with the model class {(Xu, Yu)}u∈U. A robust filter is optimal in some sense over the uncertainty class. If we think in terms of a system, then u characterizes the state of the system. Owing to the one-to-one correspondence, we can use the terms “model” and “state” interchangeably. In the context of filtering, where an unobserved ideal signal is estimated based on an observed signal, for each u ∈ U there is an observed-ideal signal pair (Xu, Yu) and an optimal filter cu. Robust filter design goes back to the late 1970s, with robust Wiener filtering involving minimax optimality in regard to uncertain power spectra (Kuznetsov, 1976; Kassam and Lim, 1977; Poor, 1980; Vastola and Poor, 1984). Robust design was extended to nonlinear filters and placed into a Bayesian framework by assuming a prior probability distribution governing the uncertainty class, the aim being to find a filter with minimal expected error across the uncertainty class (Grigoryan and Dougherty, 1999). Only recently have fully optimal solutions been found in this framework (Dalton and Dougherty, 2014), and it is to this topic that we now turn.
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