Discrete universal filtering via hidden Markov modelling

We consider the discrete universal filtering problem, where the components of a discrete signal emitted by an unknown source and corrupted by a known DMC are to be causally estimated. We derive a family of filters which we show to be universally asymptotically optimal in the sense of achieving the optimum filtering performance when the clean signal is stationary, ergodic, and satisfies an additional mild positivity condition. Our schemes are based on approximating the noisy signal by a hidden Markov process (HMP) via maximum-likelihood (ML) estimation, followed by use of the well-known forward recursions for HMP state estimation. We show that as the data length increases, and as the number of states in the HMP approximation increases, our family of filters attain the performance of the optimal distribution-dependent filter.