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The problem of discrete universal filtering, in which the components of a discrete signal emitted by an unknown source and corrupted by a known discrete memoryless channel (DMC) are to be causally estimated, is considered. A family of filters are derived, and are shown 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 comprised of approximating the noisy signal using a hidden Markov process (HMP) via maximum-likelihood (ML) estimation, followed by the use of the forward recursions for HMP state estimation. It is shown that as the data length increases, and as the number of states in the HMP approximation increases, our family of filters attains the performance of the optimal distribution-dependent filter. An extension to the case of channels with memory is also established.