On the optimality of symbol-by-symbol filtering and denoising

We consider the problem of optimally recovering a finite-alphabet discrete-time stochastic process {X_t} from its noise-corrupted observation process {Z_t}. In general, the optimal estimate of X_t will depend on all the components of {Z_t} on which it can be based. We characterize nontrivial situations (i.e., beyond the case where (X_t,Z_t) are independent) for which optimum performance is attained using "symbol-by-symbol" operations (a.k.a. "singlet decoding"), meaning that the optimum estimate of X_t depends solely on Z_t. For the case where {X_t} is a stationary binary Markov process corrupted by a memoryless channel, we characterize the necessary and sufficient condition for optimality of symbol-by-symbol operations, both for the filtering problem (where the estimate of X_t is allowed to depend only on {Z_t\´}_t\´≤t) and the denoising problem (where the estimate of X_t is allowed dependence on the entire noisy process). It is then illustrated how our approach, which consists of characterizing the support of the conditional distribution of the noise-free symbol given the observations, can be used for characterizing the entropy rate of the binary Markov process corrupted by the binary-symmetric channel (BSC) in various asymptotic regimes. For general noise-free processes (not necessarily Markov), general noise processes (not necessarily memoryless), and general index sets (random fields) we obtain an easily verifiable sufficient condition for the optimality of symbol-by-symbol operations and illustrate its use in a few special cases. For example, for binary processes corrupted by a BSC, we establish, under mild conditions, the existence of a δ^*>0 such that the "say-what-you-see" scheme is optimal provided the channel crossover probability is less than δ^*. Finally, we show how for the case of a memoryless channel the large deviations (LD) performance of a symbol-by-symbol filter is easy to obtain, thus characterizing the LD behavior of the optimal schemes when these are singlet decoders (and constituting the only known cases where such explicit characterization is available).