Anytime capacity of Markov channels

Several new expressions for the anytime capacity of Sahai and Mitter are presented for a time-varying rate-limited channel with noiseless output feedback. These follow from a parametric characterization obtained in the case of Markov channels, and include an explicit formula for the r-bit Markov erasure channel, as well as formulas for memoryless rate processes including Binomial, Poisson, and Geometric distributions. Beside the memoryless erasure channel and the additive white Gaussian noise channel with input power constraint, these are the only cases where explicit anytime capacity formulas are obtained. At the basis of these results is the study of the threshold function for mth moment stabilization of a scalar linear system controlled over a Markov time-varying digital feedback channel that depends on m and on the channel's parameters. This threshold is shown to be a continuous and strictly decreasing function of m and to have as extreme values the Shannon capacity and the zero-error capacity as m tends to zero and infinity, respectively. Its operational interpretation is that of achievable communication rate, subject to a reliability constraint.

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