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Consider a two-hop half-duplex relay channel (source–relay–destination) with a Markovian-constrained relay. The capacity of such channel is shown to be equal to the well-known cut-set upper bound. For the case where the relay-to-destination link is noise-free, the optimal state transition probabilities that give rise to the capacity are determined. This result links the relay channel to Shannon’s entropy maximization by introducing a relay adjacency matrix. For the case where both source-to-relay and relay-to-destination links are noisy, lower bounds on the achievable information rates for various constrained sequences are computed. We conjecture that our numerical bounds are tight. The numerically computed capacities and optimized information rates are significantly higher than the rate achieved by the traditional predetermined time-sharing scheme.

On the Capacity of a Two-Hop Half-Duplex Relay Channel With a Markovian Constrained Relay