Causal relay networks and new cut-set bounds

In this paper, we study causal discrete memoryless relay networks with multiple sources and destinations. Unlike the classical relay networks where the transmit signal of a relay node depends only on its past received symbols, our network model consists of two types of relays, i.e., relays with one sample delay (classical relays) and relays without delay (causal relays) whose transmit signal depends not only on the past received symbols but also on the current received symbol. For this network, we derive two new cut-set bounds, one when the causal relays have their own messages and the other when not. Our new cut-set bounds recover the classical cut-set bound when there is no causal relay. Using two examples, i.e., causal vector Gaussian two-way relay channel and causal vector Gaussian relay channel, we show that the new cut-set bounds can be achieved by a simple amplify-and-forward type relaying. Our result for the causal relay channel strengthens the previously known capacity result for the same channel by El Gamal, Hassanpour, and Mammen.