Capacity theorems for relay channels with ISI

In this paper we initially study degraded relay channels with finite-length intersymbol interference (ISI). For such channels, we show that the decode-and-forward strategy achieves the capacity, and prove a special structure for the capacity achieving distributions of the source and relay signals. We also prove that a general memoryless relay channel used with delayed feedback from the destination node to the relay node is an instance of a degraded relay channel with ISI, and observe that the delayed feedback from the destination node to the relay node does not decrease the capacity compared to instantaneous feedback. In all cases where the channel is used with delayed feedback from the destination node to the relay node the decode-and-forward scheme is optimal and the capacity is not decreased by delaying the feedback from the destination node. We extend these results to general (non-degraded) relay channels with ISI to obtain upper and lower bounds on their capacities.