A converse for the wideband relay channel with physically degraded broadcast

We investigate the multipath fading relay channel in the limit of a large bandwidth, and in the non-coherent setting, where the channel state is unknown to all terminals, including the relay and the destination. We derive a lower bound on the capacity by proposing and analyzing a peaky frequency binning scheme. The achievable rate obtained coincides with the block-Markov lower bound on the capacity of the wideband frequency-division Gaussian relay channel. When the broadcast channel is physically degraded, this achievable rate meets the cut-set upper-bound, and thus reaches the capacity of the noncoherent wideband multipath fading relay channel. In this case, a hypergraph model of the multipath fading relay channel is proposed, and the relaying scheme of concern is shown to reach its min-cut. Even if the source treats the broadcast channel as physically degraded when it is stochastically degraded, the achievable hypergraph bound is the min-cut.