Duality and Capacity Region of AF Relay MAC and BC

We consider multi-hop multiple access (MAC) and broadcast channels (BC) where communication takes place with the assistance of relays that amplify and forward (AF) their received signals. For a two hop parallel AF relay MAC, assuming a sum power constraint across all relays we characterize optimal relay amplification factors and the resulting capacity regions. We find that the parallel AF relay MAC with total transmit power of the two users $P_1+P_2=P$ and total relay power $P_R$ is the dual of the parallel AF relay BC where the MAC source nodes become the BC destination nodes, the MAC destination node becomes the BC source node, the dual BC source transmit power is $P_R$ and the total transmit power of the AF relays is $P$. The duality means that the capacity region of the AF relay MAC with a sum power constraint $P$ on the transmitters is the same as that of the dual BC. The duality relationship is found to be useful in characterizing the capacity region of the AF relay BC as the union of MAC capacity regions. The duality extends to distributed relays with multiple antennas and more than 2 hops as well.