Capacity and Duality 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 P₁+P₂=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.