Duality and Rate Optimization for Multiple Access and Broadcast Channels With Amplify-and-Forward Relays

In this paper, we consider multihop 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 optimal rate regions. We find the maximum sum rate and the maximum rate for each user in closed form and express the optimal rate pair (R₁, R₂) that maximizes mu₁R₁+mu₂R₂ as the solution of a pair of simultaneous equations. 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. The duality means that the rate 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 rate region of the AF relay BC as the union of MAC rate regions. The duality is established for distributed multiple antenna AF relay nodes and multiple (more than 2) hops as well.