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 the sum rate capacity and the maximum rate for each user in closed form and express the optimal rate pair (R1, R2) that maximizes μ1R1+μ2R2 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 PR 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 PR 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. For the M hop AF relay MAC where the m hop consists of Rm relays (possibly with multiple antennas) with total transmit power Pm and source power is P0, the dual BC is obtained by reversing the direction of communication so that the transmit power of the BC source is PM and the relays participating in the i th hop have transmit power PM−i. Thus, even though the transmitters and receivers are switched, each link in the original MAC is associated with the same transmit power as in the dual BC.