On the Duality of MIMO MAC and BC with AF Relays

In this work, we consider a two-hop multiuser amplify-and-forward relay network with MIMO nodes. The results are three-fold. First, for any relay amplification matrix D in the multiple-access channel (MAC), we show that duality holds when kappaD^dagger is employed in the broadcast channel (BC), and vice versa, where k is obtained from switching the total source and relay power constraints. Second, under a total network power constraint, we show that MAC-BC duality holds when D and D^dagger are the relaying matrices in the MAC and BC respectively. Third, for any D in the MAC and cD^dagger in the BC where c is any positive real scalar, duality under total network power constraint holds only for the above two cases.