Duality of MIMO multiple access channel and broadcast channel with amplify-and-forward relays

In this work, we consider a two-hop multiuser amplify-and-forward relay network with multi-antenna 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 ¿D^¿ is employed in the broadcast channel (BC), and vice versa, where re 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^¿ are the relaying matrices in the MAC and BC respectively. Third, for any D in the MAC and cD^¿ in the BC where c is any positive real scalar, MAC-BC duality under total network power constraint holds only for the above two cases.