Closed Form of Optimum Cooperative Distributed Relay Amplifying Matrix

This paper presents a closed form of an optimal cooperative amplify-and-forward (AF) relay amplifying matrix for a distributed relay network of M-source-M-destination pairs and N relays, called a cooperative distributed AF relay network. The objective of this paper is to derive closed forms of minimum mean square error (MMSE)-based and zero-forcing (ZF)-based optimal AF relay amplifying matrices for the cooperative distributed AF relay network under the transmitter power constraint (TPC) at the relays, the receiver power constraint (RPC) at the destinations, and the no-power constraint (NPC) condition. Additionally, by substituting the derived optimum AF relay amplifying matrices into the original cost functions (CFs), the behavior of the optimum CFs and the total optimum signal component power (SCP) at the destinations are compared to each other for different cases. Finally, using the MMSE criterion, a novel relay selection scheme is proposed for the cooperative distributed AF relay network.