Optimum 4-QAM relay amplification for the amplify-and-forward half-duplex cooperative wireless system

We consider an amplify-and-forward half-duplex co-operative wireless system. Specifically, we examine the role of the relay and design an optimum amplification factor for the relay. In the simple case of the message lasting for two symbols, and from the consideration of transmitting from a 4-QAM constellation, we develop an explicit expression of the pair-wise error probability. An optimum design problem can then be formulated so that the optimum amplification factor for the relay can be obtained. The resulting optimum value depends on the signal-to-noise ratio and its relation with the allowable amplification power.