This thesis deals with the achievable rate of amplify and forward (AF) relay system under sum power constraint. We propose the power control methods for two-way amplify-and-forward (AF) relay systems. Assuming that two source nodes have instantaneous channel state information (CSI), ther power of each node that maximizes the minimum and sum rate is derived. Generally fairness problem (maximization of minimum rate) and maximization of sum rate are separately considered and also under individual power constraint, both optimization problems give different results of the power allocation, But under sum power constraint, there is more freedom in allocating of each node power than under individual one and this leads only one optimal solution, which satisfies maximization of minimum rate and sum rate simultaneously. Obtained optimal power allocation under sum power constraint is assigning half the total power to the relay node and allocating powers that makes the rate of each direction same to two source nodes. By analysis and optimization method, we obtained the solution and using computer simulation we showed that power allocation scheme under sum power constraint outperforms conventional maximum power transmitting and power allocation under individual power constraint.
