  In this paper, we propose a relay selection amplify-and-forward (RS-AF)
protocol in general bi-directional relay networks with two sources and $N$
relays. In the proposed scheme, the two sources first transmit to all the
relays simultaneously, and then a single relay with a minimum sum symbol error
rate (SER) will be selected to broadcast the received signals back to both
sources. To facilitate the selection process, we propose a simple sub-optimal
Min-Max criterion for relay selection, where a single relay which minimizes the
maximum SER of two source nodes will be selected. Simulation results show that
the proposed Min-Max selection has almost the same performance as the optimal
selection with lower complexity. We also present a simple asymptotic SER
expression and make comparison with the conventional all-participate
amplify-and-forward (AP-AF) relaying scheme. The analytical results are
verified through simulations. To improve the system performance, optimum power
allocation (OPA) between the sources and the relay is determined based on the
asymptotic SER. Simulation results indicate that the proposed RS-AF scheme with
OPA yields considerable performance improvement over an equal power allocation
(EPA) scheme, specially with large number of relay nodes.
