  In this paper, we consider a general bi-directional relay network with two
sources and N relays when neither the source nodes nor the relays know the
channel state information (CSI). A joint relay selection and analog network
coding using differential modulation (RS-ANC-DM) is proposed. In the proposed
scheme, the two sources employ differential modulations and transmit the
differential modulated symbols to all relays at the same time. The signals
received at the relay is a superposition of two transmitted symbols, which we
call the analog network coded symbols. Then a single relay which has minimum
sum SER is selected out of N relays to forward the ANC signals to both sources.
To facilitate the selection process, in this paper we also propose a simple
sub-optimal Min-Max criterion for relay selection, where a single relay which
minimizes the maximum SER of two source nodes is selected. Simulation results
show that the proposed Min-Max selection has almost the same performance as the
optimal selection, but is much simpler. The performance of the proposed
RS-ANC-DM scheme is analyzed, and a simple asymptotic SER expression is
derived. The analytical results are verified through simulations.
