  In this letter, we analyze the ergodic capacity of bidirectional
amplify-and-forward relay selection (RS) with imperfect channel state
information (CSI), i.e., outdated CSI and imperfect channel estimation.
Practically, the optimal RS scheme in maximizing the ergodic capacity cannot be
achieved, due to the imperfect CSI. Therefore, two suboptimal RS schemes are
discussed and analyzed, in which the first RS scheme is based on the imperfect
channel coefficients, and the second RS scheme is based on the predicted
channel coefficients. The lower bound of the ergodic capacity with imperfect
CSI is derived in a closed-form, which matches tightly with the simulation
results. The results reveal that once CSI is imperfect, the ergodic capacity of
bidirectional RS degrades greatly, whereas the RS scheme based on the predicted
channel has better performance, and it approaches infinitely to the optimal
performance, when the prediction length is sufficiently large.
