Transmit Precoding in a Noncoherent Relay Channel With Channel Mean Feedback

This paper investigates the optimal source transmit precoding to maximize the ergodic mutual information between the input and the output of a noncoherent amplify-and-forward relay channel, with multiple antennas at the source and with a single antenna at both the relay and the destination. It is assumed that only the source-relay channel mean information is available at the source. The challenge here is that relaying introduces a nonconvex structure in the objective function. Therefore, previous methods dealing with channel mean feedback, which generally require the concavity of the objective function, cannot be applied. To circumvent the difficulty at hand, a different approach based on stochastic ordering is employed. The stochastic optimization problem here is ultimately transformed into the comparison of two nonnegative random variables in the Laplace transform order. It is shown that the optimal source transmit strategy is to transmit along the known channel mean and its orthogonal eigen-channels. Our result subsumes as an asymptotic case the optimal precoding for multiple-input single-output (MISO) channels without relaying under mean feedback. Furthermore, the analysis can be partially extended to the case with multiple antennas at the relay. Numerical examples are provided to complement and corroborate the analysis.