We consider a relay precoding problem in a non-regenerative multiple-input multiple-output (MIMO) relay system, where the relay has the perfect channel state information of the source-relay link and only the covariance information of the relay-destination link. The average capacity seen by the relay is maximized under a relay transmit power constraint. The relay precoding problem is transformed to its equivalent power allocation problem at the relay. The optimal power allocation solution is then derived by formulating the optimization problem via difference of convex (DC) functions programming. To reduce the computational complexity, we also propose a suboptimal scheme by modifying the naive Jensen method. Furthermore, we propose the maximum eigenmode relaying (MER) scheme, which allocates the relay transmit power only to the maximum eigenmode. A necessary and sufficient condition for the MER to be asymptotically optimal is derived in the high signal-to-noise ratio (SNR) regime of the source-relay link. The performance of the proposed schemes is investigated through extensive numerical simulations.
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