Signal-to-noise ratio optimisation for multi-input multi-output relay systems with direct source-destination path

The authors consider a relay communication system, where all nodes are equipped with multi-input multi-output antennas, and there is a direct path/channel between the source and the destination. Assuming a linear non-regenerative relaying, the relay matrix is designed by maximising the signal-to-noise ratio of the system. The authors derive the optimum relaying matrix for different power constraints in the relay, including the fixed relay power constraint and constraint on the maximum transmitting power from the relay. Under the constraint of fixed total transmit power, they derive the optimum power budgets of the relay and source that only depend on four positive quantities. They prove that there always exists a rank-one relaying matrix, which transmits signal in one-dimensional subspace and leaves the other subspaces clean. In addition, when the quality of the source-destination (SD) channel is poor, this matrix is the best rank-one relaying transform that maximises mutual information between the source and the destination. Finally, they conclude that the relaying is beneficial only if the link quality for the source-relay is twice better than that of the SD, where the link quality is proportional to the ratio of the channel power gain to the received noise variance.