Optimal multi-input??multi-output relay transform matrix with a bounded condition number

A relay communication system where its nodes are equipped with multiple antennas is considered. The relay applies a transformation matrix on its received signal and forwards it to the destination. The authors show that, given a limited relay transmit power, the transformation matrix which maximises the signal-to-noise power ratio (SNR) is almost surely a rank-one matrix. Such a singular transformation matrix prevents accurate channel estimation. As a remedy, they impose a constraint on the condition number (CN) of the relay transformation matrix. They prove that the optimal relay matrix under this new constraint has up to two distinct singular values. Thus, the relay transmits some power in all directions, and this small energy allows better channel estimation. The author´s simulation results show that imposing the CN constraint on the relay matrix results in some small loss in the SNR which is quantified in this study; however, the optimal relay matrix enables the receivers to track the channel state information.