Optimality range of Largest Eigenmode Relaying with partial channel state information in the relay

Optimal precoding in the relay with partial channel state information in the relay is studied in this paper. It is assumed that the antennas of the relay are correlated. Transmission only via the eigenvector corresponding to the largest eigenvalue of the correlation matrix is investigated in this paper which is called Largest Eigenmode Relaying (LER). We derive necessary and sufficient condition under which the transmission achieves capacity only by LER. Although the derived condition requires computationally expensive Monte Carlo simulations to be validated, an alternative method is introduced to circumvent the Monte Carlo simulations: we evaluate a system with infinite number of antennas in the source and derive a closed-form necessary and sufficient condition under which LER is optimal. It is observed, by simulations, that the optimality region of LER is almost independent of the number of antennas in the source; hence, the LER optimality condition derived for infinite number of the antennas in the source can, alternatively, be exploited for systems with finite number of antennas in the source with insignificant difference.