A Precoder Design for Two-Way Amplify-and-Forward MIMO Relay Systems with Linear Receivers

In this paper, we propose a 2×2 relay precoder design for two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relay systems. By exploring the asymptotic behavior of the sum of mean-squared errors (sum-MSE) of the received signals at the two source nodes, we find that it is dominated by the smallest singular values (or the smallest eigenvalues) of the effective MIMO channels of the corresponding relay system. This implies that the sum-MSE performance strongly depends on the sum of the condition numbers of the effective MIMO channels, where the condition number of a MIMO channel is defined as the ratio of the largest to the smallest singular value. With this observation, unlike the conventional precoder designs that are obtained in an iterative manner based on minimizing the sum-MSE, the proposed design is derived by using the Gram-Schmidt process based on minimizing the sum of the condition numbers of the effective MIMO channels. Compared to conventional iterative methods, the proposed approach achieves close performance with much lower computational complexity for both perfect and imperfect channel estimations.