This article considers the design of an optimal beamforming weight matrix of multiple-antenna multiple-relay networks. It is assumed that each relay utilizes the amplify and forward strategy, i.e., it multiplies the received signal vector by a matrix, dubbed the relay weight matrix, and forwards the resulting vector to the destination. Furthermore, we assume that the source and the destination have the same number of antennas and that each transmit antenna is virtually paired to a different destination antenna. The relay weight matrices are concurrently designed to optimize the mean square error (MSE) criterion at the destination, assuming each relay node is subject to a power constraint. Accordingly, it is demonstrated that this problem can be cast as a convex optimization problem in which the individual power constraints are tackled by employing the method of Lagrange multipliers in two stages. First, the relay gain matrix is computed analytically in terms of Lagrange dual variables, thereby converting the original problem into a scalar optimization problem. Then, these scalar variables are computed numerically. The proposed scheme is evaluated through simulation with various numbers of relays and antennas to obtain MSE and bit error rate (BER) metrics and it is shown that the resulting MSE and BER achieved through using the proposed method outperforms that of MMSE-MMSE method introduced by Oyman et.al., which is regarded as the best known method for the underlying problem.

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A multi-antenna multiple relay (MAMR) network is considered and a variation of two-hop zero-forcing amplify-forward relaying method is proposed. Deploying ZF method together with application of diagonal power allocation matrices at the relays, it is shown that the overall MAMR network is simplified to M independent single antenna multiple relay (SAMR) networks, where M is the number of source and destination antennas. This enables to incorporate network beamforming proposed for SAMR networks. Accordingly, using the BER as the performance metric, we present simulation results to show the proposed approach outperforms the common ZF method addressed in the literature