AF relaying for Multiple Antenna Multiple Relay Networks under individual power constraint at each relay

This paper investigates the optimal beamforming weight matrix for Multiple-Antenna Multiple-Relay (MAMR) Networks. It is assumed that each relay makes use of the Amplify and Forward (AF) 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. The relay weight matrices have to be concurrently designed to optimize a desired criterion at the destination, assuming each relay node is subject to a power constraint. In this work, the Mean Square Error (MSE) metric is assumed to be the corresponding cost function. In this regard, it is demonstrated that the aforementioned problem can be cast as a convex optimization problem in which the individual power constraints are tackled by employing the method of Lagrange multipliers. Then, it is demonstrated that the optimal solution can be tackled in two-fold. First, an elegant analytical method for the corresponding dual problem is devised; rendering the current complex vector optimization problem can be translated to a scalar optimization problem. Then, these scalar variables are computed numerically. Numerical results are provided, showing the Bit Error Rate (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 such problem.