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In this paper, we study a far-end channel estimation problem in a three-node multiple-input–multiple-output (MIMO) relay system. The estimation is accomplished in two phases. In the first phase, the source node keeps silence, the relay transmits pilots, and the destination estimates the relay-to-destination channel. In the second phase, the source transmits pilots, and the relay amplifies the received pilots with a precoder and forwards the resultant pilots to the destination. The destination then estimates the source-to-relay channel based on the received signals and the estimated relay-to-destination channel in the first phase. We aim to conduct a robust design deriving the optimum source pilots and precoding matrix with the minimum mean square error (MMSE) criterion. The design can be easily formulated as an optimization problem; however, the problem is not convex and is difficult to solve. We then propose using a lower bound of the objective function in the optimization problem. We show that the optimization of the lower bound is equivalent to the original problem when channel correlation matrices have certain structures. With the correlation matrices, we can derive optimum pilot structures so that the proposed optimization can be transferred to a scalar-valued concave optimization, and a closed-form solution can be obtained via Karush–Kuhn–Tucker (KKT) conditions. Simulations show that our proposed method outperforms existing nonrobust methods. [ABSTRACT FROM PUBLISHER]

We consider the precoding problem in an amplifyand-forward (AF) multiple-input-multiple-output (MIMO) relay system in which multiple antennas are equipped at the source, the relay, and the destination. Most existing methods for this problem only consider the design of the relay precoder, and some even ignore the direct link. In this paper, we consider a joint source/relay precoder design problem, taking both the direct and the relay links into account. Using a minimum-mean-square-error (MMSE) criterion, we first formulate the problem as a constrained optimization problem. However, it is found that the mean square error (MSE) is a highly nonlinear function of the precoders, and a direct optimization is difficult to conduct. We then design the precoders to diagonalize the MSE matrix in the cost function. To do that, we pose certain structural constraints on the precoders and derive an MSE upper bound. It turns out that minimization with respect to this bound becomes simple and straightforward. Using the standard Lagrange technique, we can finally obtain the solution with an iterative water-filling method. Simulation results show that the proposed method, with an additional precoder, outperforms the existing methods, in terms of either the MSE or the bit error rate (BER). [ABSTRACT FROM AUTHOR]