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The problem of jointly optimizing the source precoder, relay transceiver, and destination equalizer has been considered in this paper for a multiple-input-multiple-output (MIMO) amplify-and-forward (AF) relay channel, where the channel estimates of all links are assumed to be imperfect. The considered joint optimization problem is nonconvex and does not offer closed-form solutions. However, it has been shown that the optimization of one variable when others are fixed is a convex optimization problem which can be efficiently solved using interior-point algorithms. In this context, an iterative technique with the guaranteed convergence has been proposed for the AF MIMO relay channel that includes the direct link. It has been also shown that, for the double-hop relay case without the receive-side antenna correlations in each hop, the global optimality can be confirmed since the structures of the source precoder, relay transceiver, and destination equalizer have closed forms and the remaining joint power allocation can be solved using Geometric Programming (GP) technique under high signal-to-noise ratio (SNR) approximation. In the latter case, the performance of the iterative technique and the GP method has been compared with simulations to ensure that the iterative approach gives reasonably good solutions with an acceptable complexity. Moreover, simulation results verify the robustness of the proposed design when compared to the nonrobust design that assumes estimated channels as actual channels.

Joint Linear Processing for an Amplify-and-Forward MIMO Relay Channel with Imperfect Channel State Information