Robust Energy Efficiency Optimization for Amplify-and-Forward MIMO Relaying Systems

We investigate the energy efficiency (EE) of multiple-input-multiple-output (MIMO) amplify-and-forward relaying networks relying on the realistic imperfect channel state information (CSI). Specifically, the relay jointly optimizes the source covariance and relay beamforming matrices by maximizing the EE under additive or multiplicative relay-destination CSI errors. The optimal channel-diagonalizing structure is derived for the source covariance and relay beamforming matrices under the spectral-norm constrained additive or multiplicative CSI error. Then, the existence of a saddle point is proved, which shows that the channel-diagonalizing transmission strategy is optimal in the robust EE maximization under these two types of CSI errors, and the original matrix-valued fractional robust EE problem is transformed into a scalar fractional problem. We propose the Dinkelbach method-based alternating optimization scheme for this transformed robust EE problem, which is capable of finding a locally optimal solution of the original robust EE problem efficiently, and show that the semi-closed-form solution to each of the two associated subproblems can be obtained. We then prove that the channel-diagonalizing transmission strategy remains optimal when the statistically imperfect source-relay channel is additionally imposed. We also extend our work into multi-hop MIMO relaying scenarios and prove that the channel-diagonalizing structure is optimal for the source covariance matrix and the multiple relay beamforming matrices.