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In this paper, the authors establish in closed-form the capacity and characterize the optimal input covariance matrix at the source and the optimal power allocation scheme between source and relay for a half-duplex single-relay Amplify-and-Forward (AF) system with static channel gains. Different from Multiple-Input Multiple-Output (MIMO) systems, the channel matrix of the AF system is a function of the parameters to be optimized and hence water filling over the square of the singular values of this matrix is no longer optimal. Furthermore, given that the mutual information of the AF system is not a convex function, conventional optimization methods cannot be applied to find the optimal input covariance, the power allocation, and the capacity.
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