On Optimal Gaussian Signaling in MIMO Relay Channels With Partial Decode-and-Forward

For Gaussian multiple-input multiple-output (MIMO) relay channels with partial decode-and-forward, the optimal type of input distribution is still an open question in general. Recent research has revealed that in some other scenarios with unknown optimal input distributions (e.g., interference channels), improper (i.e., noncircular) Gaussian distributions can outperform proper (circular) Gaussian distributions. In this paper, we show that this is not the case for partial decode-and-forward in the Gaussian MIMO relay channel with Gaussian transmit signals, i.e., we show that a proper Gaussian input distribution is the optimal one among all Gaussian distributions. In order to prove this property, an innovation covariance matrix is introduced, and a decomposition is performed by considering the optimization over this matrix as an outer problem. A key point for showing optimality of proper signals then is a reformulation that reveals that one of the subproblems is equivalent to a sum rate maximization in a two-user MIMO broadcast channel under a sum covariance constraint, for which the optimality of proper signals can be shown.