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This paper considers the so-called partial decode-and-forward (DF) strategy for the Gaussian multiple-input multiple-output (MIMO) relay channel. Unlike for the DF strategy or point-to-point (P2P) transmission from source to destination, for which Gaussian channel inputs are known to maximize the achievable rates, the input distribution that attains the maximum achievable partial DF rate for the Gaussian MIMO relay channel has remained unknown so far. For some special cases, e.g., for relay channels where the partial DF strategy reduces to the DF or P2P transmission, it could be deduced that Gaussian inputs maximize the rate that can be achieved with the partial DF strategy. For the general case, however, the problem has remained open until now. In this paper, we solve this problem by proving that the maximum achievable partial DF rate for the Gaussian MIMO relay channel is always attained by Gaussian channel inputs. Our proof relies on the channel enhancement technique, which was originally introduced by Weingarten et al. to derive the (private message) capacity region of the Gaussian MIMO broadcast channel. By combining this technique with a primal decomposition approach, we first establish that jointly Gaussian source and relay inputs maximize the achievable partial DF rate for the aligned Gaussian MIMO relay channel. Subsequently, we use a limiting argument to extend this result from the aligned to the general Gaussian MIMO relay channel. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]