Optimal partial decode-and-forward rates for the Gaussian MIMO relay channel using the GSVD

In this paper, we consider the partial decode-and-forward (PDF) strategy for the Gaussian multiple-input multiple-output (MIMO) relay channel. The input distribution that maximizes the achievable PDF rate for this channel is still unknown in general. Therefore, it has so far only been possible to determine the maximum PDF rate if the best PDF strategy is equivalent to the decode-and-forward (DF) strategy, point-to-point (P2P) transmission from source to destination, or if PDF achieves the cut-set bound (CSB), i.e., for special cases where Gaussian channel inputs are known to be optimal. In this work, we exploit the properties of the generalized singular value decomposition (GSVD) to show that the maximum PDF rate for the Gaussian MIMO relay channel is also achieved by Gaussian inputs if the row spaces of the source-relay and the source-destination channel gain matrices are disjoint. Furthermore, we show that the optimal PDF rate can be determined as the solution of a convex optimization problem in that case.