Capacity Region of Reversely Degraded Gaussian MIMO Broadcast Channel

We consider the problem of broadcasting a common message and two individual messages to two users on a product channel of two reversely degraded Gaussian multiple-input multiple-output (MIMO) broadcast channels. Though El Gamal provided a single letter characterization for the general discrete memoryless problem in 1980, this characterization in fact does not include a channel cost constraint, and thus does not apply directly to the Gaussian MIMO setting. We show El Gamal's single letter characterization can indeed be generalized to include channel cost constraints, however special care has to be taken and the characterization holds only with certain class of cost functions. This characterization has an equivalent form, and by utilizing this form, as well as the enhancement technique and an extremal inequality which were only discovered recently, we show that indeed Gaussian codebooks are optimal for this MIMO setting.