The Capacity Region of the Class of Three-Receiver Gaussian MIMO Multilevel Broadcast Channels With Two-Degraded Message Sets

Nair and El Gamal established the capacity region of the three-receiver multilevel broadcast channel (MBC) with two-degraded message sets. For the three-receiver MBC with two-degraded message sets, the output at receiver 2 is a degraded version of the output at receiver 1. However, no order of degradedness is imposed on the output at receiver 3. The transmitter sends a common message to all three receivers and a private message to receiver 1. By considering a specific discrete-memoryless example, Nair and El Gamal showed that a direct extension of the Körner-Marton region (for the general two-receiver broadcast channel with degraded message sets) is strictly suboptimal. They also considered a three-receiver Gaussian product MBC and showed that, restricted to Gaussian inputs, the direct extension of the Körner-Marton region is again strictly suboptimal. However, whether Gaussian inputs are optimal remained unresolved. In this paper, we show that Gaussian inputs, along with time-sharing between rate points obtained with Gaussian inputs, achieve the capacity region of the three-receiver Gaussian multiple-input multiple-output MBC (this includes the three-receiver Gaussian product MBC considered by Nair and El Gamal) with two-degraded message sets. Our proof relies on the channel enhancement technique introduced by Weingarten as well as the perturbation approach employed by Liu and Viswanath.