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

摘要

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.