On the Equivalence of Degraded Gaussian MIMO Broadcast Channels

In this paper, we discuss the equivalence of different types of degradedness for Gaussian multiple-input multipleoutput (MIMO) broadcast channels. While there are four distinct types of degradedness for discrete memoryless broadcast channels, we show that there is basically only one such type for Gaussian MIMO broadcast channels. In particular, Gaussian MIMO broadcast channels are never physically degraded (except maybe for the practically irrelevant case where at least one destination does not experience any thermal noise), and we prove that a Gaussian MIMO broadcast channel is stochastically degraded if and only if it is both more capable and less noisy. In addition, we provide a very simple condition in terms of the channel gain matrices and noise covariances that characterizes stochastically degraded Gaussian MIMO broadcast channels.