The capacity region of a new class of K-receiver degraded compound broadcast channels

The compound broadcast channel models the situation where each receiver has a number of possible realizations and its message is to be decoded regardless of the actual realization. Weingarten et al. established the capacity region for the two-user degraded case where the realizations exhibit a degradedness order defined through a fictitious receiver. In this paper, we consider a K-receiver degraded compound broadcast channel where, instead of specifying fictitious receivers, the receivers exhibit a pair-wise degradedness order, i.e., each realization from a weaker receiver is stochastically degraded with respect to each realization from a stronger receiver. There is a restriction on the number of possible realizations for all the receivers to two. We first prove the capacity region for this discrete memoryless class of broadcast channels. The achievability follows readily from superposition coding and successive decoding. To facilitate the proof of the converse, we give an alternative characterization of the achievable rate region. The main contribution in this paper is to bypass the use of the Csiszέr-sum lemma in order to prove the converse for an arbitrary number of receivers. We also make use of our converse proof as well as Geng and Nair´s technique to prove an extremal entropy inequality. This is then finally used to prove the capacity region of the equivalent class of Kreceiver aligned Gaussian MIMO degraded compound broadcast channels.