On the capacity region of parallel Gaussian broadcast channels with common information

We consider a broadcast scenario in which a single transmitter wishes to send common, partially common and particular messages to several receivers over the product of unmatched parallel scalar Gaussian subchannels with a total power constraint. This scenario is a generalization of the 2-user 2-subchannel scenario that was studied earlier in the literature. In order to expose the signal structure and the difficulties that arise in generalizing the results on the 2-user 2-subchannel case to the case of $K$ users and $N$ subchannels, we consider a representative scenario with 3 users and 2 subchannels. For this case, we characterize the achievable rate region, and express the boundary points thereof as the solution of an optimization problem. This problem is not convex in the general case, but it provides insight that leads to tight inner and outer bounds on the capacity region that can be obtained efficiently via the solution of a convex Geometric Program (GP). (The GP also generates the corresponding power loads and partitions.) In addition to these bounds, we provide a (precise) GP formulation for the optimal power allocation problem for the 2-user 2-subchannel case.