On Power Allocation for Parallel Gaussian Broadcast Channels with Common Information

This paper considers a broadcast system in which a single transmitter sends a common message and (independent) particular messages to K receivers over N unmatched parallel scalar Gaussian subchannels. For this system the set of all rate tuples that can be achieved via superposition coding and Gaussian signalling (SPCGS) can be parameterized by a set of power loads and partitions, and the boundary of this set can be expressed as the solution of an optimization problem. Although that problem is not convex in the general case, it will be shown that it can be used to obtain tight and efficiently computable inner and outer bounds on the SPCGS rate region. The development of these bounds relies on approximating the original optimization problem by a (convex) Geometric Program (GP), and in addition to generating the bounds, the GP also generates the corresponding power loads and partitions. There are special cases of the general problem that can be precisely formulated in a convex form. In this paper, explicit convex formulations are given for three such cases, namely, the case of 2 users, the case in which only particular messages are transmitted (in both of which the SPCGS rate region is the capacity region), and the case in which only the SPCGS sum rate is to be maximized.