Optimal Power Allocation for Parallel Gaussian Broadcast Channels with Independent and Common Information

We consider a set of parallel, two-user Gaussian broadcast channels, where the transmitter wishes to send independent information to each of the receivers and common information to both receivers. The ca- pacity region of this channel has been implicitly characterized in the past, but we provide an explicit characterization of the power and rate allocation schemes that achieve the boundary of the three-dimensional rate region. Unlike the broadcast channel with only independent information, we find that the optimal power allocation policy cannot be viewed as a generaliza- tion of single-user water-filling. We also consider MIMO bro adcast chan- nels, which are non-degraded in general. We propose an achievable region based on dirty paper coding, and discuss the maximum common informa- tion rate achievable over these channels. I. I NTRODUCTION As wireless networks evolve, it is apparent that multi-cast (i.e. sending a common message to all users on a downlink chan- nel) is an important mode of communication that systems will require in the future. In cellular networks, for example, multi- cast information could be common information such as news updates or location-based information. It is reasonable to as- sume that networks will want to transmit a mixture of common information to all users and independent information to each of the users. With this in mind, we consider broadcast channels with both common and independent information. We consider parallel two-user Gaussian broadcast channels, where the transmitter wants to send independent information to users 1 and 2 at rates R1 and R2, respectively, and common in- formation (decodable by both users) at rate R0. For degraded broadcast channels, the common information rate and the inde- pendent information rate to the degraded user are interchange- able, because the strongest user can decode anything that the de- graded user can. However, we consider parallel channels where in some channels User 1 is the degraded user, but in other chan- nels User 2 is the degraded user. The capacity region of this channel (for both discrete memoryless channels and for Gaus- sian channels) was characterized in (1) in terms of a union of regions, where the union was taken over different power distri- butions between the different channels. We first derive an eq uiv- alent expression for this capacity region that is more amenable to optimization techniques. We then pose the problem of character- izing the optimal power and rate allocation schemes that achieve the boundary of the three-dimensional region using Lagrangian techniques. We then apply the utility function approach used for the broadcast channel (2) without common information, but we find that this approach does not work in general. We use a more direct approach to maximize the Lagrangian function and obtain the capacity region with common information using this approach. Using this method, the optimal allocation is found by performing a finite maximization in each channel. Finally, we consider MIMO broadcast channels, which in general are not degraded. Thus, the capacity region with or without common information is not known for this channel. We propose an achievable region based on dirty paper coding. We also consider the maximum common rate achievable on these channels, i.e. the common information capacity.