Capacity and optimal power allocation for fading broadcast channels with minimum rates

We derive the capacity and optimal power allocation scheme for a multi-user fading broadcast channel in which minimum rates must be maintained for each user in all fading states, assuming perfect channel state information at the transmitter and at all receivers. We show that super-position coding can achieve the capacity of such channels and explicitly characterize the boundary of the capacity region. The optimal power allocation scheme is a two-step process: We first allocate the minimum power required to achieve the minimum rates in all fading states, and we then optimally allocate the excess power to maximize the ergodic rates averaged over all fading states in excess of the minimum rate requirements. The optimal allocation of the excess power is a multi-level water-filling relative to effective noise that incorporates the minimum rate constraints. Numerical results are provided for different fading broadcast channel models.