Exact closed-form expressions for the sum capacity and individual users’ rates in broadcast ergodic rayleigh fading channels

We derive closed-form expressions for the sum capacity of broadcast ergodic Rayleigh fading channels considering two power allocation policies along time: constant transmit power and optimal power allocation. In addition, we derive closed- form expressions for the individual users´ rates when the sum capacity is achieved. We use these expressions to obtain the sum capacity and individual rates in a variety of broadcast channels. Among other facts, the numerical results reveal that the resulting rates using both power allocation policies are quite similar, except for some specific cases that are discussed.