| The broadcast channel is an information-theoretic model of a communication system whereby a transmitter communicates to a number of receivers. These receivers, because of their geographical locations or other practical limitations cannot cooperate in decoding their individual messages. Examples of such a channel model are downlink communication in a wireless cellular network or downstream transmission in a Digital Subscriber Line system. Certainly, a full understanding of the communication limits over broadcast channels provides practical insights on the design of more efficient wireless or wire-line networks.; In this dissertation, an alternative converse proof for the capacity region of the Gaussian vector broadcast channel is presented. First, the achievable region by the dirty paper coding (DPC) scheme is reviewed, and the multiple-access broadcast channel duality is revisited. By combining some ideas from earlier works on the problem with duality, primarily, the optimality of the DPC scheme is proven under a total transmit-power constraint. Afterward, by employing a more comprehensive notion of the multiple-access broadcast channel duality, the optimality proof for the DPC scheme is extended to general convex constraints on the transmit covariance matrix.; The capacity expression for the Gaussian vector broadcast channel is used subsequently to establish the capacity region for a family of parallel Gaussian vector broadcast channels with total average transmit-power constraint. It is shown that using independent code-books over the individual sub-channels is capacity achieving. Moreover, the problem of optimal power allocation over the sub-channels is studied from two major information-theoretic standpoints. In the first problem, the power allocation policy that uses the minimum total transmit power to achieve a given point in the capacity region is investigated. By transforming the underlying optimization problem to the Lagrange dual domain, an efficient numerical algorithm is proposed to find this power allocation policy. In the second problem, the policy that achieves a boundary point of the capacity region is studied. It is shown that this power allocation problem is essentially the same as the minimum power allocation problem in the dual domain. Based on this similarity, a numerical algorithm is developed to solve this power allocation problem. |
