Maximizing Weighted Sum-Rate Over Gaussian Broadcast Channels

A power assignment over Gaussian broadcast channels splits the power budget at the access point among all user-channel pairs subject to per-channel upper-bounds on the sum-power, and is optimal if it maximizes the weighted sum-rate (WSR). In this paper we first present a geometric algorithm for computing an optimal power assignment over single Gaussian broadcast channel, which has linear complexity if all users are presorted either by weight or by noise. We also provide an intuitively appealing water-filling interpretation of this geometric algorithm. By leveraging such water-filling interpretation, we develop a water-filling algorithm for computing an optimal power assignment over parallel Gaussian broadcast channels, whose complexity is linear in the number of user-channel pairs if all users are presorted by weight. From these algorithmic studies, we derive clean and simple expressions of the maximum WSR in both integral forms and sum forms. By exploiting the rich property of those forms, we further give a linear-complexity algorithm for computing a power budget at the access point, subject to a given upper bound, which maximizes the difference between the maximum WSR and a linear cost of the power budget. The algorithmic studies in this paper also reveal that a single Gaussian broadcast channel can be decomposed into parallel Gaussian single-user channels which preserve the maximum WSR.