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Title

Optimal Power Allocation Over Gaussian Broadcast Channels

Creator

Chen, Pengpeng

Date

2024

Description

For Gaussian broadcast channel, the maximal capacity region can be achieved. A power assignment over Gaussian broadcast channels splits the...
Show moreFor Gaussian broadcast channel, the maximal capacity region can be achieved. 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). The capacity region reaches its maximum when the weighted sum rate over Gaussian broadcast channels is maximum, making this a classic and significant problem within the wireless communication community.This thesis begins by addressing the problem of maximizing the WSR. In the single Gaussian broadcast channel, the traditional methods for computing optimal power assignment have utilized Lagrange multipliers for convex optimizations, with computational complexities ranging from $O(|U|^3)$ to $O(|U|^2 \log(|U|))$. A more recent approach has reduced this to $O(|U|^2)$. We propose a new geometric algorithm for optimal power assignment over a single Gaussian broadcast channel. This algorithm operates with linear complexity, provided all users are pre-sorted by weight or noise. Our method offers an intuitive water-filling interpretation, which subsequently allows us to develop a water-filling algorithm for optimal power assignment over parallel Gaussian broadcast channels. The complexity of this algorithm remains linear in terms of the number of user-channel pairs, assuming pre-sorted users by weight. For computing an optimal power assignment over parallel Gaussian broadcast channels, no explicit time complexity is known in the literature. Due to power constraints imposed by the base station, energy efficiency is an essential metric to evaluate the performance of these transmissions. Following the WSR optimization, we then tackle the problem of maximizing the weighted energy efficiency (WEE). In this context, a power assignment over parallel Gaussian broadcast channels involves splitting a power budget at the access point among all channel-user pairs, subject to per-channel upper bounds on the sum power. This allocation yields specific rate allocations to each channel-user pair. The WEE is defined as the ratio of the weighted sum rate to the sum-power plus a fixed positive overhead. The Max-WEE problem seeks a power assignment that maximizes the WEE. Although special variants of Max-WEE, such as those with unit weights or two users per channel, have been extensively studied, existing algorithms for these variants lack known bounds on running time. This is primarily because they rely on general-purpose methods for fractional programming.In this thesis, we derive fundamental properties and closed-form expressions for the maximum WEE. Building on these theoretical foundations, we devise a simple yet effective water-filling algorithm for solving the Max-WEE problem. Our algorithm, under the assumption that all users are pre-sorted by weight, has the linear complexity in terms of the number of channel-user pairs. Furthermore, under a mild pre-sorting condition, we develop an additional linear-complexity algorithm for the Max-WEE problem, subject to rate demand constraints.Overall, our research presents novel and efficient algorithms for power assignment over Gaussian broadcast channels, including the algorithms for maximizing the WSR, and the algorithms for maximizing the WEE. These algorithms demonstrates significant potential for practical applications in wireless communication systems. Our work offers a comprehensive solution to fundamental problems in the field and paves the way for further advancements in wireless communications.
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