Linear Beamforming for the Spatially Correlated MISO broadcast channel

TL;DR

This paper investigates the optimal linear beamforming strategies for a MISO broadcast channel with spatial correlation, deriving explicit solutions for two users and asymptotic solutions for many users, to maximize ergodic sum-rate.

Contribution

It characterizes the structure of optimal beamforming vectors in correlated MISO broadcast channels, providing closed-form solutions for two users and fixed-point equations for large systems.

Findings

Optimal beamforming vectors are the dominant generalized eigenvectors for two users.

Closed-form ergodic sum-rate expressions are derived for the two-user case.

Asymptotic analysis yields fixed-point equations for large numbers of users.

Abstract

A spatially correlated broadcast setting with M antennas at the base station and M users (each with a single antenna) is considered. We assume that the users have perfect channel information about their links and the base station has only statistical information about each user's link. The base station employs a linear beamforming strategy with one spatial eigen-mode allocated to each user. The goal of this work is to understand the structure of the beamforming vectors that maximize the ergodic sum-rate achieved by treating interference as noise. In the M = 2 case, we first fix the beamforming vectors and compute the ergodic sum-rate in closed-form as a function of the channel statistics. We then show that the optimal beamforming vectors are the dominant generalized eigenvectors of the covariance matrices of the two links. It is difficult to obtain intuition on the structure of the optimal beamforming vectors for M > 2 due to the complicated nature of the sum-rate expression. Nevertheless, in the case of asymptotic M, we show that the optimal beamforming vectors have to satisfy a set of fixed-point equations.