On multicast beamforming for minimum outage

The multicast beamforming problem is considered from the viewpoint of minimizing outage probability subject to a transmit power constraint. The main difference with the point-to-point transmit beamforming problem is that in multicast beamforming the channel is naturally modeled as a Gaussian mixture, as opposed to a single Gaussian distribution. The Gaussian components in the mixture model user clusters of different means (locations) and variances (spreads). It is shown that minimizing outage probability subject to a transmit power constraint is an NP-hard problem when the number of Gaussian kernels, J, is greater than or equal to the number of transmit antennas, N. Through dimensionality reduction, it is also shown that the problem is practically tractable for 2 - 3 Gaussian kernels. An approximate solution based on the Markov inequality is also proposed. This is simple to compute for any J and N, and often works well in practice.