Far-field multicast beamforming for uniform linear antenna arrays

The problem of transmit beamforming to multiple cochannel multicast groups is considered for the important special case when the channel vectors are Vandermonde. This arises when a uniform linear antenna antenna (ULA) array is used at the transmitter under far-field line-of-sight propagation conditions, as provisioned in 802.16e and related wireless backhaul scenarios. Two design approaches are pursued: i) minimizing the total transmitted power subject to providing at least a prescribed received signal-to-interference-plus-noise-ratio (SINR) to each intended receiver; and ii) maximizing the minimum received SINR under a total transmit power budget. Whereas these Problems have been recently shown to be NP-hard, in general, it is proven here that for Vandermonde channel vectors, it is possible to recast the optimization in terms of the autocorrelation sequences of the sought beamvectors, yielding an equivalent convex reformulation. This affords efficient optimal solution using modern interior point methods. The optimal beamvectors can then be recovered using spectral factorization. Robust extensions for the case of partial channel state information, where the direction of each receiver is known to lie in an interval, are also developed. Interestingly, these also admit convex reformulation. The various optimal designs are illustrated and contrasted in a suite of pertinent numerical experiments.