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.
Bibliographical noteFunding Information:
Manuscript received October 23, 2006; revised February 1, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Eran Fishler. An earlier version of part of this work appears in conference form in the Proceedings of the International Conference on Acoustics, Speech and Signal Processing (ICASSP), Toulouse, France, May 14–19, 2006, pp. 973–976. The work of E. Karipidis was supported in part by the 03ED918 research project, implemented within the framework of the Reinforcement Programme of Human Research Manpower (PENED) and co-financed by National and Community Funds (75% from the E.U.-European Social Fund and 25% from the Greek Ministry of Development-General Secretariat of Research and Technology). The work of N. D. Sidiropoulos was supported in part by the U.S. ARO under ERO Contract N62558-03-C-0012, and the EU under FP6 project WIP. The work of Z.-Q. Luo was supported in part by the National Science Foundation, Grant No. DMS-0312416.
- Convex optimization
- Downlink beamforming
- Semidefinite relaxation