This paper considers the problem of downlink transmit beamforming for wireless transmission and downstream precoding for digital subscriber wireline transmission, in the context of common information broadcasting or multicasting applications wherein channel state information (CSI) is available at the transmitter. Unlike the usual "blind" isotropic broadcasting scenario, the availability of CSI allows transmit optimization. A minimum transmission power criterion is adopted, subject to prescribed minimum received signal-to-noise ratios (SNRs) at each of the intended receivers. A related max-min SNR "fair" problem formulation is also considered subject to a transmitted power constraint. It is proven that both problems are NP-hard; however, suitable reformulation allows the successful application of semidefinite relaxation (SDR) techniques. SDR yields an approximate solution plus a bound on the optimum value of the associated cost/reward. SDR is motivated from a Lagrangian duality perspective, and its performance is assessed via pertinent simulations for the case of Rayleigh fading wireless channels. We find that SDR typically yields solutions that are within 3-4 dB of the optimum, which is often good enough in practice. In several scenarios, SDR generates exact solutions that meet the associated bound on the optimum value. This is illustrated using measured very-high-bit-rate Digital Subscriber line (VDSL) channel data, and far-field beamforming for a uniform linear transmit antenna array.
Bibliographical noteFunding Information:
Manuscript received November 1, 2004; revised May 11, 2005. The work of N. D. Sidiropoulos was supported in part by the U.S. ARO under ERO Contract N62558-03-C-0012, the E.U. under FP6 U-BROAD STREP # 506790, and the GSRT. The work of T. N. Davidson was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Canada Research Chairs program. The work of Z.-Q. Luo was supported in part by the U.S. National Science Foundation, Grant no. DMS-0312416. An earlier version of part of this work appears in Proceedings of the IEEE SAM 2004 workshop, vol. 1, pp. 489–493, Sitges, Barcelona, Spain, July 18–21, 2004. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Daniel Fuhrman.
- Convex optimization
- Downlink beamforming
- Minimization of total radiation power
- Semidefinite programming
- Semidefinite relaxation (SDR)
- Very-high-bit-rate Digital Subscriber line (VDSL) precoding