Transmit beamforming for physical-layer multicasting

TLDR

This paper studies downlink transmit beamforming for wireless and wireline multicasting where channel state information is available at the transmitter, enabling transmit optimization beyond blind broadcasting. The study aims to minimize transmission power while meeting prescribed minimum SNR constraints at each intended receiver. The authors formulate min‑power and max‑min SNR problems, solve them with semidefinite relaxation derived from Lagrangian duality, and evaluate the approach through simulations on Rayleigh fading channels and VDSL data with uniform linear arrays. Both problems are NP‑hard, but SDR yields near‑optimal solutions within 3–4 dB of the optimum and sometimes exact solutions, demonstrating practical effectiveness.

Abstract

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.