Distributed Beamforming for Relay Networks Based on Second-Order Statistics of the Channel State Information

• Published in: IEEE transactions on signal processing Vol. 56; no. 9; pp. 4306 - 4316
• Main Authors: Havary-Nassab, V., Shahbazpanahi, S., Grami, A., Zhi-Quan Luo
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Array signal processing; Beamforming; Channel state information; Channels; Closed-form solution; Convex feasibility problem; Detection, estimation, filtering, equalization, prediction; distributed beamforming; distributed signal processing; Exact sciences and technology; Exact solutions; Information, signal and communications theory; Mathematical analysis; Mathematical models; Miscellaneous; Optimization; Quadratic programming; Quality of service; Receivers; Relay; relay networks; Relays; semidefinite programming; Signal and communications theory; Signal processing; Signal to noise ratio; Signal, noise; Statistical distributions; Studies; Telecommunications and information theory; Transmitters; Wireless networks; semi-definite programming; Distributed beamforming; relay networks; convex feasibility problem; distributed signal processing; Second order; Parameter estimation; Relay; Wireless telecommunication; Transmitter; Convex programming; Beam forming; Optimization; semidefinite programming; Relaxation; Distributed processing; Convex feasibility problem; Channel estimation; Interior point method; Order statistic; Quadratic programming; Algorithm; Statistical method; Signal processing; Numerical simulation; Feasibility; distributed beamforming; Service quality; Signal to noise ratio
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2008.925945

In this paper, the problem of distributed beamforming is considered for a wireless network which consists of a transmitter, a receiver, and relay nodes. For such a network, assuming that the second-order statistics of the channel coefficients are available, we study two different beamforming design approaches. As the first approach, we design the beamformer through minimization of the total transmit power subject to the receiver quality of service constraint. We show that this approach yields a closed-form solution. In the second approach, the beamforming weights are obtained through maximizing the receiver signal-to-noise ratio (SNR) subject to two different types of power constraints, namely the total transmit power constraint and individual relay power constraints. We show that the total power constraint leads to a closed-form solution while the individual relay power constraints result in a quadratic programming optimization problem. The later optimization problem does not have a closed-form solution. However, it is shown that using semidefinite relaxation, this problem can be turned into a convex feasibility semidefinite programming (SDP), and therefore, can be efficiently solved using interior point methods. Furthermore, we develop a simplified, thus suboptimal, technique which is computationally more efficient than the SDP approach. In fact, the simplified algorithm provides the beamforming weight vector in a closed form. Our numerical examples show that as the uncertainty in the channel state information is increased, satisfying the quality of service constraint becomes harder, i.e., it takes more power to satisfy these constraints. Also our simulation results show that when compared to the SDP-based method, our simplified technique suffers a 2-dB loss in SNR for low to moderate values of transmit power.