ABSTRACT
CONTENTS 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 21.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503 21.3 Polynomial-Time Approximation Algorithm for the Robust Beamforming Problem . 505
21.3.1 An equivalent QCQP Reformulation of the Robust Optimal Beamforming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
21.3.2 Randomized Approximation Algorithm for the Robust Beamforming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
21.3.3 Solvable Subclasses of the Robust Beamforming Problem via SDP Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
21.4 Another Randomized Approximation Algorithm via Complex-Valued S-Lemma and Convex Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 21.4.1 Complex-Valued S-Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 21.4.2 Convex Relaxation for the Robust Beamforming Problem . . . . . . . . . . . . . . . . . . 513 21.4.3 Relations between the Two Relaxation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 514 21.4.4 Another Randomized Approximation Algorithm via Convex SDP
Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 21.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516
21.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 21.7 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
21.1 Introduction In the spectrum sharing cognitive radio (CR) networks, downlink beamforming design for the secondary transmission has been an intensive research topic in the past decade. Spectrum sharing allows secondary and primary users to access the same channel concurrently, and beamforming techniques can be applied in order to avoid excessive interference caused to the primary users while steering power towards the secondary receivers, and by equipping the secondary transmitter (e.g., a base station or access point) with antenna arrays. For a comprehensive coverage of the recent advances on CR communications and networking, readers are referred to the survey paper [1] and the magazine paper [2]. In particular, [3]– [5] provided readers some recent specific works on optimal CR transmit beamforming.
