SNR Maximization and Distributed Beamforming in Multiuser Multi-relay Networks Duy H. N. Nguyen and Ha H. Nguyen Department of Electrical and Computer Engineering University of Saskatchewan 57 Campus Dr., Saskatoon, SK, Canada S7N 5A9 duy.nguyen@usask.ca, ha.nguyen@usask.ca Abstract—This paper studies optimal distributed beamforming designs to jointly maximize the signal-to-noise (SNR) margin in a multiuser multi-relay network. Considered are optimization problems with two different types of power constraints: sum relay power constraint and per-relay power constraints. Although these two problems can be readily solved by the bisection method via a sequence of second-order conic feasibility programs, we propose simple and fast converging iterative algorithms to directly solve the two optimization problems under consideration. I. I NTRODUCTION A wireless relay network allows multiple relays to cooperate with each other, emulate a virtual array of transmit antennas, and assist pairs of source-destination in their communications. It is widely known that such cooperation can significantly improve the transmission reliability of the source signals [1]. When a relay has full knowledge of its locally-bidirectional channel state information (CSI), i.e., source-to-relay (S → R) and relay-to-destination (R → D) channels, it can beam the signals such that the received signals at the destination are coherently constructed. Moreover, the relay can also adjust its transmitted power. This cooperative strategy, referred to as distributed beamforming, was investigated in [2]–[5] to maximize the achievable signal-to-noise ratio (SNR) at the destination. In particular, reference [2] shows that, depending on its own bidirectional channels and other relays’ channels, each relay may not transmit at its maximum power to achieve the optimal SNR. References [3], [4] show that the SNR maxi- mization problem can be solved efficiently through a sequence of convex feasibility problems using the bisection method. With a sum power constraint at the relays, a closed-form solution to the optimal distributed beamformer is presented in [4], [5]. However, most of the early works in distributed beamforming only consider the system with one source and one destination. In [6], we examined the optimal distributed beamforming strategy in a multiuser multi-relay network with a guaranteed Quality of Service (QoS) in terms of SNR at the destina- tions. While guaranteed QoS is important to maintain the connectivity for the users, it is also critical to perform a strict power control at the relays due to the following reasons. First, the relays, which can be any nodes in the network, are often operating on a small power budget. Power conservation at the relays helps to extend the lifetime of the relays and hence the network as well. Second, if one user’s channel is in deep fades, the relays may consume an excessive amount of power to maintain the user’s connectivity, and thus may induce significant interference to adjacent networks. Built upon our early works in [6], [7], this paper studies the optimal distributed beamforming designs to jointly maximize the SNR margin at the destinations subject to two different types of power constraints: the sum relay power constraint and per- relay power constraints. It is first shown that the two problems can be effectively solved by the bisection method via second- order conic programming (SOCP) feasibility problems [8]. We then propose a simple but yet efficient fixed-point iteration algorithm, which does not rely on any external convex solution package, to directly solve the problems. Notations: Superscripts (·) T , (·) ∗ , and (·) H stand for trans- pose, complex conjugate, and complex conjugate transpose operations, respectively; x ⋆ denotes the optimal value of the variable x; CN (0,σ 2 ) denotes a circularly symmetric complex Gaussian random variable with variance σ 2 . II. SYSTEM MODEL 1 g N g 1 f N f 11 f 12 f 1R f 21 g 1 R g 11 g Fig. 1. Block diagram of a distributed beamforming system with R relays and N users. We consider a R-relay network with N pairs of source- destination (users) (S n -D n , n =1,...,N ) as illustrated in Fig. 1. All users share and compete for the transmitted power at the relays to maximize their SNRs. It is assumed that all the relays work in a half-duplex mode and the communication between the two terminals of each user occurs over two stages of transmissions. In the first stage, each user’s source This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings. 978-1-4244-4148-8/09/$25.00 ©2009