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Collaborative-Relay Beamforming With Perfect CSI: Optimum and Distributed Implementation Gan Zheng, Kai-Kit Wong, Arogyaswami Paulraj, and Björn Ottersten

Abstract—This letter studies the collaborative use of amplify-and-forward (AF) relays to form a virtual multiple-input single-output (MISO) beamforming system with the aid of perfect channel state information (CSI) in a flat-fading channel. In particular, we optimize the relay weights jointly to maximize the received signal-to-noise ratio (SNR) at the destination terminal with both individual and total power constraints at the relays. We show that the optimal collaborative-relay beamforming (CRB) solution achieves the full diversity of a MISO antenna system. Another main contribution of this letter is a distributed algorithm that allows each individual relay to learn its own weight, based on the Karush–Kuhn–Tucker (KKT) analysis. Index Terms—Collaborative beamforming, distributed implementation, relay.

I. INTRODUCTION

C

OMMUNICATIONS over wireless channels continues to be a major challenge of today’s technologies. Although multiple-input multiple-output (MIMO) antenna systems offer a means to create more capacity [1], [2], its use is tightly constrained by the limited space of a mobile device. On the other hand, however, recent researches have demonstrated that users can cooperate to form a distributed multi-antenna system by relaying [3]–[8]. Among them, amplify-and-forward (AF) relaying, which scales the received noisy signal from the sender and forwards it to the destination, is arguably the most attractive strategy, due to its low implementation complexity. While previous works focused mostly on the use of fixed-gain AF relays, recent attempts have turned to the joint optimization of power allocation at the relays with the aid of some channel state information (CSI) [9]–[12]. If CSI is exploited, the relays can mimic a MIMO beamforming system. This letter studies a dual-hop wireless relay network where the direct link between the source and the destination is broken. Our aim is to maximize the destination signal-to-noise ratio (SNR) by jointly optimizing the complex relay weights, subject to both individual relays’ power constraints and a total power constraint, with the aid of perfect CSI.

Manuscript received August 11, 2008; revised November 09, 2008. Current version published February 11, 2009. This work was supported by the EPSRC under Grant EP/E022308/1, U.K. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Aria Nosratinia. G. Zheng and K.-K. Wong are with the Adastral Park Research Campus, University College London, Martlesham IP5 3RE, U.K. (e-mail: [email protected]; [email protected]). A. Paulraj is with the Information Systems Laboratory, Stanford University, Stanford, CA 94305 USA (e-mail: [email protected]). B. Ottersten is with the School of Electrical Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2008.2010810

Fig. 1. CRB channel.

A similar problem appears in [10] where an analytical solution is derived. Our proposed method can be viewed as an alternative under the convex optimization framework. The advantage of our work over [10] is that the presented results cope with both total and individual relay power constraints, while [10] has not addressed the total power constraint. The similar problem was also studied in [13] where only second-order statistics of the CSI is available with total and individual power constraints separately but not jointly. Our contributions are threefold. • We show that the optimal collaborative-relay beamforming (CRB) solution can be found by second-order cone programming (SOCP) under the convex optimization framework. • We prove that the optimal CRB system, even with individual relays’ power constraints, attains the same diversity order of a multiple-input single-output (MISO) channel. • A distributed algorithm that permits each relay to learn its own optimal weight is proposed. II. SYSTEM MODEL Consider a communication channel with a source , a destination , and relays depicted in Fig. 1. We assume that there is no direct link between and and the use of relays is necessary to establish the communication link. We also assume that the relays work synchronously in an AF manner by the complex weights to produce a virtual beam pointing to . as , and the We denote the channel between and channel between and as . We consider a more-orless local cluster of relays with source and destination , so and should be about the propagation losses contained in equal for equal relay sensitivities. The received signal at is given by , with the source symbol with , being the noise following a Gaussian distribution with and zero mean and variance of . The received signal is then and forwarded to . The scaling factor weighted by (1) is used to normalize the output power of .

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, so that

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The received signal at the destination is the superposition of the weighted noisy signals being forwarded by the relays , which is expressed as1

It has been proved in [12] that this solution achieves the full diversity order of , i.e., (8)

SNR

(2) where is the Gaussian noise at with zero mean and variance of . A boldface letter denotes the vector . has diagonal elements given by the vector The matrix , and denotes the transpose of a matrix or vector. The metric of interest is the received SNR at

(3)

which is to be maximized by optimizing aid of perfect CSI.

jointly with the

III. OPTIMAL CRB A. With a Total Power Constraint Given a total power constraint over the relays, mathematically, we aim to solve , where denotes the Frobenius norm of a vector. At the optimum, we can further write

(4) where denotes conjugate transposition. Then, (4) is recognized as a generalized eigenvalue problem and has the optimal solution

(5) where is chosen to ensure , , is therefore

and the relay weight at

(6) where denotes the conjugate of a complex number. The corresponding maximum SNR is

where

includes both

for some target rate

and

, and

.

B. With Individual and Total Power Constraints In a multiuser network such as the relay system we study in this letter, it becomes necessary to consider individual power constraints at the relays, which can be reflected by having or where we have used to denote element-wise norm-square operation and is a column vector containing the elements . In what follows, the problem of interest will be to (9) , the total constraint can be igNote that if nored, since it is always satisfied. The total power constraint only infringes on the individual power limits when their sum exceeds . Equation (9) is quasi-convex and therefore is closely related to (10)

in (9) [i.e., the maximum achievable In particular, if SNR of (9)], the optimal of (10) is also optimal for (9). As such, the optimal solution of (9) can be obtained by solving (10) repeatedly using a bisection search over . It remains to solve (10), with the knowledge of , , and . To do so, we note that the phase of the optimal should be chosen to match the conjugate of the channel , as this will ensure that the signal components are added constructively at . The resulting problem then becomes a power allocation problem to obtain the transmit power at the relay with the equivalent (real) channel and noise . The problem is then recognized as power an SOCP problem, convex, and can be solved optimally. The following theorem describes the diversity of the optimal CRB system. Theorem 1: The optimal -relay CRB system with individual relays’ power constraints achieves the full diversity of as in a MISO antenna system. Proof: Consider the CRB vector where is given by (5) with and is chosen to . Then, and the following SNR is meet achievable:

(7) 1Although the model assumes a quasi-static flat-fading channel, the results in this letter apply also to frequency-selective fading channels, if orthogonal frequency-division multiplexing (OFDM) is adopted.

(11)

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where is recognized as the maximum SNR for a CRB system with a total power constraint . As asymptotically achieves the full diversity of , the SNR achievable by can also achieve the full diversity, and so does the optimal CRB system with individual relays’ power constraints . IV. DISTRIBUTED IMPLEMENTATION It is assumed that the optimization is performed at , which needs to know all the parameters necessary. In practice, can learn by training and sent from the relays, respecand know tively. It can also measure its own noise level from the relays. The power constraints and the (e.g., battery levels) are assumed known to , which means that knows the SNRs and . After has computed the optimal relay weights , it informs each individual relay , which in practice about its own amplification coefficient, will take some bandwidth to do so. Remarkably, in this section, we show that a distributed algorithm, which allows each relay to learn its own weight based on the local CSI, is possible. We assume that individual CSI is available at each relay, i.e., each and from by training and from by relay can learn . The alfeedback, respectively, and measure its noise level gorithm is a result of Lemma 1 described below. Lemma 1: The optimal is in the null space of (12) where and are, respectively, the vectors containing the and , denotes the maximum SNR elements achievable for (9), and are the solutions of the dual of (10) (13) is unique up to its norm. and that Proof: See the Appendix. From Lemma 1, we have closed-form solution , with

, which leads to the

(14) which can further be written as (15) is the th element of . To where is a constant, and determine the scaling factor , it is noticed that at the optimum, there is at least one relay that uses its full power or the total power constraint is active. As such, is chosen to satisfy the relay power constraints, i.e.,

(16) From the Karush–Kuhn–Tucker (KKT) conditions, in (22), we have . In other words, at the optimum, for , if , then and the optimal power allocation is given . Otherwise, by and . Summarizing this, the distributed algorithm is presented as follows. 1) The destination obtains and from solving (9) using the bisection search. Then, is computed from (16).

Fig. 2. Received SNR at D against the total relay SNR for 10-relay CRB sys(dB) and p =N : P =MN 8m (solid line), tems with P =N and P =N P =MN (dashed line). (dB) and p =N

=0

= 10

=5

= 15

2) The destination broadcasts the common information ( , , and ) to all relays. , the optimal weight can be found locally as 3) At (17) V. SIMULATION RESULTS Simulations are conducted to evaluate the performance of the proposed system in independent Rayleigh flat-fading channels, for and where we assumed that for . The source SNR is defined as and is the performance metric. Unless otherwise specified, we consider the relays to have the same power budgets: and the same noise level as the destination, i.e., . To generate the results, we used Sedumi [14] together with YALMIP [15] to solve the SOCP problems. In Fig. 2, the destination SNR results against the total relay SNR of are provided for a 10-relay CRB system with (dB) and (solid line), and (dB) and (dashed line). As can be seen, the optimal CRB solution achieves 1 and (dB) higher SNR than the fixed power allocation in the low and medium-to-high SNR regimes, respectively. Results also show that the performance with both total and individual power constraints is close to that with total power constraint only. The consideration of both power constraints appears to be important as compared with individual power constraints only, especially when the individual power limit is relatively large (dashed line) and an apparent gap is observed. The SNR outage performance results, measured by , are plotted in Fig. 3, with the . As expected, the optimal assumption solution with both individual and the total power constraints attains the same (full) diversity order as that with only the total power constraint. Results also illustrate that the fixed power allocation solution cannot achieve the full diversity order, due to the fact that the power allocation is not adaptive to the CSI variation. VI. CONCLUSION This letter has addressed the SNR maximization of a CRB channel with perfect CSI. The optimal CRB solution can be

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Also, we have the complementary conditions (22)

Without loss of generality, suppose that the optimal solution of , with . Because of (19) is the conditions, should lie in the null space of . Now, we show that any vector, say, , that lies in the null space of is unique up to its norm. This can be done by seeing that implies

Fig. 3. Received SNR at D against the individual relay SNR for various number 8 . with of collaborative relays

M

p =N = P =N m

found by SOCP and realized by a distributed algorithm that permits each individual relay to learn its own weight. Simulation results have illustrated that the proposed CRB system yields significant performance gains over the fixed power allocation system. APPENDIX PROOF OF LEMMA 1

Note that is a positive matrix and is the eigenvector of this matrix associated with the positive eigenvalue 1. Then according to the Perron–Frobenius theorem [16], for any positive matrix, we can conclude that is unique up to its norm. This further implies that the solution of (19) must be a positive rank-1 solution , which is optimal for (18), and obviously that lies in the null space of . REFERENCES

Proof: Rewrite the power minimization problem (10)

(18)

Introducing

(23)

, we study the SDP relaxation problem

trace

(19) where is the th element of and denotes a matrix with all zeros except its th diagonal entry being 1. Due to the rank relaxation, (19) in general yields a lower bound of (18). However, we shall prove that (19) always gives a rank-1 solution, and thus the optimal solution for (19) is also optimal for (18). To do so, we know that the Lagrangian of (19) is given by

(20) with its dual

(21)

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