IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 8, OCTOBER 2013 4093

with MRC and the scheduled STBC systems, respectively. Using thepdfs, we derived the approximated closed-form expressions of theaverage postprocessing SNR and ergodic capacity for the scheduledCR systems with MIMO transmission. Through the analysis, weprovide that the scheduled MIMO systems have both SNR gainsand ergodic capacity gain compared with a nonscheduled system byMUD. Through numerical results, we found that the performanceimprovement of scheduled TAS with MRC by MUD is higher thanthat of the scheduled STBC. In addition, both scheduled systems underimperfect CSI degrade the ergodic capacity over the perfect CSI.

REFERENCES[1] T. Skinner and J. Cavers, Selective diversity for rayleigh fading chan-

nels with a feedback link, IEEE Trans. Commun., vol. COM-21, no. 2,pp. 117126, Feb. 1973.

[2] S. Thoen, L. V. Perre, B. Gyselinckx, and M. Engels, Performanceanalysis of combined transmit-SC/receive-MRC, IEEE Trans. Commun.,vol. 49, no. 1, pp. 58, Jan. 2001.

[3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time blockcodes from orthogonal designs, IEEE Trans. Inf. Theory, vol. 45, no. 5,pp. 744765, Jul. 1999.

[4] S. Alamouti, A simple transmitter diversity, technique for wireless com-munications, IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 14511458,Sep. 1998.

[5] J. Mitola and G. Q. Maguire, Cognitive radio: Making software ra-dios more personal, IEEE Pers. Commun., vol. 6, no. 4, pp. 1318,Apr. 1999.

[6] A. Ghasemi and S. S. Elvino, Fundamental limits of spectrum-sharingin fading environments, IEEE Trans. Wireless Commun., vol. 6, no. 2,pp. 649658, Feb. 2007.

[7] X. Kang, Y. C. Liang, H. K. Garg, and R. Zhang, Optimal power alloca-tion for fading channels in cognitive radio networks: ergodic capacity andoutage capacity, IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 940950, Feb. 2009.

[8] Z. Chen, C. Wang, X. Hong, J. Thompson, S. Vorobyov, F. Zhao, andX. Ge, Interference mitigation for cognitive radio MIMO systems basedon practical precoding, Phys. Commun., in press.

[9] K. Cumanan, R. Zhang, and S. Lambotharan, A new design paradigm forMIMO cognitive radio with primary user rate constraint, IEEE Commun.Lett., vol. 16, no. 5, pp. 706709, May 2012.

[10] T. Ban, W. Choi, B. Jung, and D. Sung, Multi-user diversity in a spectrumsharing system, IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 102106, Jan. 2009.

[11] M. F. Hanif, P. J. Smith, D. P. Taylor, and P. A. Martin, MIMO cognitiveradios with antenna selection, IEEE Trans. Wireless Commun., vol. 10,no. 11, pp. 36883699, Nov. 2011.

[12] D. Li, Effect of channel estimation errors on arbitrary transmit antennaselection for cognitive MISO systems, IEEE Commun. Lett., vol. 15,no. 6, pp. 656658, Jun. 2011.

[13] Asaduzzaman and H. Y. Kong, Ergodic and outage capacity of interfer-ence temperature-limited cognitive radio multi-input multi-output chan-nel, IET Commu., vol. 5, no. 5, pp. 652659, Mar. 2011.

[14] D. Lee and K. Kim, Error probability analysis of combining space-timeblock coding and scheduling in MIMO systems, IEEE Signal Process.Lett., vol. 16, no. 12, pp. 10711074, Dec. 2009.

[15] X. Zhang, Z. Lv, and W. Wang, Performance analysis of multiuser di-versity in MIMO systems with antenna selection, IEEE Trans. WirelessCommun., vol. 7, no. 1, pp. 1521, Jan. 2008.

[16] N. Balakrishman and A. Cohen, Order Statistics and Inference, Estima-tion Methods. New York, NY, USA: Academic, 1991.

[17] N. Johnson, S. Kots, and N. Balakrishnan, Contiunous Univari-ate Distributions, 2nd ed. New York, NY, USA: Wiley-Interscience,1995.

[18] N. Ma, Complete multinomial expansions, Appl. Math. Comput.,vol. 124, no. 3, pp. 365370, Dec. 2001.

[19] J. Perez, J. Ibanez, L. Vielva, and I. Santanmaria, Closed-form approx-imation for the outage capacity of orthogonal STBC, IEEE Commun.Lett., vol. 9, no. 11, pp. 961963, Nov. 2005.

A New Beamforming Design for Multicast Systems

Wookbong Lee, Haewook Park, Han-Bae Kong, Jin Sam Kwak,and Inkyu Lee, Senior Member, IEEE

AbstractIn this paper, we study a transmit beamforming techniquefor multiple-inputmultiple-output (MIMO) downlink multicast systems.Assuming that channel state information (CSI) is available at a trans-mitter, we try to maximize the lowest transmit data rate among usersfor a given transmit power level. Since finding the optimal beamformingvector requires prohibitively high computational complexity, we proposea reduced complexity scheme by applying greedy vector search and asimple power-allocation (PA) algorithm. Simulation results show that theproposed scheme outperforms the conventional semidefinite relaxation(SDR) technique and provides a performance quite close to the optimalperformance with much reduced complexity.

Index TermsDownlink beamforming, multicast channels,multicasting.

I. INTRODUCTION

Multicasting is one of the important applications in wireless sys-tems, particularly for multimedia services. A traditional approach inthe multicasting is radiating transmission power isotropically [1] orusing open-loop techniques, such as spacetime coding [2], so that allusers in a cell can be covered. Unlike broadcasting, which distributesdata to all users in a cell, multicasting transmits a common signal onlyto a group of intended users. Thus, we do not necessarily design atransmission scheme to support all users in a cell. In the meantime, wecan utilize channel state information (CSI) for designing the transmitbeamformer to improve the system capacity or coverage in closed-loopmultiple-inputmultiple-output (MIMO) systems [3][5].

There have been two major approaches for determining the transmitbeamforming vector in MIMO downlink multicasting systems. One isbased on finding the minimum transmit power level, which guaranteesthe transmit-data-rate requirement for all users. The other is maximiz-ing the lowest transmit data rate among users for a given transmitpower level. Between these approaches, we focus on the latter casein this paper since the solution of the former problem can be derivedby the solution of the latter problem with slight modification [6].

It is known that optimizing a transmit beamformer in the multicastscenario is an NP-hard problem [6]. Therefore, various suboptimalprecoding techniques have been studied in the literature [6][8].To solve the optimization problem, Sidiropoulos et al. [6] relaxedthe problem by employing semidefinite relaxation (SDR) techniques.Another method was proposed in [7], which slowly changes thetransmit beamforming vector so that the worst user data rate can beimproved. In addition, a beamforming scheme was introduced in [8],which employs Lagrangian formulation to solve the problem for each

Manuscript received July 20, 2012; revised October 11, 2012; acceptedDecember 6, 2012. Date of publication December 28, 2012; date of currentversion October 12, 2013. This work was supported in part by the NationalResearch Foundation of Korea under Grant 2010-0017909. The review of thispaper was coordinated by Dr. C. Yuen.

W. Lee, H. Park, H.-B. Kong, and I. Lee are with the School of ElectricalEngineering, Korea University, Seoul 136-701, Korea (e-mail: wookbong@korea.ac.kr; jetaime01@korea.ac.kr; kongbae@korea.ac.kr; inkyu@korea.ac.kr).

J. S. Kwak is with the Department of Electrical Engineering, Korea Ad-vanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail:samji0908@gmail.com).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2012.2236583

0018-9545 2012 IEEE

4094 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 8, OCTOBER 2013

Fig. 1. Wireless multicasting systems with transmit beamforming.

permuted user channel vector. In this case, K! iterations are required tosearch all possible user permutations, where K is the number of users.

In this paper, we propose a simple algorithm that achieves the near-optimal performance of MIMO multicasting systems. To develop anefficient algorithm, we first consider a beamforming vector, whichconsists of basis vectors and transmit power, and then present a newalgorithm by applying basis vector set selection using a greedy methodand power allocation (PA) for the given vector set. It is confirmedfrom simulations that the proposed scheme can achieve near-optimalperformance with much reduced complexity, and it outperforms theconventional SDR scheme.

The following notations are used throughout this paper. We employuppercase boldface letters for matrices and lowercase boldface forvectors. For any general matrix A, AH and AT denote Hermitian andtranspose, respectively. In addition, for any complex scalar variable a,(a) represents the real part of a. CMN and RMN+ indicate a set ofcomplex and positive real matrices of size M by N , respectively.

The rest of this paper is organized as follows. In Section II, weintroduce MIMO multicast systems and formulate the optimizationproblem of interest. The proposed scheme is shown in Section III.Section IV presents the performance results of the proposed scheme.Section V concludes this paper.

II. SYSTEM MODEL AND PROBLEM FORMULATION

Consider a wireless multicasting system that consists of a singlebase station with M transmit antennas and K users equipped withN receive antennas,1 as shown in Fig. 1. Under frequency-flat fadingchannels, the received signal vector yk of length N at user k (k =1, . . . ,K) is given by

yk = Hkwx+ zk (1)

where Hk CNM stands for the complex channel response matrixfrom the base station to the kth user, w CM1 represents thetransmit beamforming vector with transmit power constraint wHw P , x is the transmit data symbol with unit variance, and zk CN1indicates the complex additive white Gaussian noise vector at the kth

1Although we assume that all users are equipped with the same number ofreceive antennas, all work in this paper can be simply extended to the case of adifferent number of antennas at each users.

user with an identity covariance matrix. In this case, the received SNRat the kth user, i.e., gk (k = 1, . . . ,K), can be written as

gk = Hkw2. (2)We assume that perfect CSI for all users are available at the trans-

mitter. The goal of our multicasting problem is maximizing the lowesttransmit data rate, i.e., max

wmin

k=1,...,Klog(1 + gk) or, equivalently,

maximizing the minimum SNR, which can be formulated as

maxw

mink=1,...,K

gk s.t. wHw P. (3)

Since any complex vector of lengthM can be represented by a linearcombination of M basis vectors, a solution of the problem (3) can bewritten as

w = Vc (4)where we have basis matrix V = [v1,v2, ,vM ] CMM andcoefficient vector c = [c1, c2, . . . , cM ]T CM1. Here, vi CM1denotes the ith basis vector, and ci represents the complex coefficientof the ith basis vector. Note that any vectors {vi} can be employedas long as w spans CM1 space, but to make (3) tractable, we chooseorthonormal basis vectors, i.e., vHmvn = mn. Now, let us define ci =Pie

ji , where i and Pi indicate the phase and the transmit powerlevel for the ith basis vector, respectively, and

Mi=1

Pi P to meetthe power constraint in (3).

One method to solve problem (3) is exhaustive search by examiningall possible continuous values of {P1, . . . , PM} and {1, . . . , M},which is too complicated. Instead, we implement an exhaustive searchmethod by identifying a solution over discrete values. Note that thebeamforming vector ejw produces the same performance as wfor any . In addition, all Pi are positive values, and one of thepower value is determined automatically to meet the power constraint.Thus, the complexity of the exhaustive search becomes O((K/(M 1)!)DM11 D

M12 ), where D1 and D2 represent the number of

the discrete values within the search range of {P1, . . . , PM} and{1, . . . , M}, respectively. It is clear that the complexity becomesprohibitive and exponentially increases with respect to M .

III. PROPOSED MULTICAST BEAMFORMING SCHEME

Here, we propose a low-complexity multicast beamforming scheme.We first revisit the optimal beamforming vector in (4). It iseasy to show that the solution w of the optimization problem(3) is spanned by the stacked channel matrix H = [hH1, 1, . . . ,hH1, N , . . . ,h

HK, 1, . . . ,h

HK,N ] CMNK , where hk, l is the lth row

vector of the kth users channel matrix Hk. This can be explainedby contraction as follows. Suppose that there exists a certain vectorvn orthogonal to HH with cn = 0. Then, since vHn HHk Hkvn = 0for any k, it follows that gk = Hkw2 = Hkw2, where w =M

i=1,i =n civi. This is equivalent to solving (3) with constraintwHw = P , where P is given as P =

Mi=1,i =n |ci|2. However, the

objective function of (3) increases linearly with the maximum powerP . Thus, P should be equal to P to maximize the objective function.In other words, cn needs to be 0, which contradicts to the assumption.This means that the basis vector of w corresponding to a nonzerocoefficient should be spanned by H. Since w is a linear combinationof the basis vectors, it is also spanned by H. Furthermore, the optimumvalue of (3) is obtained when wHw = P .

Therefore, we can rewrite (4) asw = Vc (5)

where V = [v1, v2, , vR] CMR, and c = [c1, c2, . . . , cR]T C

R1. Here, cH c = P , and R (R M) is the rank of the stacked

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 8, OCTOBER 2013 4095

channel matrix H. Note that the space spanned by V is the same asthat spanned by H. Since V consists of orthonormal basis vectors, wecan obtain V by GramSchmidt orthogonalization from the columnvectors of the stacked channel matrix H.

Solving problem (3) using the beamforming vector in (5) is stilldifficult, since there exist too many parameters to optimize. To over-come this issue, we restrict ci to be real valued, i.e., i = 0. From thisrelaxation, we can formulate the solution of problem (3) as

w = Vp (6)where p = [

P1,

P2, . . . ,

PR]

T RR1+ , and pT p = P .In the following, we describe a method to select the basis vector set

of V, and then we present how to determine the PA vector p for thegiven basis matrix V.

A. Selection of the Basis Vector SetAs we restrict the coefficient of basis vectors to be real valued in

(6), the space spanned by the vectors does not fully cover the wholespace of the original structure in (4). In this case, a different orderingof column vectors in H for GramSchmidt orthogonalization producesdifferent subspaces, which affects performance. However, examiningall possible orderings of channel vectors requires R!

(NKR

)operations,

which becomes prohibitive as N , K, or R grows. Thus, we propose agreedy user selection algorithm similar to [9].

Denoting (k, l) as the index of the lth receive antenna at thekth user, we can represent the set of all channel vector indicesas U = {(1, 1), . . . , (1, N), . . . , (K, 1), . . . , (K,N)}. Let us de-fine V(R) = [v1, v2, . . . , vR], p(R) = [

P1

P2, . . . ,

PR]

T, and

SR = {s1, . . . , sR} U with |SR| = R as the basis matrix with rankR, the PA vector of length R, and the set of selected channel vectorindices, respectively. Here, V(R) can be computed by performing theGramSchmidt orgthogonalization with [hHs1 ,h

Hs2, . . . ,hHsR ].

Now, we explain our algorithm starting with R = 1. First, we findthe best vector hHk, l/hk, l among (k, l) U , which provides thebest performance of the objective function of (3). We set V(1) = [v1],and SR = {s1}, where s1 is the index corresponding to the bestvector, and v1 = hHs1/hs1. Next, we increase R by 1. Then, allthe vectors v(k, l) are examined, which can generate the best per-formance of the objective function with V(R)k, l = [V(R1)v(k, l)] =[v1, . . . , vR1 v(k, l)] and the corresponding PA vector p(R)k, l . Notethat v(k, l) can be calculated using the GramSchmidt orthogo-nalization with [V(R1)hHk, l], where index (k, l) is determined asU \ SR1. Moreover, for the given V(R)k, l , p(R)k, l can be determinedby the PA algorithm for a given vector set which will be describedin Section III-B. We repeat this until no additional performance gain isobtained by adding one more basis vector. We summarize the proposedalgorithm as follows.

Proposed Scheme

Set R = 0, V(0) = [ ], and S0 = .While R < M

Update R R+ 1.sR = argmax(k, l)U\SR1 minm=1,...,K HmV(R)k, l p(R)k, l 2.Set V(R) = [V(R1)v(sR)] and SR = SR1 {sR}.If minm HmV(R)p(R)2

minm HmV(R1)p(R1)2Set R R 1 and break.

EndEnd

Fig. 2. Exemplary plot of gk with respect to Pm.

B. PA Algorithm for a Given Vector SetBased on the beamforming vector (6), we can rewrite SNR at the

kth user, i.e., gk, as

gk =HkVp2

=|Akp2 (7)where Ak = HkV = [ak, 1, . . . ,ak,R] CNR and ak,i = Hkvi C

N1. To find the optimal PA, we first check the crossover points pc,

pc RR1+ , which satisfy gk(pc) = gl(pc) for l = k. The crossoverpoints can be obtained by solving

pTc Bklpc = 0 s.t. pTc pc = P

where Bkl = AHk Ak AHl Al. Unfortunately, it is difficult to solvethis quadratic equation in general. Instead, we employ an alternatingoptimization method that computes a local optimal solution. In this al-gorithm, we recursively determine a power level pair (Pm, Pn) (m =n) at a time.

To illustrate the key idea of our proposed PA method, we describehow to optimize Pm and Pn for fixed Pi (i = m, n). Let us denotepmn as a vector that is equal to p, except that the mth and the nthelements are zero. Then, using the power constraint pT p = P , Pncan be expressed as Pn = P mn Pm, where P mn = P pTmnpmn.In this case, we can rewrite (7) as a function of Pm as

gk(Pm) = (aHk,mak,m aHk, nak, n)Pm

+ 2(aHk,mak, n)

Pm(P mn Pm)

+ 2(pTmnAHk ak,m)

Pm

+ 2(pTmnAHk ak, n)

P mn Pm + Ck (8)for 0 Pm P mn, where Ck is a constant term independent ofPm. Fig. 2 depicts an exemplary plot of gk with respect to Pm. Asshown in Fig. 2, the worst user SNR for a given Pm is changed onlyafter the crossover point, and clearly, a local optimal point is one ofthe boundary points (Pm = 0 or P mn), the crossover points Pm, c,where gk(Pm, c) = gj(Pm, c), or the inflection points Pm, i, wheregk(Pm, i) = 0, with gk (Pm, i) < 0. From this observation, we cancome up with the following algorithm.

First, for R = 1, it is clear that p =P . In the case of R = 2, the

optimal PA is calculated as follows. We start from the minimum point

4096 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 8, OCTOBER 2013

of Pm, i.e., Pm = 0. Let us assume that the range of Pm is Pm,min Pm Pm,max. Then, suppose that the nth user has the minimumSNR gn(Pm) among K users at Pm = Pm,min, and Pm, c is theminimum crossover point between user n and j (j = n), whichsatisfies gn(Pm, c) = gj(Pm, c) and Pm,min Pm, c Pm,max.In this case, it is clear that gn(Pm) is the minimum among SNR ofK users for Pm,min Pm Pm, c. Among these minimum points,the maximum gn(Pm) can be obtained at either the boundary points(Pm = Pm,min or Pm, c) or the inflection points with gn(Pm, i) < 0.If gn(Pm, c) > gj(Pm, c), then the jth users SNR becomes theminimum at Pm, c since gj(Pm) < gn(Pm) for Pm, c < Pm. Ifgn(Pm, c) < g

j(Pm, c), then the nth user still has the minimum

SNR at Pm, c. With the same logic, we repeat the given procedureuntil there is no crossover point between Pm,min and Pm,max. Notethat the crossover point or the inflection point is computed by solving apolynomial equation of at most fourth order, which has a closed-formsolution.

To address the problem for R > 2, let us denote S as a set ofall possible combinations of (m,n) (m and n = 1, . . . , R). Then,we can find the PA vector as in the following example. In case ofR = 3, we have S = {(1, 2), (2, 3), (1, 3)}, and we first identify theoptimal P1 and P2 for given P3 using the algorithm described earlierfor R = 2. Then, P2 and P3 are calculated for fixed P1, and P1and P3 are computed for fixed P2. Since the formulated problem isnonconvex, the proposed alternating PA algorithm results in a localoptimal point. To improve the local optimal point, we can run thealgorithm with multiple random initial points p and choose the bestpoint among solutions generated by these initial points. As will beshown in the simulation section, we only need about three initial pointsto achieve the near-optimal performance. Note that we need to run thealgorithm for R = 2|S| times for each initial point. Denoting P m andf(P m) are the optimal power level of Pm for a given condition andthe corresponding minimum SNR among K users, we summarize theproposed PA algorithm for the given basis matrix V as follows.

PA Algorithm

Set max = 0 and initialize p with a random vector.For S = {(1, 2), (2, 3), . . .}

Find (P m, P n).If max < f(P m)

Update Pm = P m, Pn = P n , and max f(P m).End

End

Since the PA algorithm searches for at mostK1

k=1(K k)

crossover points and K inflection points for each stage, the complexityof the PA algorithm becomes less than O((Ni/4)(K2 +K)(R2 R)) for R > 1, where Ni is the number of initial points. In addition,the vector set selection algorithm finds (NK R+ 1) indices fora given R. Thus, the complexity of the proposed scheme in theworst case is bounded by O((Ni/12)NK3M3). Simulations showthat, with M = 3, K = 10, and SNR = 10 dB, the computationalcomplexity of our scheme is five orders of magnitude lower thanthat of the near-optimal solution, which is determined with a 0.1-dBresolution and a 0.1-rad resolution.

IV. SIMULATION RESULTS

Here, we demonstrate the efficiency of our proposed algorithmcompared with the conventional SDR scheme in [6] and the near-optimal solution introduced in Section II through Monte Carlo sim-ulations. In our simulations, it is assumed that the channel coefficients

Fig. 3. Performance of the PAGBM algorithm.

Fig. 4. Performance comparison for N = 1

are sampled from independent and identically distributed complexGaussian random variables with zero mean and unit variance.

First of all, we evaluate the performance of the PA algorithm inFig. 3, which plots the minimum user throughput performance withNi = 3. We assume that all users are equipped with two receiveantennas (N = 2), whereas a BS has three transmit antennas (M =3). In addition, we include the performance of an exhaustive powersearch method that serves as an upper bound to validate our algo-rithm. The exhaustive search method searches for all possible powerlevel combinations with a 0.1-dB resolution. Here, we compare bothmethods with the vector set selection proposed in Section III-A. Fromthe plot, we confirm that the PA algorithm with three initial pointsprovides similar performance to that of the exhaustive scheme withmuch reduced complexity.

In Figs. 4 and 5, we illustrate the average worst user rate of theproposed scheme and the conventional SDR scheme for the caseof N = 1 and 2, respectively. We assume that two or four transmitantennas are equipped at the BS. For M = 4, we run the PA algorithmwith three initial points. Note that we do not need random initial pointsin the case of M = 2 since |S| is always 1 in this case. We also includethe performance of the optimal solution (4), which is obtained witha 0.1-dB resolution for the power level and a 0.1-rad resolution for

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 8, OCTOBER 2013 4097

Fig. 5. Performance comparison for N = 2

the phase. Here, the optimal solution for M = 4 case is not includeddue to its extremely high complexity. One interesting point is that, asK grows, our proposed scheme approaches the performance of theoptimal solution, whereas the performance of the conventional schemedegrades. This is due to the fact that the subspace covered by the vectorset selection increases as K grows, whereas the performance of theconventional scheme deteriorates when K is large [10]. In addition,in the case of N = 1, about 15% and 52% gains at K = 40 can beobtained over the conventional scheme for M = 2 and 4, respectively.In addition, about 6% and 20% gains are observed in the case ofN = 2 for M = 2 and 4, respectively. Clearly, the gain of the proposedscheme over the conventional scheme increases as M increases.

V. CONCLUSION

In this paper, we have studied a transmit beamforming techniquefor MIMO downlink multicast systems. We have focused on max-imizing the lowest SNR among intended users, assuming that CSIis available at a transmitter. To solve the problem, we proposed anew beamforming method that reduces the complexity of the system.Simulation results show that the proposed scheme outperforms theconventional SDR-based scheme, and achieves near-optimal perfor-mance with much reduced complexity.

REFERENCES[1] Evolved Universal Terrestrial Radio Access (E-UTRA): Physical chan-

nels and modulation, Cedex, France, 3GPP TS 36.211, Dec. 2011.[2] WirelessMAN-Advanced Air Interface for Broadband Wireless Access

Systems, IEEE P802.16.1/D6, Apr. 2012.[3] H. Lee, S. Park, and I. Lee, Orthogonalized spatial multiplexing for

closed-loop MIMO systems, IEEE Trans. Commun., vol. 55, no. 5,pp. 10441052, May 2007.

[4] S.-H. Park, H.-S. Han, and I. Lee, A decoupling approach for low-complexity vector perturbation in multiuser downlink systems, IEEETrans. Wireless Commun., vol. 10, no. 6, pp. 16971701, Jun. 2011.

[5] H. Sung, S.-H. Park, K.-J. Lee, and I. Lee, Linear precoder designs forK-user interference channels, IEEE Trans. Wireless Commun., vol. 9,no. 1, pp. 291301, Jan. 2010.

[6] N. D. Sidiropoulos, T. N. Davidson, and Z.-Q. Luo, Transmit beamform-ing for physical-layer multicasting, IEEE Trans. Signal Process., vol. 54,no. 6, pp. 22392251, Jun. 2006.

[7] A. Lozano, Long-term transmit beamforming for wireless multicasting,in Proc. IEEE ICASSP, Apr. 2007, pp. 417420.

[8] M. A. Khojastepour, A. Khajehnejad, K. Sundaresan, and S. Rangarajan,Adaptive beamforming algorithms for wireless link layer multicasting,in Proc. IEEE PIMRC, Sep. 2011, pp. 19941998.

[9] G. Dimic and N. D. Sidiropoulos, On downlink beamforming withgreedy user selection: Performance analysis and a simple new algorithm,IEEE Trans. Signal Process., vol. 53, no. 10, pp. 38573868, Oct. 2005.

[10] Z.-Q. Luo, N. D. Sidiropoulos, P. Tseng, and S. Zhang, Approxi-mation bounds for quadratic optimization with homogeneous quadraticconstraints, SIAM J. Optim., vol. 18, no. 1, pp. 128, Feb. 2007.

Maximum-Likelihood Detector for DifferentialAmplify-and-Forward Cooperative Networks

Peng Liu, Student Member, IEEE, Il-Min Kim, Senior Member, IEEE,and Saeed Gazor, Senior Member, IEEE

AbstractThe exact maximum-likelihood (ML) detector for amplify-and-forward (AF) cooperative networks employing M -ary differentialphase-shift keying (DPSK) in Rayleigh fading is derived in a single-integralform, which serves as a benchmark for differential AF networks. Twoalgorithms are then developed to reduce the complexity of the ML detector.Specifically, the first algorithm can eliminate a number of candidates inthe ML search, while causing no loss of optimality of ML detection. Inhigh signal-to-noise ratios (SNRs), this algorithm almost surely identi-fies a single candidate that amounts to the ML estimate of the signal.For low to medium SNRs with multiple candidates determined, we thenderive an accurate closed-form approximation for the integral involvedin the likelihood function, which only requires a five-sample evaluationper symbol candidate. Finally, combining these algorithms, we propose aclosed-form approximate ML detector, which achieves an almost identicalbit-error-rate (BER) performance to the exact ML detector at practicalcomplexity. In particular, it is shown that the proposed approximate MLdetector is far less complex than the well-known diversity combiner in highSNRs, while achieving approximately 1.7-dB gain in the 105 BER whenthe relay is closer to the destination.

Index TermsAmplify-and-forward (AF), cooperative networks, differ-ential phase-shift keying (DPSK), maximum-likelihood (ML).

I. INTRODUCTION

Noncoherent signalings lend simplicity to receiver designs by elim-inating the need for instantaneous channel state information (CSI),thus constituting an attractive approach particularly for fast-fadingchannels [1], [2]. Recently, noncoherent modulations have been stud-ied for amplify-and-forward (AF) cooperative networks. In particular,Annavajjala et al. derived a maximum-likelihood (ML) detector forbinary noncoherent frequency-shift keying (NCFSK) in Rayleigh fad-ing [3]. However, the ML detector in [3] involves integrals that haveno closed-form solutions and, hence, are very complex for implemen-tation. In addition, suboptimum detectors with lower complexity butdegraded performance have been developed for NCFSK [3][5]. SinceFSK modulation is not a bandwidth-efficient noncoherent solution,it might not be suitable for bandwidth-limited applications. For such

Manuscript received October 30, 2012; revised March 28, 2013; acceptedApril 19, 2013. Date of publication April 24, 2013; date of current versionOctober 12, 2013. This work was supported by the Natural Sciences and Engi-neering Research Council of Canada. The review of this paper was coordinatedby Prof. Y. Ma.

The authors are with the Department of Electrical and Computer En-gineering, Queens University, Kingston, ON K7L 3N6, Canada (e-mail:peng.liu@queensu.ca; ilmin.kim@queensu.ca; gazor@queensu.ca).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2013.2259856

0018-9545 2013 IEEE

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