Coordinated Beamforming for the Multicell Multi-Antenna Wireless ...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1Coordinated Beamforming for theMulticell Multi-Antenna Wireless SystemHayssam Dahrouj Student Member, IEEE, and Wei Yu, Senior Member, IEEEAbstract—In a conventional wireless cellular system, signalprocessing is performed on a per-cell basis; out-of-cell interferenceis treated as background noise. This paper considers thebenefit of coordinating base-stations across multiple cells in amulti-antenna beamforming system, where multiple base-stationsmay jointly optimize their respective beamformers to improvethe overall system performance. Consider a multicell downlinkscenario where base-stations are equipped with multiple transmitantennas employing either linear beamforming or nonlineardirty-paper coding, and where remote users are equipped witha single antenna each, but where multiple remote users maybe active simultaneously in each cell. This paper focuses onthe design criteria of minimizing either the total weightedtransmitted power or the maximum per-antenna power acrossthe base-stations subject to signal-to-interference-and-noise-ratio(SINR) constraints at the remote users. The main contribution ofthe paper is an efficient algorithm for finding the joint globallyoptimal beamformers across all base-stations. The proposedalgorithm is based on a generalization of uplink-downlink dualityto the multicell setting using the Lagrangian duality theory.An important feature is that it naturally leads to a distributedimplementation in time-division duplex (TDD) systems. Simulationresults suggest that coordinating the beamforming vectorsalone already provide appreciable performance improvements ascompared to the conventional per-cell optimized network.Index Terms—Beamforming, dirty-paper coding, uplinkdownlinkduality, multiple-input multiple-output (MIMO), multicellsystemsI. INTRODUCTIONCONVENTIONAL wireless systems are designed with acellular architecture in which base-stations from differentcells communicate with their respective remote terminalsindependently. Signal processing is performed on a per-cellbasis; intercell interference is treated as background noise;intercell coordination is limited to the handling of mobilehandoff. Conventional cellular networks are also typicallydesigned to operate in the intercell-interference limited regime.Consequently, the performance of a conventional network canbe significantly improved if joint signal processing is enabledManuscript received June 23, 2009; revised November 27, 2009; acceptedJanuary 26, 2010. The associate editor coordinating the review of this paperand approving it for publication was A. Hjørungnes.The material in this paper was presented in part at the Conference onInformation Sciences and Systems (CISS), Princeton, NJ, U.S.A., March2008.The authors are with The Edward S. Rogers Sr. Department of Electricaland Computer Engineering, University of Toronto, 10 Kings College Road,Toronto, Ontario M5S 3G4, Canada (e-mail: hayssam.dahrouj@utoronto.ca,weiyu@comm.utoronto.ca).This work was supported by LG Electronics and by Natural Science andEngineering Research Council (NSERC) under the Canada Research Chairsprogram.Digital Object Identifier 10.1109/TWC.2010.0909361536-1276/10$25.00 c○ 2010 IEEEacross the different base-stations to minimize or even to cancelintercell interference.This paper evaluates the benefit of one particular type ofbase-station coordination for the multicell downlink system.The setup here is a scenario in which the base-stations areequipped with multiple antennas and the remote receiversare equipped with a single antenna each. Within each cell,multiple remote users may be active simultaneously and areseparated via spatial multiplexing using beamforming. In aconventional system, the beamforming vectors in each cellare set independently. The main point of this paper is thatit is possible to improve the overall network performance byallowing beamforming vectors from different base-stations tobe coordinated.The system setup in this paper differs from many of thecoordinated joint processing systems studied in the literature(e.g. [1], [2], [3], [4]) where antennas from multiple basestationsact as a single antenna array. Such a fully coordinatedcase is sometime referred to as “network MIMO” (multipleinputmultiple-output), where much larger performance gainsare conceivable. However, network MIMO comes at a costof signal-level coordination, i.e. data streams intended fordifferent mobile users belonging to different cells need tobe shared among the base-stations. In contrast, the systemconsidered in this paper only requires coordination at thebeamforming level, and is therefore much easier to implement.Downlink beamforming for multi-antenna wireless systemshas been studied extensively in the literature. A conceptknown as uplink-downlink duality has emerged as a maintool. This paper extends duality to the multicell setting byestablishing that the multicell downlink beamforming problemfor minimizing either the total weighted transmit power or themaximum per-antenna power subject to the received signal-tonoise-and-interference-ratio(SINR) constraints can be solvedvia a dual uplink problem. The duality holds for both lineartransmit beamforming and for the nonlinear dirty-paper codedsystem. The main contribution of this paper is an efficientalgorithm, which is capable of finding the globally optimaldownlink beamforming vector across all base-stations. Thisalgorithm is a multicell generalization of a similar algorithmproposed in [5] for the single-cell case. A key advantage ofthe proposed algorithm is that it naturally leads to a distributedimplementation in a time-division duplex (TDD) system.As an alternative to the transmit power minimization problemmentioned above, one can also formulate a rate-regionmaximization problem subject to power constraints at the basestations.Both problems are of practical interest. Although itis possible to show that uplink-downlink duality continues to

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATIONhold for the rate region maximization problem under suitableconditions (e.g. sum power constraint across all the transmitters),numerical optimization of achievable rate regionsbecomes much more difficult. For this reason, the remainderof this paper restricts its attention to the power minimizationproblem subject to SINR constraints.A. Related WorkThe benefit of base-station coordination in a cellular networkhas been a subject of many recent studies. Motivatedby the joint detection and cooperation techniques for intracellinterference mitigation, [1], [2], [3], [4], [6], [7], [8] study thecapacity improvement due to the joint encoding or decodingacross the base-stations for intercell interference mitigation.Base-station coordination is practically conceivable becausein a cellular network base-stations are connected by highcapacitybackhaul links. However, the amount of backhaulcommunications required to achieve joint processing is alsosubstantial. This motivates network models with either constrainedbackhaul capacity [9], [10], [11] or with cooperationonly among neighboring base-stations [12], [13], [14] oramong cluster of base-stations [15].The problem setting in this paper differs from the aboveseries of works in that rather than joint processing at asignal level, we consider coordination at the beamformer level,which requires much less overhead and is more practical toimplement. In particular, the data stream for each user onlyneeds to be (pre-)processed at its own base-station (and notacross all the base-stations). Further, the base-stations neednot be symbol synchronized as required with signal-levelcoordination. With these practical considerations, this paperfocuses on the joint optimization of transmit beamformersacross the base-stations for minimizing the transmit powersubject to quality-of-service constraints at the mobiles. Thissetup is suitable for constant bit-rate traffic with stringentdelay constraints. As the simulation results of this papershow, appreciable performance gain can already be obtainedwith this limited form of coordination. Related works basedon the minimization of packet loss probability [16] and themaximization of capacity [17], [18] have also been reportedin the literature.The transmit beamforming design problem goes back tothe classic work of [19], where an iterative algorithm isproposed for the optimization of the beamforming vectorsand power allocations to satisfy a set of target SINRs for anarbitrary set of transmission links. The main contribution of[19] is a beamformer-power update algorithm based on anuplink-downlink duality that converges to a feasible solutionto the problem. In the single-cell multi-user downlink case,the optimality of this duality-based approach is proved in[20] and [21], [22]. More recently, [5] shows that the singlecelldownlink beamforming problem can be formulated as asecond-order cone-programming problem. This crucial insightallows an interpretation of duality via Lagrangian theory inconvex optimization [23].The single-cell uplink-downlink duality can be immediatelygeneralized to the multicell setting if signal-level coordinationbetween the base-stations is assumed. This is shown in [24]in a CDMA context, where a beamformer-power iterationalgorithm similar to that of [19] is proposed.Less obvious is the question of whether uplink-downlinkduality continues to hold in a multicell network withbeamforming-level coordination only. This paper uses a Lagrangianduality approach to establish that duality indeedexists in this case. In addition, this paper proposes an optimizationprocedure based on power iteration alone. Theproposed algorithm has a key advantage of being amenableto distributed implementation, which is highly desirable ina multicell network. Further, this paper takes realistic powerconstraint into account by solving the problem of minimizingthe maximum per-antenna power constraint across all the basestations.Finally, this paper provides a generalization to includedirty-paper coding within each cell.The multicell uplink-downlink duality considered in thispaper is related to the concept of network duality proposedin [25]. However, the network duality established in [25] isderived based on a linear programming approach, which isdifferent from the Lagrangian approach taken in this paper.Consequently, [25] arrives at a different set of numericalalgorithms for the coordinated beamforming problem, whichhappen to be not as easily implementable in a distributedfashion. In fact, to obtain distributed solutions, [25] has toresort to suboptimal algorithms. As a further note, [25] dealswith the more general problem of joint transmit and receivebeamforming optimization for a multicell network where boththe base-station and the remote users are equipped with multipleantennas. Although uplink-downlink duality continues tohold, one consequence of this more general setup is that theproof of global optimality is no longer available, in contrastto the simpler single-antenna-per-remote-user case consideredin this paper. Finally, the coordinated beamforming problemcan also be solved using yet another different approach basedon treating all the base-stations as a single transmitter, thenmodifying the corresponding channel matrix and determiningthe corresponding optimal beamforming vectors [26]. However,distributed implementation of the resulting algorithm isnot yet available.Coordination at the beamforming-level for the multicellmulti-antenna channel has also been explored in [27], [28]from a viewpoint of egotistic vs. altruistic strategies. Theseworks focus on the Pareto boundary of the achievable rateregion of the multicell network under fixed power constraint,which is complementary to the problem of power minimizationunder fixed SINR constraints considered in this paper. Theachievable rate region of multicell systems has also beenexplored in [29] using duality, and in [30]. In particular,[30] deals with a more general setting with different possiblelevels of coordination between the base-stations. The problemsetting of the current paper corresponds to a particular limitedcoordination scenario in [30]. Further, [30] also addresses theissue of imperfect channel knowledge at the base-stations.The problem setting of this paper assumes that the setof active mobile users with each cell and their respectiveSINR constraints are fixed. User scheduling (e.g. [31], [32],[33]) and congestion control strategies (e.g. [26]) are assumedto be performed separately. Throughout the paper, perfectchannel side information for mobile users within each cell

DAHROUJ and YU: COORDINATED BEAMFORMING FOR THE MULTICELL MULTI-ANTENNA WIRELESS SYSTEM 3is assumed to be available at each base-station. In a practicalimplementation, channel side information may be estimatedvia channel reciprocity for a TDD system or via a feedbackmechanism; see [34], [35].B. OrganizationThe remainder of the paper is organized as follows. SectionII contains the problem formulation and the system model.In Section III, we establish uplink-downlink duality for themulticell network for problem of minimizing either the totalweighted transmit power or the maximum per-antenna powersubject to SINR constraints. Section IV contains distributedalgorithms for multicell downlink beamforming. Section Vprovides simulation results. Concluding remarks are made inSection VI.Notations: R and C denote the real and complex spaces. Theidentity matrix is denoted as I. The transpose and Hermitiantranspose of a matrix are denoted as (.) T and (.) H respectively.II. PROBLEM FORMULATIONA. System ModelThis paper considers a multicell multi-user spatial multiplexsystem with N cells and K users per cell with N t antennasat each base-station and a single antenna at each remoteuser. Multiuser downlink transmit beamforming is employedat each base-station. Let x i,j be a complex scalar denotingthe information signal for the jth user in the ith cell, andw i,j ∈ C Nt×1 be its associated beamforming vector. Thereceived signal at the jth remote user in the ith cell, denotedas y i,j ∈C, is a summation of the intended signal, intracellinterference, and intercell interference:y i,j = ∑ h H i,i,jw i,l x i,l + ∑h H m,i,jw m,n x m,n + z i,j (1)lm≠i,nwhere h l,i,j ∈C Nt×1 is the vector channel from the basestationof the lth cell to the jth user in the ith cell, and z i,jis the additive white circularly symmetric Gaussian complexnoise with variance σ 2 /2 on each of its real and imaginarycomponents. Fig. 1 illustrates the system model for a networkwith seven cells and three users per cell.B. Transmit Beamforming ProblemThe beamformer design problem is that of minimizing somefunction of transmit power across all base-stations subjectto SINR constraints at the remote users. With w i,j as thebeamforming vectors, the SINR for the jth user in the ithcell can be expressed as:|wi,j H Γ i,j =h i,i,j| 2∑l≠j |wH i,l h i,i,j| 2 + ∑ m≠i,n |wH m,n h m,i,j| 2 + σ 2 (2)Let γ i,j be the SINR target for the jth user in the ith cell.We can then formulate, for example, a total transmit powerminimization problem as follows:∑minimize wi,jw H i,j (3)i,jsubject to Γ i,j ≥ γ i,j , ∀i =1···N, j =1···Kwhere the minimization is over the w i,j ’s. Throughout thispaper, we assume that the set of SINR targets are feasible.Fig. 1.Multicell wireless network.C. Conventional SystemsIn a conventional wireless cellular system, the multiuserbeamforming problem is solved on a per-cell basis; out-ofcellinterference is regarded as a part of background noise.In particular, for a fixed base-station î, a conventional systemfinds the optimal set of wî,j , j =1···K, assuming that allother (N − 1)K beamformers are fixed:∑minimize w H w î,j î,j(4)jsubject to Γî,j ≥ γî,j , ∀j =1···Kwhere Γî,j is given by (2). This single-cell downlink problemhas a classic solution as given in [19], [20], [21], [5].Note that in a conventional system, the choice of beamformersat each base-station affects the background noise levelat neighboring cells, and hence the setting of beamformersin neighboring base-stations. Thus, the above per-cell optimizationis in practice performed iteratively until the systemconverges to a per-cell optimal solution.D. Motivating Example for Joint OptimizationThis paper is motivated by the observation that the percelloptimization above does not necessarily lead to a jointoptimal solution. Significant performance improvement maybe obtained if base-stations coordinate in jointly optimizing allof their beamformers at the same time. The following exampleillustrates this point.Consider a multicell network but with only a single user percell. The per-cell optimization reduces to the optimal transmitbeamforming problem for a multi-input single-output (MISO)system with a background noise level which includes outof-cellinterference. Note that regardless of the level of thebackground noise, the optimal per-cell transmit beamformer is

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATIONa vector that matches the channel. Thus, in this example, percelloptimization across the cells converges in one iteration—every base-station uses a transmit beamformer that matchesthe MISO channel.This channel-matching solution is not necessarily the jointoptimum. For example, when two users belonging to twodifferent cells are near each other at the cell edge, it maybe advantageous to steer the beamforming vectors for the twobase-stations away from each other so as to reduce mutualinterference. Such a joint optimal beamforming solution maylead to higher received SINRs at a fixed transmit power, orconversely a lower transmit power at fixed SINRs.One of the first algorithms for solving the multicelljoint beamforming optimization problem is given by Rashid-Farrokhi, Liu and Tassiulas [19]. They showed that the optimaldownlink beamforming problem under SINR constraints canbe solved efficiently by an iterative uplink beamformer andpower update algorithm. It is well known that the uplinkbeamforming problem is much easier to solve [36]. Thus, bytransforming the downlink problem into the uplink domain,the downlink problem may be solved efficiently as well.The global optimality of the beamformer-power iterationalgorithm has been shown for the single-cell case in [20],[21], [5]. This paper gives a rigorous derivation of duality forthe multicell case, then proposes a new algorithm for solvingthe joint multicell downlink beamforming problem.III. UPLINK-DOWNLINK DUALITY FOR MULTICELLSYSTEMSA. Minimization of Weighted Transmitted PowerWe begin by formulating a slightly more general version ofthe transmit beamforming problem (3). In a multicell system,each base-station (or sometime each antenna) typically has itsown power constraint. Thus, it is useful to consider a problemof minimizing the weighted total transmit power, with thetransmit power at the ith base-station weighted by a factorα i . In this case, (3) becomes∑minimize α i wi,j H w i,j (5)i,jsubject to Γ i,j ≥ γ i,j , ∀i =1···N, j =1···KThe SINR target constraints in (5) may appear nonconvex.In a study of single-cell downlink beamforming problem, [5]showed that SINR constraints of this type can be transformedinto a second-order-cone constraint (see also [37]). This crucialobservation enables methods for solving (5) via convexoptimization.Uplink-downlink duality refers to the fact that the minimumtransmit power needed to achieve a certain set of SINRconstraints in a downlink channel is the same as the minimumtotal transmit power needed to achieve the same set of SINRtargets in an uplink channel, where the uplink channel isobtained by reversing the input and the output of the downlink.The main goal of this section is to show that uplink-downlinkduality, previously established for the single-cell case, carriesover to the multicell setting. The following theorem is amulticell generalization of the single-cell duality result asstated in [23]. The proof is based on a Lagrangian technique,similar to the approach used in [23].Theorem 1: The optimal transmit beamforming problem(5) for the downlink multiuser multi-cellular network canbe solved via a dual uplink channel in which the SINRconstraints remain the same and the noise power is scaledby α i . Mathematically, a Lagrangian dual of the optimizationproblem for the downlink (5) is the following uplink problem:∑minimize λ i,j σ 2 (6)subject toi,jΛ i,j ≥ γ i,jwhere the minimization is over λ i,j ,andλ i,j |ŵi,j H Λ i,j =maxh i,i,j| 2ŵ i,j∑(m,l)≠(i,j) λ m,l|ŵi,j H h i,m,l| 2 + α i ||ŵ i,j || . 2The optimal ŵ i,j has the interpretation of being the receiverbeamformer of the dual uplink channel, and is a scaled versionof the optimal w i,j . The optimal λ i,j has the interpretation ofbeing the dual uplink power, and it corresponds to the dualvariable associated with the SINR constraint of (5).Proof: The proof hinges upon the fact that the SINRconstraints can be reformulated as a second-order coneprogrammingproblem as shown in [5]. Therefore, strongduality holds for (5). This allows us to characterize the solutionof (5) via its Lagrangian:L(w i,j ,λ i,j )= ∑ α i wi,j H w i,j − ∑ [|wHi,jh i,i,j | 2λ i,j −γi,ji,ji,j∑|wi,l H h i,i,j| 2 − ∑]|wm,n H h m,i,j| 2 − σ 2 (7)l≠jm≠i,nRearranging (7), we get:L(w i,j ,λ i,j )= ∑ λ i,j σ 2 + ∑ [wi,jH α i I−i,ji,j(1+ 1 )λ i,j h i,i,j h H i,i,jγ + ∑ ]λ m,n h i,m,n h H i,m,n w i,ji,jm,n(8)The dual objective isg(λ i,j )=minL(w i,j ,λ i,j ) (9)w i,j( )It is easy to see that if α i I − 1+ 1γ i,jλ i,j h i,i,j h H i,i,j∑+m,n λ m,nh i,m,n h H i,m,nis not a positive definite matrix, thenthere exists a set of w i,j that would make g(λ i,j )=−∞.Thus, the Lagrangian dual of (5), which is the maximum ofg(λ i,j ),is∑maximize λ i,j σ 2 (10)wheresubject toi,jΣ i ≽(1+ 1γ i,j)λ i,j h i,i,j h H i,i,jΣ i α i I + ∑ m,nλ m,n h i,m,n h H i,m,n (11)

DAHROUJ and YU: COORDINATED BEAMFORMING FOR THE MULTICELL MULTI-ANTENNA WIRELESS SYSTEM 5Next, we show that the above dual is equivalent to (6). Theproblem (6) corresponds to an uplink channel with receivebeamformers ŵ i,j , where the noise power of the dual channelis scaled by α i . The optimal receive beamformers ŵ i,j thatmaximize the SINR are the minimum-mean-squared-error(MMSE) receivers, which can be expressed as:⎛ŵ i,j = ⎝ ∑ λ m,l σ 2 h i,m,l h H i,m,l + α iσ 2 I⎠m,l⎞−1h i,i,j (12)where the (j, n)-th entry of each K × K sub-matrix F im isdefined as follows:⎧1⎪⎨F imγ i,j|ŵi,j H h i,i,j| 2 if m = i and n = j,jn = −|ŵi,n ⎪⎩H h i,i,j| 2 if m = i and n ≠ j, (17)−|ŵm,n H h m,i,j| 2 if m ≠ iThe δ i,j ’s can be found by taking the inverse of the matrix F:δ = F −1 1σ 2 (18)Plugging ŵ i,j into the SINR constraint of (6), one can showthat the SINR constraint is equivalent toα i I + ∑ (λ m,n h i,m,n h H i,m,n ≼ 1+ 1 )λ i,j h i,i,j h H i,i,jγm,ni,jThus, one can rewrite (6) as follows:∑minimize λ i,j σ 2 (13)subject toi,jΣ i ≼(1+ 1γ i,j)λ i,j h i,i,j h H i,i,jNote that the problems in (10) and (13) are identical exceptthat the maximization is replaced by minimization and theinequality constraints are reversed. It can be shown thatthe optimal solutions for both problems are such that theconstraints are satisfied with equality. Thus, (10) and (13) givethe same solutions.In addition, it can be shown that w i,j and ŵ i,j are scaledversions of each other. Thus, one would be able to find w i,jby first finding ŵ i,j , then updating it through scalar multiplesδ i,jw i,j = √ δ i,j ŵ i,j . (14)The δ i,j can be found through a matrix inversion using thefact that the SINR constraints in (5) are satisfied with equality.Plugging (14) into the SINR constraint of (5), one can rewritethe SINR contraint as:1|ŵi,j H γ h i,i,j| 2 δ i,j − ∑ |ŵi,n H h i,i,j| 2 δ i,ni,jn≠j− ∑|ŵm,nh H m,i,j | 2 δ m,n = σ 2 (15)δ im≠i,nDefine δ i =[δ i,1 ,δ i,2 , ··· ,δ i,K ] T for i =1···N and δ =[δ T 1 ,δ T 2 , ··· ,δ T N ]T . Based on (15), one can writeFδ =1σ 2 . (16)Here, 1 is the NK × 1 all ones-vector and F is the followingNK × NK matrix:⎡⎤F 11 F 12 ··· F 1NF 21 F 22 ··· F 2NF =.⎢ .⎥⎣ .⎦F N1 F N2 ··· F NNB. Minimization of Maximum Antenna PowerThe weighting factors α i in Theorem 1 provide a mechanismto trade off the power consumptions at different basestationsin a multicell network. It is also straightforward to introduceadditional weighting factors to account for power consumptiontradeoff at the per-antenna level. However, choosingthe right weights is often not easy. But if we consider apractical scenario of minimizing the maximum antenna poweracross all the base-stations, the weights adjustment can bedone automatically using a further extension of duality. Theoptimization problem in this case is formulated as follows:minimize τ (19)subject to Γ i,j ≥ γ i,j , ∀i, j⎡ ⎤⎣ ∑ w i,j wi,jH ⎦ ≤ τ, ∀i, mjm,mwhere [·] m,n denotes the (m, n)-th entry of a matrix. Thefollowing theorem is a multicell generalization of the singlecellper-antenna power minimization problem treated in [23].Theorem 2: The optimal maximum antenna power transmitbeamforming problem (19) for the downlink multiuser multicellularnetwork can be solved via a dual uplink channelin which the SINR constraints remain the same and thenoise is uncertain. Mathematically, a Lagrangian dual of theoptimization problem (19) is the following max-min problem:wheremaxQ iminλ i,jsubject to∑λ i,j σ 2 (20)i,jΛ i,j ≥ γ i,j ∀ i, jtr(Q i ) ≤ N t , Q i diagonal,Q i ≽ 0 ∀ iλ i,j |ŵi,j H Λ i,j =maxh i,i,j| 2ŵ i,j∑(m,l)≠(i,j) λ m,l|ŵi,j H h i,m,l| 2 + ŵi,j H Q .iŵ i,jThe optimal ŵ i,j has the interpretation of being the receiverbeamformer of the dual uplink channel, and is a scaled versionof the optimal w i,j . The optimal λ i,j has the interpretation ofbeing the dual uplink power, and it corresponds to the dualvariable associated with the SINR constraint of (19). The optimaluplink noise covariance matrix Q i = diag(q i,1 , ..., q i,Nt )is a diagonal matrix of dual variables associated with thedownlink per-antenna power constraint in (19).

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATIONProof: The proof mirrors that of Theorem 1. It againhinges upon the fact that the SINR constraints can be reformulatedas a second-order cone-programming problem.Therefore, strong duality holds for (19). This allows us tocharacterize the solution of (19) via its Lagrangian. First, onecan rewrite (19) as follows:minimize NN t τ (21)subject to Γ i,j ≥ γ i,j , ∀i, j⎡ ⎤⎣ ∑ w i,j wi,jH ⎦ ≤ τ, ∀i, mjThe Lagrangian of (21) ism,mL(w i,j ,λ i,j , Q i ,τ)= ∑ λ i,j σ 2 + ∑ wi,jH Q i −i,ji,j(1+ 1 )λ i,j h i,i,j h H i,i,jγ + ∑ ]λ m,n h i,m,n h H i,m,n w i,ji,jm,n− τ ∑ i[[tr(Q i ) − N t ] (22)Using the same argument as in the proof of Theorem 1, wecan write the Lagrangian dual of (19) aswheremaxQ imaxλ i,jsubject to∑λ i,j σ 2 (23)i,j(Θ i ≽ 1+ 1 )λ i,j h i,i,j h H i,i,jγ i,jtr(Q i ) ≤ N t , Q i diagonal,Q i ≽ 0 ∀ iΘ i Q i + ∑ m,nλ m,n h i,m,n h H i,m,n (24)Also, following the same steps as in the proof of Theorem1, one can rewrite (20) as followsmaxQ iminλ i,jsubject to∑λ i,j σ 2 (25)i,j(Θ i ≼ 1+ 1 )λ i,j h i,i,j h H i,i,jγ i,jtr(Q i ) ≤ N t , Q i diagonal,Q i ≽ 0 ∀ iThe problems in (25) and (23) are identical except that themaximization is replaced by minimization and the inequalityconstraints are reversed. Thus, as in the previous proof, (25)and (23) give the same solutions.Comparing (6) and (20), it is now clear that the weightedpower minimization problem with α i as weights correspondsto the setting of Q i = α i I in (20). An outer maximization overQ i provides a way to set the weights optimally to minimizethe maximum per-antenna power.C. Beamforming With Dirty-Paper CodingUplink-downlink duality can be extended to include thepossibility of implementing dirty-paper coding (DPC) withineach cell. DPC refers to an information theoretical operationwhere the downlink intracell interference can be pre-subtractedat the base-station. Dirty-paper coding may be implementedin practice using Tomlinson-Harashima precoding-like techniques.It can be thought of as the dual operation of receiverbasedinterference cancellation for the uplink.Assuming a particular pre-subtraction order 1, 2, ..., K ineach cell, i.e. the jth user of the ith cell, is encoded at basestationi by subtracting the intracell interference caused by thefirst (j − 1) users of the same cell. The SINR of the downlinknow becomesΓ ′ |wi,j Hi,j =h i,i,j| 2∑l>j |wH i,l h i,i,j| 2 + ∑ m≠i,n |wH m,nh m,i,j | 2 + σ 2It is not difficult to see that the dual uplink problem is exactlythe same as in the linear beamforming case, except the uplinkSINRs are modified asΛ ′ λ i,j |ŵi,j Hi,j =maxh i,i,j| 2∑ŵ i,j (m,l)≻(i,j) λ m,l|ŵi,j H h i,m,l| 2 + α i ||ŵ i,j || 2where the notation (m, l) ≻ (i, j) denotes that either m>ior m = i and l>m. A similar modification applies to themaximum per-antenna power minimization problem.IV. DISTRIBUTED DOWNLINK BEAMFORMINGThe derivation of uplink-downlink duality via Lagrangiantheory forms the basis for numerical algorithms for computingthe optimal coordinated beamformers for the downlinkmulticell system. Our algorithms are based on the idea ofiterative function evaluation, first proposed for the singlecellcase in [5]. This paper generalizes the algorithm to themulticell system.An important consideration for algorithms design in amulticell system is the issue of distributed implementation. Inthe second half of this section, we show that for TDD systems,the proposed algorithms naturally lead to a distributed per-cellimplementation.A. Iterative Function Evaluation AlgorithmWe first present numerical algorithm for finding the optimalbeamformer for the weighted sum power minimization problem(5). The main idea is to solve the downlink beamformingproblem in the dual uplink domain by first finding the optimalλ i,j , then the corresponding ŵ i,j . To find the optimal λ i,j ,wefirst take the gradient of the Lagrangian (8) with respect tow i,j and set it to zero:[α i I − (1 + 1 )λ i,j h i,i,j h H i,i,jγ +i,j]∑λ m,n h i,m,n h H i,m,n w i,j =0 (26)ThusΣ i w i,j =m,n(1+ 1γ i,j)λ i,j h i,i,j h H i,i,j w i,j (27)

DAHROUJ and YU: COORDINATED BEAMFORMING FOR THE MULTICELL MULTI-ANTENNA WIRELESS SYSTEM 7where Σ i is as defined in (11).Now, multiplying both sides by h H i,i,j Σ i −1 ,weget:(h H i,i,jw i,j = 1+ 1 )λ i,j h Hγi,i,jΣ −1 i h i,i,j h H i,i,jw i,j (28)i,jFinally, cancelling out the h H i,i,j w i,j factor on both sides of theequation, we obtain a necessary condition for optimal λ i,j :1λ i,j = ( )(29)1+ 1γ i,jh H i,i,j Σ i −1 h i,i,jwhich can be used iteratively to obtain the optimal λ i,j .The algorithm is summarized as follows:1) Find the optimal uplink power allocation λ i,j using theiterative function evaluation:1λ i,j = ( )(30)1+ 1γ i,jh H i,i,j Σ i −1 h i,i,jwhereΣ i = α i I + ∑ m,nλ m,n h i,m,n h H i,m,n (31)2) Find the optimal uplink receive beamformers based onthe optimal uplink power allocation λ i,j :⎛⎞ŵ i,j = ⎝ ∑ −1λ m,l σ 2 h i,m,l h H i,m,l + σ2 α i I⎠m,lh i,i,j(32)3) Find the optimal transmit downlink beamformers byscaling ŵ i,j :w i,j = √ δ i,j ŵ i,j (33)The global convergence of this algorithm is guaranteed bythe duality result discussed in the previous section togetherwith the convergence of the iterative function evaluation whichcan be justified by a line of reasoning similar to that in [5].The proof is based on the property of standard functions [38].In particular, one can stack the dual variables λ i,j into onevector Υ. Then (30) can be rewritten asλ (t+1)i,j = f i,j (Υ (t) ), i =1···N, j =1···K (34)The function f i,j satisfies the following properties:1) If λ i,j ≥ 0 ∀i, j, thenf i,j (Υ) > 0.2) If λ i,j ≥ λ ′ i,j ∀i, j, thenf i,j(Υ) ≥ f i,j (Υ ′ )3) For ρ>1, wehaveρf i,j (Υ) >f i,j (ρΥ) ∀i, j.as shown in the Appendix. These properties guarantee thatf i,j is a standard function. Thus, starting with some initialΥ (0) , the iterative function evaluation algorithm converges toa unique fixed point, which must be the optimal downlinkpower by duality.B. Minimizing the Maximum Antenna PowerTo solve the problem of minimizing the maximum antennapower (19), we use Theorem 2 to solve the dual uplink beamformingproblem with uncertain noise (20). Unlike the totalweighted transmitted power minimization problem, solving thedual of the maximum antenna power minimization problemrequires finding both the uncertain noise covariance matricesQ i and the transmit uplink powers λ i,j . The idea is to solve thedual problem by iteratively computing the inner minimizationon (λ i,j , ŵ i,j ) and the outer maximization on Q i .The inner minimization can be solved using the iterativefunction evaluation approach presented in the previous section.As to the outer maximization, we use a subgradient projectionapproach similar to the one presented in [23]. Consider thefunction φ(Q 1 , ··· , Q N ) which is a subproblem of (20) withfixed Q i :whereφ(Q 1 , ··· , Q N ) =minimizeλ i,jsubject to∑λ i,j σ 2 (35)i,jΛ i,j ≥ γ i,j ∀ i, jλ i,j |ŵi,j H Λ i,j =maxh i,i,j| 2∑ŵ i,j (m,l)≠(i,j) λ m,l|ŵi,j H h i,m,l| 2 + ŵi,j H Q iŵ i,jIt is shown in [23] that φ is concave in (Q 1 , ··· , Q N ).Further, let w i,j be the optimal downlink beamforming vector,then diag[ ∑ j w i,jwi,j H ] is a subgradient for φ with respect toQ i . Therefore, the outer maximization can be done based ona subgradient projection approach, with projection onto theconstraint set S Qi = {Q i : tr(Q i ) ≤ N t , Q i ≽ 0}.The algorithm is summarized as follows:1) Initialize Q (0) i for i =1, 2, ..., N2) Fix Q (n) i . Find the optimal uplink power allocation λ i,jusing the iterative function evaluation:where1λ i,j = ( )(36)1+ 1γ i,jh H i,i,j [Θ i (n) ] −1 h i,i,jΘ i (n) Q i (n) + ∑ m,nλ m,n h i,m,n h H i,m,n (37)3) Find the optimal uplink receive beamformers based onthe optimal uplink power allocation λ i,j :⎛⎞ŵ i,j = ⎝ ∑ −1λ m,l σ 2 h i,m,l h H i,m,l + σ2 (n)Q i⎠m,lh i,i,j4) Find the optimal transmit downlink beamformers byscaling ŵ i,j :w i,j = √ δ i,j ŵ i,j (38)5) Update Q (n) i using the subgradient projection methodwith step size t n :[ ∑]}Q (n+1) i = P SQi{Q (n) i + t n diag w i,j wi,jHj(39)The projection operation is a simple renormalization, i.e.multiplication by a constant so that tr(Q (n+1) i )=N t .6) Increment n. Return to step 2) until convergence.The global convergence of this algorithm is guaranteedby the duality result discussed earlier, the convergence ofthe iterative function evaluation and the convergence ofthe subgradient projection method due to the concavity ofφ(Q 1 , ··· , Q N ) [39]. Note that the algorithm can be extendedin a straightforward fashion when dirty-paper coding is used.

8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATIONC. Distributed ImplementationAn important feature of the iterative function evaluationalgorithms proposed in subsection IV-A and IV-B is that theycan be implemented in a distributed fashion in a TDD system,where uplink and downlink channels are reciprocals of eachother. In this case the virtual dual uplink is the real uplink.Consider first the iterative function evaluation step of thealgorithm. The function iteration (30) on the uplink power λ i,jinvolves channel vectors h i,i,j within each cell only, which thebase-station typically has the knowledge of, and the matrixΣ i . Observe that Σ i is essentially the covariance matrix ofthe received signal in the uplink direction at the base-stationi, which includes the intended signal, the interference, and ascaled version of the background noise. In a TDD system,this covariance matrix may be estimated locally at each basestationin the uplink direction. Channel reciprocity implies thatthe signal and interference covariance matrices at the uplinkreceiver are exactly as required. In addition, the backgroundnoise level is typically known, so the scaling factor α i canbe easily compensated for. Thus, we can perform the iterativeupdates of λ i,j and the estimation of Σ i on a per-cell basis, i.e.the iterative function evaluation process can be done locallywithout the need of explicit inter-base-station coordination.This uplink per-cell iteration process always converges. Itconverges to the optimal uplink power. Note that base-stationcoordination is achieved implicitly via uplink power control(i.e. the update of λ i,j ’s, which affect all other Σ i ’s). Thechannel side information is needed only for users within eachcell, and not for out-of-cell interferers.Secondly, the beamforming vectors can also be easilyobtained in the dual uplink. This is essentially receiver MMSEbeamforming at the base-station.Finally, to use the uplink beamformer for downlink transmission,one needs to scale it by the right δ i,j . This involvesa matrix inversion (18). But, this process is equivalent toa downlink power control problem on the effective matrixchannel for achieving a desirable set of SINRs, which, by theclassic result of Foschini and Miljanic [40], has a distributedimplementation using a per-user power update algorithm. Eachstep of the algorithm sets a δ i,j to satisfy its correspondingSINR constraint with equality assuming all other δ i,j ’s arefixed. The convergence of the algorithm can be proved eitherusing the method of [40] or by a standard function argument[38].In fact, both the uplink per-cell iterative function evaluationand the downlink power control part of the algorithm caneven be implemented asynchronously at each base-stationusing possibly outdated power information. The convergenceof such asynchronous update is still guaranteed by the standardfunction argument as shown in Theorem 4 of [38]. The onlynecessary synchronization is that the base-stations must all bein the uplink phase or the downlink phase together, so that thethree steps of the algorithm can be executed consecutively.The above remarks are also applicable to the maximum perantennapower minimization problem. The only modificationsare the projection operation on Q i , i.e. (39), and the modificationof the received covariance matrix (37) based on Q i ,both of which can be done on a per-cell basis.D. Beamformer-Power Iteration AlgorithmAn alternative approach for solving the downlink beamformingproblem is to iteratively update the beamformers andthe power, as proposed in [19], [36]. For the minimization ofthe total weighted transmitted power, the algorithm goes asfollows:1) Initialize ŵ i,j ;2) Find the λ i,j to satisfy the SINR constraints of (6) withequality;3) Find the uplink receive beamformers based on the uplinkpower allocation λ i,j :⎛⎞ŵ i,j = ⎝ ∑ −1λ m,l σ 2 h i,m,l h H i,m,l + σ 2 α i I⎠m,l4) Go to step 2 until convergence;5) Update the transmit downlink beamformersh i,i,j(40)w i,j = √ δ i,j ŵ i,j (41)The convergence of the iterations involving Steps 2 and 3 isshown in [36]. By our multicell duality result, the algorithmmust also converge to the global optimal solution for thedownlink.Both the iterative function evaluation algorithm and thebeamformer-power iteration algorithm give the optimal solutionfor the multicell downlink beamforming problem. Bothcan be implemented directly in a TDD system. However, animplementation of the beamformer-power iteration algorithmrequires repeated updates of uplink powers in Step 2 above,which would itself require either centralized processing (e.g.by performing a matrix inversion) or a separate iterativeprocess (e.g. by fixed-point iteration). As both Steps 2 and3 above need to be repeatedly executed by all users at thesame time, the beamforming-power iteration algorithm wouldrequire a higher level of synchronization among all the usersand all the base-stations than the iterative function evaluationalgorithm proposed in the previous section.V. SIMULATIONSWe begin by investigating the benefit of coordinated beamformingin a two-cell configuration shown in Fig. 2. Standardcellular network parameters are used in simulation: the noisepower spectral density is set to -162 dBm/Hz; the channelvectors are chosen according to the distance-dependent pathloss L = 128.1 +37.6log 10 (d), whered is the distance inkilometers, with 8dB log-normal shadowing, and a Rayleighcomponent. The distance between neighboring base-stationsis set to be 2.8km. An antenna gain of 15dBi is assumed.For illustration purposes, the weighting factors correspondingto the base-station antenna power constraints are set to beα i =1.Fig. 3 shows the benefit of coordinated beamforming whenthere are two users per cell. Fig. 4 shows a similar casewith three users per cell. In both cases, one of the users ineach cell is located at distance d away from its own basestationon the straight line connecting the two base-stations.The other users are located randomly elsewhere in the cell.

DAHROUJ and YU: COORDINATED BEAMFORMING FOR THE MULTICELL MULTI-ANTENNA WIRELESS SYSTEM 98070Fig. 2. A two-cell configuration with two users located between two basestationsat distance d.The base-stations are equipped with four antennas. It is clearfrom the figures that the coordinated beamforming systemsignificantly outperforms the conventional per-cell optimizedsystem, especially at the high SINR target range. The benefitis the largest when the users are close to the cell edge asexpected.Fig. 3 and Fig. 4 also illustrate that the performance gaindue to coordinated beamforming is larger in the two-userper-cellcase as compared to the three-user-per-cell case.Intuitively, when the number of users per cell is small ascompared to the number of base-station antennas, there arespare dimensions available for interference suppression. Thisis when coordinated beamforming shows the most benefit.It is interesting to note that in the two-cell two-user-per-cellcase, the conventional system eventually becomes infeasible asthe SNR target increases. Yet, the coordinated beamformingsystem is always feasible. This is because the base-stationsare equipped with four antennas. With a total of four usersbetween the two cells, the coordinated system has the capabilityof zero-forcing the out-of-cell users, thus completelyeliminating the out-of-cell interference. In contrast, out-ofcellinterference is always present in the conventional system.Note that complete zero-forcing is no longer possible in thethree-users-per-cell configuration. Nevertheless, coordinatedbeamforming still produces significant power saving.Fig. 5 and Fig. 6 show the performance of coordinatedbeamforming in a 7-cell network with 3 randomly locatedusers per cell (as in Fig. 1) under the minimum total transmitpower and the minimum maximum antenna power criteria,respectively. Again, each base-station is equipped with 4antennas. It is again observed that while the joint optimizationalgorithm outperforms the conventional per-cell update in therange of high SINR targets. This is due to the fact that athigh SINRs, the multicell network becomes predominantlyinterference limited. The figures also show the dirty papercoding gain for both the joint optimization and the per-cellupdate algorithms.Fig. 7 illustrates the power saving in the maximum antennapower by running the per-antenna optimization algorithm. Forthis 7-cell 3-user-per-cell case, the power saving is in the 1-2dB range.To illustrate the convergence behavior of the proposed iterativeevaluation algorithm, we compare it with the beamformerpoweriterative algorithm of [19]. Fig. 8 shows the normresidue of the uplink transmitted power (in mW) versus theTotal Transmitted Power in dBm60504030Joint Optimization Algorithm for d=1 kmPer−cell Update Algorithm for d=1 km20Joint Optimization Algorithm for d=0.6 kmPer−cell Update Algorithm for d=0.6 kmJoint Optimization Algorithm for d=0.3 kmPer−cell Update Algorithm for d=0.3 km100 5 10 15 20 25 30SINR Target in dBFig. 3. Total transmitted power versus the SINR targets for the jointoptimization of a coordinated beamforming system and the per-cell updateof a conventional system for two-cell network with two users per cell. Oneof the users are at various distances d away from its own base-stations. Thebase-stations are equipped with four antennas.Total Transmitted Power in dBm7065605550454035Joint Optimization Algorithm for d=1 km30Per−cell Update Algorithm for d=1 kmJoint Optimization Algorithm for d=0.75 kmPer−cell Update Algorithm for d=0.75 km25Joint Optimization Algorithm for d=0.3 kmPer−cell Update Algorithm for d=0.3 km200 5 10 15 20 25SINR Target in dBFig. 4. Total transmitted power versus the SINR targets for the jointoptimization of a coordinated beamforming system and the per-cell updateof a conventional system for two-cell network with three users per cell. Oneof the users are at various distances d away from its own base-stations. Thebase-stations are equipped with four antennas.number of iterations. The norm residue is defined as:R (n) = σ 2 ||Υ (n) − Υ ∗ || 2 (42)where Υ ∗ represents the optimal power vector. It is observedthat while the beamformer-power iterative algorithm convergesmore rapidly at the beginning, the distributed iterative functionevaluation algorithm in fact provides faster convergenceasymptotically. Note that for the distributed iterative functionevaluation algorithm, each iteration step requires a covariancematrix estimation of the received signal at the base-stations.The algorithm convergence speed will therefore typically beaffected by the accuracy of the estimation process, especiallywhen channels change over time.Finally, we note that the convergence speed of the iterative

10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATIONTotal Transmitted Power in dBm555045403530252015Joint Optimization Algorithm without DPCPer−cell Update Algorithm without DPCJoint Optimization Algorithm with DPCPer−cell Update Algorithm with DPCMaximum Antenna Power in dBm252015105Minimization of Total Transmitted Power AlgorithmMinimization of Maximum Antenna Power Algorithm1050 2 4 6 8 10 12 14 16 18 20 22SINR Target in dBFig. 5. Total transmitted power versus the SINR targets for the jointoptimization of a coordinated beamforming system and the per-cell updateof a conventional system for a 7-cell network with 3 users per cell with andwithout dirty-paper coding. The base-stations are equipped with four antennas.05 6 7 8 9 10 11 12 13SINR Target in dBFig. 7. Comparing the maximum antenna power for the total power minimizationalgorithm and the maximum antenna power minimization algorithm.10 2 Number of Iterations5010 0Iterative Function Evaluation AlgorithmBeamformer−Power Iteration AlgorithmMaximum Antenna Power in dBm4540353025201510Joint Optimization Algorithm without DPCPer−cell Update Algorithm without DPCJoint Optimization Algorithm with DPCPer−cell Update Algorithm with DPCNorm Residue10 −210 −410 −610 −810 −1010 −12510 −140 50 100 150 200 250 300 350 40000 2 4 6 8 10 12 14 16 18 20SINR Target in dBFig. 6. The maximum per-antenna power versus the SINR targets for the jointoptimization of a coordinated beamforming system and the per-cell updateof a conventional system for a 7-cell network with 3 users per cell with andwithout dirty-paper coding. The base-stations are equipped with four antennas.Fig. 8. The norm residue versus the number of iterations for the twooptimization algorithms.10 310 4 Number of Iterationsfunction evaluation algorithm depends not only on the size ofthe problem (e.g. the number of antennas, the number of cellsand the number of users per cell), but also on the SINR targets.Fig. 9 shows the norm residue versus the number of iterationsfor different values of the SINR targets. It is observed thatconvergence becomes slower when the SINR target increases.VI. CONCLUSIONThis paper provides a solution to the optimal coordinateddownlink beamforming design problem for a multicell networkwith multiple users per cell. The uplink-downlink duality isgeneralized to the multicell case using the Lagrangian theoryfor two different design criteria: minimizing the total weightedtransmitted power and minimizing the maximum antennapower subject to SINR constraints. An iterative functionNorm residue10 210 110 010 −110 −210 −310 −4SINR Target=10 dBSINR Target=13 dBSINR Target=14.8 dBSINR Target=16.0 dB0 20 40 60 80 100 120 140 160 180 200Fig. 9. The norm residue versus the number of iterations for different SINRtargets.

DAHROUJ and YU: COORDINATED BEAMFORMING FOR THE MULTICELL MULTI-ANTENNA WIRELESS SYSTEM 11evaluation based algorithm which is capable of finding theglobally optimal coordinated beamformers is presented. Akey feature of the algorithm is that it can be implementedin a distributed fashion for the TDD system. The proposedalgorithm is efficient, and it outperforms conventional wirelesssystems with per-cell signal processing.APPENDIXThis appendix presents proofs of standard function propertiessatisfied by f ij in (34). The proofs are similar to the onespresented in [5], and are included here for completeness.1) If λ i,j ≥ 0 ∀i, j, thenf i,j (Υ) > 0.Proof: This property holds because if λ i,j ≥ 0−1then Σ i ≻ 0 and consequently Σ i ≻ 0. Thush H i,i,j Σ i −1 h i,i,j > 0 and consequently f i,j (Υ) > 0.2) If λ i,j ≥ λ ′ i,j ∀i, j, thenf i,j(Υ) ≥ f i,j (Υ ′ ).Proof: Assume λ i,j ≥ λ ′ i,j . Then,where1f i,j (Υ) = ( )(43)1+ 1γ i,jh H i,i,j Σ i −1 h i,i,jΣ i = α i I + ∑ m,nλ m,n h i,m,n h H i,m,n= α i I + ∑ m,nλ ′ m,n h i,m,nh H i,m,n+ ∑ m,n(λ m,n − λ ′ m,n)h i,m,n h H i,m,n (44)Now, since λ i,j ≥ λ ′ i,j , we have ∑ m,n (λ m,n −λ ′ m,n)h i,m,n h H i,m,n ≽ 0. But as shown in [5], forpositive semidefinite matrices C and D and vector xin the range of C:1x T (C + D) −1 x ≥ 1x T C −1 x(45)with equality if and only if D(C + D) −1 x =0. Thus11h H i,i,j Σ i −1 ≥h i,i,j h H i,i,j Σ ih i,i,j(46)where( )′ ∑Σ i = λ ′ m,nh i,m,n h H i,m,n + α i I (47)m,nHence, f i,j (Υ) ≥ f i,j (Υ ′ ).3) For ρ>1, ρf i,j (Υ) >f i,j (ρΥ) ∀i, j.Proof: Let ρ>1,where1ρf i,j (Υ) = ( )(48)1+ 1γ i,jh H i,i,j (ρΣ i) −1 h i,i,jρΣ i = ρα i I + ρ ∑ m,nλ m,n h i,m,n h H i,m,n=(ρ − 1)α i I + α i I + ρ ∑ m,nλ m,n h i,m,n h H i,m,nSince ρ>1, wehave(ρ − 1)α i I ≽ 0. Based on (45),we get1h H i,i,j (ρΣ i) −1 ≥h i,i,j1(α i I + ρ ∑ −1m,n λ m,nh i,m,n hi,m,n) H hi,i,jh H i,i,j(49)Thus, ρf i,j (Υ) ≥ f i,j (ρΥ). 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Yates, “A framework for uplink power control in cellular radiosystems,” IEEE J. Sel. Areas Commun., vol. 13, no. 7, pp. 1341–1347,Sep. 1995.[39] N. Z. Shor, Minimization Methods for Non-Differentiable Functions.New York: Springer, 1985.[40] G. J. Foschini and Z. Miljanic, “A simple distributed autonomous powercontrol algorithm and its convergence,” IEEE Trans. Veh. Technol.,vol. 42, no. 4, pp. 641–646, Nov. 1993.Hayssam Dahrouj (S’05) received the B.E. degreein Computer and Communications Engineering withhigh distinction from the American University ofBeirut, Beirut, Lebanon in 2005. Since September2005, he has been a graduate student in the Electricaland Computer Engineering Department at the Universityof Toronto, Toronto, Ontario, Canada, wherehe is currently pursuing his doctoral studies. Hismain research interests include convex optimization,wireless communications, multiuser information theory,signal processing, and distributed algorithms.During his graduate studies, he has been working on transmitter optimizationfor multicell wireless systems.Wei Yu (S’97-M’02-SM’08) received the B.A.Sc.degree in Computer Engineering and Mathematicsfrom the University of Waterloo, Waterloo, Ontario,Canada in 1997 and M.S. and Ph.D. degreesin Electrical Engineering from Stanford University,Stanford, CA, in 1998 and 2002, respectively. Since2002, he has been with the Electrical and ComputerEngineering Department at the University ofToronto, Toronto, Ontario, Canada, where he is nowan Associate Professor and holds a Canada ResearchChair in Information Theory and Digital Communications.His main research interests include multiuser information theory,optimization, wireless communications and broadband access networks.Prof. Wei Yu is currently an Editor for IEEE TRANSACTIONS ON COM-MUNICATIONS. He was an Editor for IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS from 2004 to 2007, and a Guest Editor for a number ofspecial issues for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNI-CATIONS and the EURASIP JOURNAL ON APPLIED SIGNAL PROCESSING.He is member of the Signal Processing for Communications and NetworkingTechnical Committee of the IEEE Signal Processing Society. He received theIEEE Signal Processing Society Best Paper Award in 2008, the McCharlesPrize for Early Career Research Distinction in 2008, the Early Career TeachingAward from the Faculty of Applied Science and Engineering, University ofToronto in 2007, and the Early Researcher Award from Ontario in 2006.