Coordinated multicast beamforming in multicell networks

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Coordinated Multicast Beamforming inMulticell Networks

Zhengzheng Xiang, Meixia Tao,Senior Member, IEEE, and Xiaodong Wang,Fellow, IEEE

Abstract—We study physical layer multicasting in multicellnetworks where each base station, equipped with multiple an-tennas, transmits a common message using a single beamformerto multiple users in the same cell. We investigate two coordinatedbeamforming designs: the quality-of-service (QoS) beamformingand the max-min SINR (signal-to-interference-plus-noiseratio)beamforming. The goal of the QoS beamforming is to minimizethe total power consumption while guaranteeing that receivedSINR at each user is above a predetermined threshold. Wepresent a necessary condition for the optimization problemto befeasible. Then, based on the decomposition theory, we proposea novel decentralized algorithm to implement the coordinatedbeamforming with limited information sharing among differ entbase stations. The algorithm is guaranteed to converge and inmost cases it converges to the optimal solution. The max-minSINR (MMS) beamforming is to maximize the minimum receivedSINR among all users under per-base station power constraints.We show that the MMS problem and a weighted peak-powerminimization (WPPM) problem are inverse problems. Basedon this inversion relationship, we then propose an efficientalgorithm to solve the MMS problem in an approximate manner.Simulation results demonstrate significant advantages of theproposed multicast beamforming algorithms over conventionalmulticasting schemes.

Index Terms—Physical layer multicasting, coordinated beam-forming, quality of service (QoS), max-min SINR (MMS),semidefinite programming (SDP).

I. I NTRODUCTION

W ITH the rapid development of wireless communicationtechnology, various kinds of traditional data service,

such as media streaming, cell broadcasting and mobile TV,have been implemented in wireless networks nowadays. Asa result, wireless multicasting becomes a central feature ofthe next generation cellular networks. Physical layer multi-casting with beamforming is a promising solution enabled byexploiting channel state information (CSI) at the transmitterover the traditional isotropic broadcasting. The problem ofmulticast beamforming for quality-of-service (QoS) guaranteeand for max-min fairness was firstly considered in [1]. Thesimilar problem of multicasting to multiple cochannel groups

Manuscript received December 29, 2011; revised April 22 andJuly 26,2012; accepted September 19, 2012. The associate editor coordinating thereview of this paper and approving it for publication was S. Aydin.

Z. Xiang and M. Tao are with the Dept. of Electronic Engineering, ShanghaiJiao Tong University, P. R. China (e-mail:{7222838, mxtao}@sjtu.edu.cn).

X. Wang is with the Department of Electrical Engineering, ColumbiaUniversity, New York, USA (e-mail: [email protected]).

This work is supported by the Joint Research Fund for Overseas Chinese,Hong Kong and Macao Young Scholars under grant 61028001, theNational973 project under grant 2012CB316100, and the New Century ExcellentTalents in University (NCET) under grant NCET-11-0331. Part of this workwill be presented in IEEE GLOBECOM 2012.

Digital Object Identifier 10.1109/TWC.2012.12.112295

was then investigated in [2]. The core problem of multicastbeamforming is essentially NP-hard [1]. Some other issuessuch as outage analysis and capacity limits were studied in[3], [4]. So far multicast beamforming has been included inthe UMTS-LTE / EMBMS draft for next-generation cellularwireless services [5], [6].

In conventional wireless systems, signal processing is per-formed on a per-cell basis. The intercell interference is treatedas background noise and minimized by applying a predesignedfrequency reuse pattern such that the adjacent cells use dif-ferent frequency bands. Due to the fast growing demand forhigh-rate wireless multimedia applications, many beyond-3Gwireless technologies such as 3GPP-LTE and WiMAX haverelaxed the constraint on the frequency reuse such that the totalfrequency band is available for reuse by all cells in the samecluster. However, this will cause the whole system limitedby the intercell interference. Consequently, cooperativesignalprocessing across the different base stations has been identifiedas a key technique to mitigate intercell interference in thenext-generation wireless systems.

The goal of this paper is to investigate multicast beam-forming for cooperative multicell networks. The base sta-tions cooperate with each other and design their transmitbeamformers in a coordinated manner. In general, there aretwo cooperation scenarios in multicell networks. In the firstscenario, different base stations are fully cooperative and actas “networked MIMO (multiple-input multiple-output)” (e.g.,[7], [8]). Namely, they coordinate at the signal level, i.e., datainformation intended for different users in different cells isshared among the base stations. Clearly, networked MIMOneeds tremendous amount of information exchange overhead.In contrast, in the second scenario, the base stations are onlyrequired to coordinate at the beamforming level which needsrather small information sharing (e.g., [9]). In this paper, wefocus on the latter case by only allowing beamforming levelcoordination.

We first formulate the problem of multicell multicast beam-forming as sum power minimization subject to the constraintthat the received signal-to-interference-plus-noise ratio (SINR)of each user is above a threshold. This problem is referredto as QoS problem. It is known that the QoS problem insingle-cell scenario is always feasible. However, this is notso in multicell networks due to the intercell interference,especially when the channel condition is bad or the SINRtarget is very stringent. We first present a necessary conditionon the optimization problem to be feasible. Then we proposea novel distributed algorithm to solve the multicell multicastcooperative beamforming design. More specifically, based

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2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

on the decomposition theory [10], we introduce a set ofinterference constraint parameters and decompose the originalproblem into several parallel sub-problems. Since the sub-problems are nonconvex and NP-hard, each base station thenapplies the semidefinite relaxation and independently solvesits own sub-problem. The interference parameters are updatedbased on a master problem. Simulation results show that thealgorithm converges in only several iterations.

In view of the user fairness issue, we also consider themulticell multiast beamforming design with the goal of max-imizing the minimum SINR (MMS) among all users underindividual power constraints. This is called MMS problem.By linking the MMS problem with a weighted peak powerminimization (WPPM) problem, we propose an efficient algo-rithm to find the near-optimal solution. Previously, the authorsin [11] [12] also studied the multicell multicast beamformingproblem with the objective to maximize the minimum SINRbut subject to a total sum power constraint across all basestations. Hence, the problem considered in [11] [12] is verysimilar to the multicast beamforming problem in single-cellmulti-group systems as in [2]. Authors in [13] also consideredthe similar problem under per-base station power constraintbut allowed data sharing among base stations, which is sim-ilar to the “networked MIMO” and requires a large amountof information exchange overhead. Thus, our beamformingdesign for max-min fairness is more practical as we considerindividual peak power constraint on each base station and onlyallow coordination at the beamforming level.

The rest of the paper is organized as follows. In Section II,the system model for multicell multicasting is presented. Sec-tion III considers the beamformer design in the QoS problemand develops a novel decentralized coordinated beamformingalgorithm. In Section IV, we consider the max-min SINRbeamforming problem under individual base station powerconstraints. Section V provides simulation results. Concludingremarks are made in Section VI.

Notations:R andC denote the real and complex spaces.The identity matrix is denoted asI and the all-one vector isdenoted as1. For a square matrixS, S � 0 means thatS ispositive semi-definite.[H]i,j denotes its element in theith rowandjth column.(·)T, (·)H, (·)† and Tr{·} stand for transpose,Hermitian transpose, Moore Penrose pseudoinverse and thetrace respectively.|x| denotes the absolute value of the scalarx and‖x‖ denotes the Euclidean norm of the vectorx.

II. SYSTEM MODEL

Consider a multicell multicast network comprisingN cellsand K mobile users per cell as shown in Fig. 1. The basestation in each cell is equipped withNt antennas and everymobile user has a single antenna. LethH

i,j,k denote thefrequency-flat quasi-static1×Nt complex channel vector fromthe base station in theith cell to thekth user in thejth cell, andwi denote theNt×1 multicast beamforming vector applied tothe base station in theith cell. We define the complex scalarsi as the multicast information symbol for the users in theithcell. The discrete-time baseband signal received by thekth

Fig. 1. Multicell multicast network.

user in theith cell is given by

yi,k = hHi,i,kwisi +

N∑

j 6=i

hHj,i,kwjsj + zi,k, (1)

wherezi,k is the additive white circularly symmetric Gaussiancomplex noise with varianceσ2

i,k/2 on each of its real andimaginary components. In (1), the second term is the intercellinterference.

Based on the received signal model in (1), the performanceof each user can be characterized by the output SINR, definedas

SINRi,k =|hH

i,i,kwi|2∑N

j 6=i |hHj,i,kwj |2 + σ2

i,k

. (2)

Notice that each user in the considered multicell multicastnetwork only suffers from the intercell interference, which isdifferent from multicell unicast systems where both inter-cellinterference and intra-cell interference exist.

In practical scenarios, the channels from a base station todifferent users, which may or may not belong to a same cell,can be correlated, especially when these users are in closeproximity to each other. For the users belonging to the samecell, we call the correlation betweenhH

i,j,k andhHi,j,k′ for k 6=

k′ as the intracell-user channel correlation. For the users fromdifferent cells, we call the correlation betweenhH

i,j,k andhHi,j′,l

for j 6= j′ as the intercell-user channel correlation.

III. Q OS BEAMFORMING

The coordinated QoS beamforming design is to minimizethe total energy consumption of the system while maintaininga target SINR for all users by properly designing the beam-formers at each base station. This is formulated as:

P(γ) : min{wi}N

i=1

N∑

i=1

‖wi‖2 (3)

s.t.|hH

i,i,kwi|2N∑j 6=i

|hHj,i,kwj |2 + σ2

i,k

≥ γi, ∀i, k (4)

whereγ = [γ1, γ2, ..., γN ]T is the target SINR vector witheach elementγi being the target SINR value to be achievedby the users in theith cell. Since the base station transmits a

XIANG et al.: COORDINATED MULTICAST BEAMFORMING IN MULTICELL NETWORKS 3

common information in a multicast manner, the informationrate for the users within one cell is the same, and hence weset a common SINR target for all the users in the same cell.

A. Feasibility analysis

Due to the SINR constraints, the QoS problem in (3) isnot always feasible, which is similar to the multiuser unicastscenario [14]. In order to verify its feasibility, we need toshowwhether there exists a set of beamformers{wi}Ni=1 for a givenγ such that

mink

SINRi,k ≥ γi, ∀i ∈ {1, 2, ..., N} (5)

For simplicity, the SINR targets for different cells are assumedto be the same. Then the above condition boils down to thefollowing

mini,k

SINRi,k ≥ γ (6)

For thekth user in each cell, we combine their channel vectorsas follows

Hk =

hH1,1,k hH

2,1,k · · · hHN,1,k

hH1,2,k hH

2,2,k · · · hHN,2,k

... · · ·hH1,N,k hH

2,N,k · · · hHN,N,k

, k = 1, 2, ...,K

(7)where eachHk is anN × (N × Nt) matrix. The followinglemma provides a necessary condition for the QoS problem tobe feasible.

Lemma 1: Given the SINR target vectorγ = γ · 1, if theproblem (3) is feasible, then the SINR targetγ should satisfythe following condition

γ ≤ min

{rank(H1)

N − rank(H1), · · · , rank(HK)

N − rank(HK)

}(8)

Proof: Please refer to Appendix A.From Lemma 1, we can see that the intercell-user channel

correlation has a negative impact on its feasibility. Morespecifically, if some of the rows inHk, say theith row andi′throw for i 6= i′, are correlated, which means that the channelsbetween thekth user in theith cell and thekth user in thei′th cell are correlated, then the rank ofHk could be less thanN and as a result, the SINR thresholdγ will have a finiteupper bound. On the other hand, if the channels of users indifferent cells are independent, we have that the matrixHk isfull rank with probability one, then the corresponding upperbound is infinite, which means no constraint onγ. Since theproblem (3) is NP-hard [2], determining its feasibility is notan easy job. The lemma gives a necessary condition from theperspective of channel correlation. In the following, we onlyconsider the problem (3) when it is feasible.

B. Decentralized coordinated beamforming

A desired feature of coordinated beamforming in multicellnetworks is that the base station at each cell can implementits beamforming design locally [17], [18]. This is due to theconstraint in practical systems that the backhaul channel haslimited capacity. In this subsection, we propose a decentralized

scheme for implementing the multicell multicast QoS beam-forming. It is assumed that each base station in the networkonly has the channel knowledge of the mobile users within itsown cell.

The distributed algorithm to problemP(γ) can not be easilyobtained primarily because all the beamformers are coupledtogether in the constraint (4). According to the decompositiontheory [10] and inspired by [18], we introduce a set of slackvariablesΓi,j,k denoting the constraint of the interference fromthe ith base station to thekth user in thejth cell. Then theproblemP(γ) can be reformulated as

P(γ,Γ) : min{wi}N

i=1

N∑

i=1

‖wi‖2 (9)

s.t.|hH

i,i,kwi|2N∑j 6=i

Γj,i,k + σ2i,k

≥ γi, ∀i, k (10)

|hHi,j,kwi|2 ≤ Γi,j,k, ∀i, j 6= i, k(11)

Here, the real-valued vectorΓ is defined as follows

Γ = (Γ1,2,1, ...,Γ1,2,K , ...,ΓN,N−1,K)T ∈ R(N(N−1)K)×1

(12)The introduction ofΓ is similar to the concept of interfer-

ence temperature (IT) in cognitive radios and hence we referto it as IT vector. It is now observed that the constraints in (10)are all decoupled. Unlike [18], our problem is still nonconvexand NP-hard. We cannot solve it by exploiting its dual problemsince the strong duality does not hold. In the following we takethe semidefinite relaxation (SDR) approach, the optimalityofwhich shall be discussed later in this section. Introducingnewvariables{Wi = wiw

Hi }Ni=1, the relaxed problem ofP(γ,Γ)

becomes:

P1(γ,Γ) : min{Wi}N

i=1

N∑

i=1

Tr{Wi} (13)

s.t. Tr{Hi,i,kWi} ≥ γieTi,i,kΓ+ γiσ

2i,k, ∀i, k (14)

Tr{Hi,j,kWi} ≤ eTi,j,kΓ, ∀i, j 6= i, k (15)

Wi � 0 (16)

Here, for notation convenience we have introduced the(N (N − 1)K) × 1 direction vectorsei,j,k and definedHi,j,k , hi,j,kh

Hi,j,k.

For a pre-fixed IT vectorΓ, since the constraints havebeen decoupled, problemP1(γ,Γ) can be decomposed intoN parallel subproblems. Theith subproblem is as follows

Psubi (γ,Γ) : min

Wi

Tr{Wi} (17)

s.t. Tr{Hi,i,kWi} ≥ γieTi,i,kΓ+ γiσ

2i,k, ∀k (18)

Tr(Hi,j,kWi) ≤ eTi,j,kΓ, ∀j 6= i, k (19)

Wi � 0 (20)

It is easily observed that in each subproblem, the base stationonly needs the local channel state information. Specifically,the ith cell’s base station needs just the channel vectorshHi,j,k, ∀j = 1 · · ·N, ∀k = 1 · · ·K. Since it is also convex, the

optimal solution can be obtained efficiently. Having solvedthe


subproblems, we then define the master problem in charge ofupdating the IT vectorΓ

Pmas(γ) : minΓ

P (Γ) (21)

where P (Γ) =N∑i=1

P ⋆i (Γ) with P ⋆

i (Γ) being the optimal

solution of problemPsubi (γ,Γ) for a givenΓ. This master

problem can be solved iteratively using a subgradient pro-jection method. Denoteg ∈ R(N(N−1)K)×1 as the globalsubgradient ofP (Γ) at Γ. The following theorem suggeststhat g can be obtained from each base station.

Theorem 1 : The global subgradientg of P (Γ) in themaster problemPmas(γ) is given by

g =

N∑

i=1

gi (22)

wheregi is the subgradient ofP ⋆i (Γ).

Proof: Please refer to Appendix B.According to Theorem 1, every base station first solves

its own subproblem, gets the subgradient vectorgi and thenbroadcasts it to other cells. Upon receiving all theN subgra-dient vectorsgi’s, the base station in each cell will sum themup to get the subgradientg and update the IT vectorΓ asbelow

Γ(n+ 1) =

[Γ(n)− µ(n) · g(n)

‖g(n)‖

]+, (23)

wheren denotes the iteration index andµ is the step size.Here, [·]+ denotes the projection onto the nonnegative or-thant. Since the problemP1(γ,Γ) is convex, this distributedalgorithm is guaranteed to converge and converge exactlyto the optimal solution ofP1(γ,Γ). Just like the steepestdescent method, the choice of step size affects the convergenceproperties of the iterative algorithm, such as the speed andtheaccuracy. Here we just take the simple diminishing step, i.e.,µ(n) = s/

√n, wheres > 0 is the initial step size.

When the algorithm converges, each base station can getits own beamformer from its resulting matrixW⋆

i . We nowdiscuss how to extract the beamformer vectorwi from eachW⋆

i and its optimality. Since the original problemP(γ)is NP-hard, generally there is no guarantee that an algo-rithm for solving the relaxed SDP problem will give thedesired rank-one solution. If theW⋆

i is rank-one, then thebase station applies the eigen-value decompsotion (EVD) toW⋆

i as W⋆i = λ⋆

iw⋆iw

⋆iH and takeswi =

√λ⋆iw

⋆i as

the optimal beamformer. Otherwise, it first generates a setof candidate beamforming vectors{wl

i} by randomizationmethod. Specifically, one can perform EVD onW⋆

i to getW⋆

i = UiΣiUHi and then generate thelth candidate vector

wli aswl

i = UiΣ1/2i vl, wherevl ∼ CN (0, I). Then based on

its own subproblemPsubi (γ,Γ), the base station needs to do

scaling to get the beamforming vector. The scaling problemis formulated as below

Li(γ,Γ) : minαi

αi (24)

s.t. αi|hHi,i,kw

′i|2 ≥ γie

Ti,i,kΓ+ γiσ

2i,k, ∀k (25)

αi|hHi,j,kw

′i|2 ≤ eTi,j,kΓ, ∀j 6= i, k (26)

αi > 0 (27)

The above scaling problem is a linear programming problemabout the one-dimension variableαi and can be solved veryquickly. Denote the minimum scaling ratio asα⋆

i and theassociated vector asw⋆

i , then the corresponding beamformeris wi =

√α⋆i · w⋆

i . Both the randomization and scalingprocedures are implemented locally by each base station anddo not break the distributed nature of the algorithm. Noticealthough the proposed algorithm cannot guarantee the optimalsolution for the original problemP(γ), it is seen from thesimulation results in Section V that this algorithm achievesthe optimal solution of the original problem in most cases,especially when the network size is small.

Finally, the decentralized algorithm is summarized below.

Algorithm 1: Decentralized algorithm for QoS beam-forming

• Step 1. Initialize the IT vectorΓ(0) with certain values,e.g.,Γ(0) = 1 and set the iteration numbern = 0.

• Step 2. Each base station locally solves its sub-problem(17) and broadcasts the resultinggi (via the backhaulsignaling) to the base stations in the otherN − 1 cells.

• Step 3. Upon receiving the subgradient vectors from allother cells, each base station updates the IT vectorΓ(n)according to (23).

• Step 4. setn = n + 1 and go to Step 2 until meet thestopping condition.

• Step 5. For each base stationi, if W⋆i is rank-one,

then the optimal beamformerwi is the eigenvector ofW⋆

i ; Otherwise the base station implements the random-ization and scaling procedures to get the near-optimalbeamformer.

The main information exchange in this decentralized beam-forming scheme is the real-valued subgradient vectors. Duringeach iteration, for each base station, it should broadcastits subgradient vectorgi, which only hasN × K nonzeroentries. The sum signaling overhead among the base stationsin one iteration is thusO

(N2K

). Denote the number of

iterations times asNb, then the total signaling isO(NbN

2K).

Through the simulations shown in Section V, we can seethat the algorithm converges very fast and can achieve majorpart of the beamforming’s gain after only several iterations.Furthermore, this distributed algorithm also works for theconventional interference channel, which is a special case(K = 1) of the multi-cell multicast networks.

IV. M AX -M IN SINR BEAMFORMING

In this section, we study the beamforming problem for theMMS scheme under individual base station power constraints.The problem is formulated as follows

S(p) : max{wi}N

i=1

min∀i,∀k

|hHi,i,kwi|2

N∑j 6=i

|hHj,i,kwj |2 + σ2

i,k

(28)

s.t. ‖wi‖2 ≤ pi, ∀i (29)

wherep = [p1, p2, ..., pN ]T is the power constraint vector forbase stations in the network.


A. Connection with power optimization

Define a weighted peak power minimization problem as

Q(γ,p) : min{wi}N

i=1

max∀i

1

pi‖wi‖2 (30)

s.t.|hH

i,i,kwi|2N∑j 6=i

|hHj,i,kwj |2 + σ2

i,k

≥ γ, ∀i, ∀k (31)

whereγ is the common SINR target for all cells. It can be seenthat the two problemsS(p) andQ(γ,p) are connected witheach other through the power vectorp. We use the notationγ = S(p) to stand for the optimal objective value of problemS(p), which means that the maximum worst-case SINR forS(p) is γ. For the problemQ(γ,p), we denote the associatedoptimum value asp = Q(γ,p). Then the following theoremtells the relationship between these two problems:

Theorem 2: The SINR optimization problem of (28) andthe power optimization problem of (30) are inverse problems:

γ = S (Q(γ,p) · p) (32)

1 = Q (S(p),p) . (33)

Proof: We first prove (32) by contradiction. Letp and{wi}Ni=1 be the optimal solution ofQ(γ,p), and γ and{wi}Ni=1 be the optimal solution ofS(p · p). We assume thatγ 6= γ. Then if γ < γ, then we can choose{wi}Ni=1 as thesolution forS(p · p), which provides a larger objective valueγ. This is a contradiction of the optimality of{wi}Ni=1 forS(p · p). Otherwise, if γ > γ, then we can find a constantc < 1 to scale the solution set{wi}Ni=1 while still satisfyingthe SINR constraints of problemQ(γ,p). Since {wi}Ni=1

satisfy the power constraints inS(p · p) which means thatmax∀i

1pi‖wi‖2 = p, the resulting set{cwi}Ni=1 achieve a

smaller objective value (weighted peak base-station power)thanp, which contradicts the assumption that{wi}Ni=1 is theoptimal solution ofQ(γ,p). Thus we must haveγ = γ.

The proof of (33) is similar and therefore omitted.In addition, we find numerically that the optimal objective

values of both the two problems are monotonically non-decreasing in the constraint parametersp andγ. Such findingis reasonable since with more power on each base station, thelarger SINR can be achieved, and vice versa.

A similar theorem on inverse property for single-cell multi-casting has been proved in [2]. Unlike [2], our max-min SINRproblem for multicell multicasting cannot be solved by directlysolving the corresponding QoS problem since each basestation is subject to an individual power constraint. Instead,we show the this max-min-SINR problem with multiple powerconstraints can be solved by solving another weighted peakpower minimization problem.

B. Inversion-property-based algorithm

Since the problemS(p) is non-convex, similar to sectionIII, we apply the semidefinite relaxation and get the relaxed

problemS1(p) as below

S1(p) : max{Wi}N

i=1

min∀i,∀k

Tr{Hi,i,kWi}N∑j 6=i

Tr{Hj,i,kWj}+ σ2i,k

(34)

s.t. Tr{Wi} ≤ pi, ∀i (35)

Wi � 0, ∀i (36)

The non-convex rank-one constraint has been dropped. Wefirst get the optimal solution ofS1(p) based on Theorem 2.The inverse problem of problemS1(p) is just the relaxedversion ofQ(γ,p), which is defined as follow

Q1(γ,p) : min{Wi}N

i=1

max∀i

1

piTr{Wi} (37)

s.t. Tr{Hi,i,kWi} ≥ γN∑

j 6=i

Tr{Hj,i,kWj}+ γσ2i,k, ∀i, ∀k (38)

Wi � 0, ∀i (39)

Introducing a slack variablex, we rewrite this problem in amore elegant way

Q1(γ,p) : min{Wi}N

i=1,x

x (40)

s.t. Tr{Hi,i,kWi} ≥ γ

N∑

j 6=i

Tr{Hj,i,kWj}+ γσ2i,k, ∀i, ∀k (41)

1

piTr{Wi} ≤ x, ∀i (42)

Wi � 0, ∀i (43)

Then we can solveS1(p) by iteratively solving its inverseproblemQ1(γ,p) for different γ’s. Notice thatQ1(γ,p) isan SDP problem with strong duality and thus can be solvedefficiently using the interior method. Due to the inversionproperty, ifγ⋆

0 is the optimal value forS1(p), then the optimalvalue for the problemQ1(γ

⋆0 ,p) should be equal to1. With the

non-decreasing monotonicity, we can find the optimal valueγ⋆0

efficiently by a one-dimensional bisection search overγ. Whenwe get the optimal solution ofS1(p), based on the resultingmatrices{W⋆

i }Ni , we apply the EVD or the randomizationand scaling to obtain the final beamformers for all the basestations. Finally, the beamforming algorithm is summarizedbelow

Algorithm 2: Inversion-property-based algorithm forMMS beamforming

• Step 1. Initialize the interval[L,U ] which containsthe optimal valueγ⋆

0 of S1(p), e.g., L = 0, U =max∀i,∀k pi · ‖hi,i,k‖2/σ2, and set the iteration numbern = 0.

• Step 2. Setγ = (L + U)/2, and solve the problemQ1(γ,p).

• Step 3. If the optimal valuep⋆0 of Q1(γ,p) is larger than1, setU = γ; Otherwise, setL = γ.

• Step 4. Setn = n+1 and go back to Step 2 until meetthe stopping condition.

• Step 5. If W⋆i is rank-1 for all i, we can obtain the

optimal solution forS(p) by EVD; Otherwise, the central


controller does randomization and scaling to obtain theapproximate solution.

Note that, in each iteration, we need to solve the weightedpeak power minimization problem, which is the main differ-ence from the algorithm in [2]. Although this algorithm givesan approximate solution forS(p), from the simulation resultsin Section V, we can see that it obtains the optimal solutionin most cases.

V. SIMULATION RESULTS

In this section, we provide numerical examples to illustratethe performance of the proposed multicell multicast beam-forming designs. Within each cell, the channel is assumed asthe normalized Rayleigh fading channel, i.e., the elementsofeach channel vector are independent and identically distributedcircularly symmetric zero-mean complex Gaussian randomvariables with unit variance. For the intercell channels, i.e.,the channels from theith cell’s base station to the users in thejth cell (j 6= i), we introduce the average large-scale fadingratio ε, 0 < ε < 1. A big ε means large intercell interference,which often occurs when the users are at the boundary of eachcell. Here, we setε = 1

2 . A common noise variance is set to beσ2i,k = 1, ∀i, k. Throughout this section, we use the notation

“N − K − Nt” to describe the system configuration, whichmeans that the network hasN cells with K mobile usersin each cell and each base station hasNt antennas. For allsimulations, 200 channel realizations are simulated and 100Gaussian randomizations are generated if the randomizationmethod is needed.

A. Performance comparison with existing schemes

In this subsection, we consider the performance of theproposed algorithms for the two transmission schemes. Forthe QoS scheme, we assume that the SINR targets for theusers in different cells are the same for simplicity. We firstillustrate the convergence of the algorithm. Fig. 2 plots thepower consumption during each iteration at different targetSINR for the(2−2−4) network. The initial step size isµ = 1and the diminishing step size isµ(n) = µ/

√(n). Since the

problemP1(γ) is convex, the proposed distributed algorithmalways converges to its optimal value. The simulation resultvalidates it. Also we can see that at the first few iterations,the algorithm converges very fast and achieves the major partof the optimal value.

In Fig. 3, we illustrate the efficiency of the SDR approachby comparing it with the lower bound, which is the solutionof problemP1(γ) in (13) where the rank-one constraint isrelaxed. It can be seen that although we take the semidefiniterelaxation, the proposed coordinated beamforming algorithmobtains the optimal performance in most cases.

We now compare the performance of the proposed SDR-based QoS beamforming inAlgorithm 1 with two conven-tional multicell beamforming algorithms: multicell blockdi-agonalization (M-BD) [19] and layered signal-to-leakage-plus-noise ratio (L-SLNR) [20]. Though these two algorithms wereproposed for multicell networks with unicast traffic, they canbe easily extended to multicasting. More specifically, for the

0 5 10 150

20

40

60

80

100

120

140

iteration number

tota

l pow

er

(Watt

)

proposed 2−2−4

SINR target = 6 dB

SINR target = 16 dB

SINR target = 10 dB

Fig. 2. Convergence behavior of the decentralized algorithm for the QoSscheme.

6 8 10 12 14 165

10

15

20

25

30

SINR target (dB)

tota

l pow

er

(dB

)

lower bound

proposed

3−3−6

2−2−4

3−2−6

Fig. 3. Comparison of the proposed algorithm with its lower bound for theQoS scheme.

M-BD algorithm, each base station chooses the beamformingvector which lies in the null space of the channels from theotherN − 1 base stations, i.e.,

wi ∈ Null(Hi

)(44)

where Hi = [h1,i,1, ...,h1,i,K , ...,hi−1,i,1, ...,hi−1,i,K ,

hi+1,i,1, ...,hi+1,i,K , ...,hN,i,1, ...,hN,i,K ]H and Null(.)

stands for the null space of a matrix. Then the QoS problemreduces to a power allocation problem and we can solve it toobtain the resulting power for each base station. For the L-

6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

SINR target (dB)

feasib

ility

proposed 2−2−4

L−SLNR 2−2−4

M−BD 2−2−4

open−loop STBC 2−2−4

Fig. 4. Comparison on feasibility of the different multicast beamformingalgorithms for the QoS scheme.


SLNR algorithm, each base station chooses its beamformingvector as

wi ∝ max.eigenvector

N∑

j 6=i

K∑

l=1

hi,j,lhHi,j,l + σ2I

−1

·[

K∑

l=1

hi,i,lhHi,i,l

])(45)

which means thatwi should have the same direction as theeigenvector corresponding to the largest eigenvalue of theabove matrix. We can similarly get the resulting beamformingvectors for the base stations by solving the reduced powerallocation problem. As a performance benchmark, open-loopspace time block coding (STBC) without requiring any CSIat each base station is also simulated. Fig. 4 shows thecomparison on feasibility1. Here, the feasibility percentageis obtained by counting the number of trials among the 200channel realizations that each algorithm is able to find thesolution to meet all the constraints. It can be seen that boththe open-loop STBC and the L-SLNR beamformer almostdo not work when SINR targetγ is large. This is expectedas the open-loop STBC does not make any use of channelstate information and serves purely as isotropic broadcasting.The L-SLNR beamformer, on the other hand, only tries tomaximize the SLNR from the transmitter perspective andcannot guarantee the SINR maximization at the receiver end.From Fig. 4 it is also seen that the M-BD beamformer and theproposed coordinated beamformer are always feasible in theconsidered SINR target region. Here, the reason that the M-BD beamformer can work well is that the considered networkis an interference-limited one and M-BD can null out allthe interference for each user. The good performance of theproposed algorithm is expected as it can obtain the optimalsolution in most cases under this particular setting (see Fig.2). Therefore, in Fig. 5 we only compare the energy efficiencyof the proposed algorithm with M-BD algorithm. We can seethat the M-BD algorithm consumes much more power thanthe proposed algorithm. In particular, at target SINR of10dB,M-BD consumes3 dB more power in the(2− 2− 4) systemand 4dB more power in the(3 − 2 − 6) system. Notice thatthe M-BD algorithm requires that the number of transmittingantennas at each base station should be larger than the totalnumber of receive antennas at all users. However, our proposedalgorithm does not have this requirement.

For the max-min SINR beamforming scheme, we assumethe power constraint for each base station is the same. Wefirst compare the the performance of the proposedAlgorithm2 with its upper bound in Fig. 6. Similar to the QoS scheme,we can see that the gap from the upper bound is very smalland it achieves the optimal solution in most cases. Fig. 7compares its performance with the existing algorithms. Wecan see that our proposed algorithm significantly outperformsthe other ones over a large range of individual power constraintparameter. In particular, at the individual power constraint of10dB in the(3−2−5) system, the worst-case SINR achieved

1Although we have assumed that the beamforming problem is feasible,each specific algorithm may not be able to find the beamformersto meet theSINR target due to its sub-optimality.

6 8 10 12 14 165

10

15

20

25

SINR target (dB)

tota

l pow

er

(dB

)

proposed 2−2−4

proposed 3−2−6

M−BD 2−2−4

M−BD 3−2−6

M−BD

proposed

Fig. 5. Comparison on energy efficiency of the different multicell multicastbeamforming algorithms for the QoS scheme.

4 7 10 13 16 194

6

8

10

12

14

16

18

20

per−base−station power (dB)

min

imum

SIN

R (

dB

)

upper bound

proposed

3−2−4

3−2−6

3−3−6

2−2−4

Fig. 6. Comparison on the minimum SINR of the proposed algorithm withits upper bound for the MMS scheme.

by the proposed algorithm is6dB higher than L-SLNR,8dBhigher than M-BD and9dB higher than open-loop STBC.From Fig. 7, it is also found that the M-BD algorithm performsthe worst when the per base station power is small but issuperior to the L-SLNR algorithm at large per base stationpower.

B. Effects of channel correlation

In practical scenarios, the channels among different usersmay be correlated with each other, especially for the userswho are very near to each other in the geographical position.

4 7 10 13 16 19−5

0

5

10

15

20

per−base−station power (dB)

min

imum

SIN

R (

dB

)

proposed 3−2−5

L−SLNR 3−2−5

M−BD 3−2−5

open−loop STBC 3−2−5

Fig. 7. Comparison of the different multicell multicast beamformingalgorithms for the MMS scheme.


6 8 10 12 14 160

0.2

0.4

0.6

0.8

1

SINR target (dB)

feasib

ility

r=0

r=0.5

r=0.7

r=0.9

3−2−4

Fig. 8. Feasibility of the proposed algorithm with correlated intercell-userchannels for the QoS scheme.

In order to model the correlated channels, we use the followingKronecker model [21] [22].

Hkro = C1/2G, (46)

whereHkro is a channel matrix whose rows are correlatedwith each other andG is a matrix with i.i.d. circularlysymmetric Gaussian entries with zero mean and unit variance(This is for intracell channels by default. If it is intercellchannel, the variance isε2). We model channel correlationmatrix C as a Hermitian Toeplitz matrix with exponentialentries [C]i,j = r|i−j| [23]. Here, r can be seen as thecorrelation ratio and0 ≤ r ≤ 1.

We first investigate the effects of intercell-user channelcorrelation on the feasibility of the QoS problem in Fig.8. The correlation ratior is set to be0.5, 0.7 and 0.9,where r = 0 stands for the independent channel. We cansee that when the intercell-users’ channels are correlated, thefeasibility decreases. This justifies our statement in SectionIII-A, i.e., the intercell-user channel correlation has a negativeimpact on the feasibility of the QoS problem.

Fig. 9 shows the effects of intracell-user channel correlationfor the QoS beamforming scheme. The correlation ratior isalso set to be 0.5, 0.7 and 0.9. From the results, we can see thatwhen the intracell-user channels are correlated, it is helpful forthe system. More specifically, the consumed power becomesless. We have observed the similar results on the usefulnessofintracell-user channel correlation on the MMS scheme, whichare ignored here due to page limit.

VI. CONCLUSION

This paper considered two coordinated mulicast beamform-ing schemes for multicell networks. For the QoS scheme, weprovided a necessary condition for the beamforming problemto be feasible when the network shares a common SINRtarget and proposed a decentralized algorithm to implementthe coordinated beamforming. For the max-min SINR scheme,we considered individual base station power constraints andalso proposed an efficient beamforming algorithm. Besides,we also investigated the impacts of intercell-user and intracell-user channel correlation on the multicast network.

There are several other issues to be further investigated formulticell multicast beamforming. First, finding the sufficient

6 8 10 12 14 160

5

10

15

20

SINR target (dB)

tota

l pow

er

(dB

)

r=0

r=0.5

r=0.7

r=0.9

2−2−4

Fig. 9. Total transmitted power with correlated intracell-user channels forthe QoS scheme.

condition for the feasibility of the QoS problem remainsopen. Second, a decentralized algorithm to implement thebeamforming design for the MMS scheme is to be designed.Last but not least, it is also interesting to consider robustbeamforming when the channel state information at each basestation is not perfect.

APPENDIX APROOF OFLEMMA 1

We first define the beamformer matrixW which is full-rankas below

W =

w1 0 0 · · · 00 w2 0 · · · 0... · · ·0 0 0 · · · wN

(47)

Based on the fact that SINRi,k ≤ SIRi,k, we have

SINRi,k =|hH

i,i,kwi|2N∑j 6=i

|hHj,i,kwj |2 + σ2

i,k

≤|hH

i,i,kwi|2N∑j 6=i

|hHj,i,kwj|2

=1

1ηi,k

− 1(48)

where we define theηi,k as below

ηi,k =|hH

i,i,kwi|2N∑j=1

|hHj,i,kwj |2

=|[HkW]i,i|2

N∑j=1

|[HkW]i,j |2=

|[HkW]i,i|2|[HkWWHHH

k ]i,i|(49)

the second equation in (49) is because (7) and (47). We alsoknow the monotonicity off(x) = 1/(1/x − 1) whenx < 1.Comparing it with (48), we can bound the SINRi,k throughbounding the associateηi,k. Let the singular value decomposi-tion(SVD) ofHkW be denoted asHkW = UkΣkV

Hk , where

both Uk andVk areK × rk quasi-unitary matrix, i.e., theircolumns are orthogonal with each other.Σk is an rk × rk


diagonal matrix whose elements are the singular values andrk = rank(HkW). Then

ηi,k =|uH

k,iΣkvk,i|2uHk,iΣ

2kuk,i

, i = 1 2 · · ·N, k = 1 2 · · ·K (50)

where uk,i and vk,i are the ith columns ofUk and Vk.According to Cauchy-Schwarz inequality, we have

|uHk,iΣkvk,i|2 ≤ ‖uH

k,iΣk‖2 · ‖vk,i‖2

=(uHk,iΣ

2kuk,i

) (vHk,ivk,i

)(51)

SincevHk,ivk,i = [(HkW)†(HkW)]i,i, we conclude that

ηi,k ≤ [(HkW)†(HkW)]i,i (52)

Thus we have

mini

ηi,k ≤ 1

N

N∑

i=1

ηi,k ≤ 1

N

N∑

i=1

[(HkW)†(HkW)]i,i

=1

NTr{(HkW)†HkW} =

rank(HkW)

N≤ rank(Hk)

N(53)

Further we can get

mini,k

ηi,k = mink

{mini

ηi,k

}

≤ min{ rank(H1)

N,

rank(H2)

N, · · · , rank(HK)

N} (54)

So if the problem (3) is feasible, then the minimum SINRshould be larger than the thresholdγ. Plugging (54) into (48),we can conclude that

γ ≤ mini,k

SINRi,k ≤{

rank(H1)

N − rank(H1), · · · , rank(HK)

N − rank(HK)

}

(55)which completes the proof of Lemma 1.

APPENDIX BPROOF OFTHEOREM 1

We begin by computing the subgradientgi from eachsubproblem. The Lagrangian of theith subproblem is givenby

L(Wi,Γ,λ) = Tr{(Wi)}

−K∑

k=1

λi,i,k

[1

γiTr{Hi,i,kWi} − eH

i,i,kΓ− σ2i,k

]

+

N∑

j 6=i

K∑

k=1

λi,j,k

[Tr{Hi,j,kWi} − eH

i,j,kΓ

](56)

Then the dual functiondi(Γ,λ) is

di(Γ,λ) = minWi

L(Wi,Γ,λ)

=

K∑

k=1

λi,i,keTi,i,k −

N∑

j 6=i

K∑

k=1

λi,j,keTi,j,k

Γ+ fi(λ) (57)

where

fi(λ)

= minWi

(Tr{Wi} −

K∑

k=1

λi,i,k

[1

γiTr{Hi,i,kWi} − σ2

i,k

]

+

N∑

j 6=i

K∑

k=1

λi,j,kTr{Hi,j,kWi}

(58)

Since the problemPsubi (γ,Γ) is convex, then the strong

duality holds which means that

P ⋆i (Γ) = max

λ�0

di(Γ,λ) (59)

Denoteλ⋆ as the optimal Lagrange multiplier for the dualproblem, then we have

P ⋆i (Γ) = di(Γ,λ

⋆)

=

K∑

k=1

λ⋆i,i,ke

Ti,i,k −

N∑

j 6=i

K∑

k=1

λ⋆i,j,ke

Ti,j,k

Γ+ fi(λ⋆) (60)

Definegi as

gi ,

K∑

k=1

λ⋆i,i,kei,i,k −

N∑

j 6=i

K∑

k=1

λ⋆i,j,kei,j,k, (61)

we have

P ⋆i (Γ) = gH

i Γ+ fi(λ⋆)

= gHi (Γ− Γ) + gH

i Γ+ fi(λ⋆) ≤ gH

i (Γ− Γ) + P ⋆i (Γ) (62)

which is equivalent to that

P ⋆i (Γ) ≥ P ⋆

i (Γ) + gHi (Γ− Γ). (63)

It means thatgi is the subgradient ofP ⋆i (Γ) and obtained for

the ith subproblem.In the same way, we can compute the global subgradientg

of Pmas(γ,Γ) as below

g =

N∑

i=1

K∑

k=1


N∑

i=1

N∑

j 6=i

K∑

k=1

λ⋆i,j,kei,j,k

=

N∑

i=1

K∑

k=1


N∑

j 6=i

K∑

k=1

λ⋆i,j,kei,j,k

=

N∑

i=1

gi (64)

which completes the proof of Theorem 1.

REFERENCES

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[2] E. Karipidis, N. D. Sidiropoulos, and Z.-Q. Luo, “Quality of serviceand max-min fair transmit beamforming to multiple cochannel multicastgroups,” IEEE Trans. Signal Process., vol. 56, no. 3, Mar. 2008.

[3] V. Ntranos, N. D. Sidiropoulos, and L. Tassiulas, “On multicast beam-forming for minimum outage,”IEEE Trans. Wireless Commun., vol. 8,no. 6, June 2009.

[4] S. Y. Park and D. J. Love, “Capacity limits of multiple antenna mul-ticasting using antenna subset selection,”IEEE Trans. Signal Process.,vol. 56, no. 6, June 2008.


[5] Motorola Inc., “Long term evolution (LTE): a technicaloverview,” Technical White Paper, http://business.motorola.com/experienceltr/pdf/LTE%20Technical%20overview.pdf

[6] A. Lozano, “Long-term transmit beamforming for wireless multicast-ing,” in Proc. 2007 ICASSP.

[7] S. Shamai (Shitz) and B. M. Zaidel, “Enhancing the cellular downlinkcapacity via co-processing at the transmitting end,’ inProc. 2001 IEEEVehicle Technology Conference, vol. 3, pp. 1745–1749.

[8] H. Zhang and H. Dai, “Cochannel interference mitigationand coop-erative processing in downlink multicell multiuser MIMO networks,”EURASIP J. Wireless Commun. Netw., no. 2, 2004.

[9] R. Mochaourab and E. Jorswieck, “Optimal beamforming ininterferencenetworks with perfect local channel information,”IEEE Trans. SignalProcess., vol. 59, no. 3, Mar. 2011.

[10] D. P. Palomar and M. Chiang, “A tutorial on decomposition methodsfor network utility maximization,”IEEE J. Sel. Areas Commun., vol. 21,no. 8, Aug 2006.

[11] M. Jordan, X. Gong, and G. Ascheid, “Multicell multicast beamformingwith delayed SNR feedback,” inProc. 2009 IEEE GLOBECOM.

[12] G. Dartmann, X. Gong, and G. Ascheid, “Low complexity cooperativemulticast beamforming in multiuser multicell downlink networks,” inProc. 2011 CROWNCOM.

[13] G. Dartmann, X. Gong, and G. Ascheid, “Cooperative beamformingwith multiple base station assignment based on correlationknowledge,”in Proc. 2010 VTC.

[14] A. Wiesel, Y. C. Eldar, and S. Shamai (Shitz), “Linear precodingvia conic optimization for fixed MIMO receivers,”IEEE Trans. SignalProcess., vol. 54, no. 1, Jan. 2006.

[15] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convexprogramming.” Available: http://stanford.edu.boyd/cvx.

[16] S. Zhang and Y. Huang, “Complex quadratic optimizationand semidef-inite programming,”SIAM J. Optim., vol. 16, no. 3, Jan. 2006.

[17] H. Dahrouj and W. Yu, “Coordinated beamforming for the multicellmulti-antenna wireless system,”IEEE Trans. Wireless Commun., vol. 9,no. 5, May 2010.

[18] R. Zhang and S. Cui, “Cooperative interference management with MISObeamforming,”IEEE Trans. Signal Process., vol. 58, no. 10, Oct. 2010.

[19] R. Zhang, “Cooperative multi-cell block diagonalization with per-base-station power constraints,”IEEE J. Sel. Areas Commun., vol. 28, no. 9,Dec. 2010.

[20] R. Zakhour and D. Gesbert, “Distributed multicell-MISO precodingusing the layered virtual SINR framework,”IEEE Trans. WirelessCommun., vol. 9, no. 8, Aug. 2010.

[21] X. Mestre and J. Fonollosa, “Capacity of MIMO channels:asymptoticevaluation under correlated fading,”IEEE J. Sel. Areas Commun., vol.21, no. 5, June 2003.

[22] K. Werner, M. Janson, and P. Stoica, “On estimation of covariancematrices with Kronecher product structure,”IEEE Trans. Signal Process.,vol. 56, no. 2, Feb. 2008.

[23] A. Paulraj, R. Nabar, and D. Gore,Introduction to Space-Time WirelessCommunications. Cambridge University Press, 2003.

Zhengzheng Xiang received the B.S. degree inelectronic engineering from Shanghai Jiao TongUniversity, Shanghai, China, in 2010. He is currentlyworking toward the Ph.D. degree with the Instituteof Wireless Communication Technology, ShanghaiJiao Tong University. His research interests includeinterference management in wireless networks, wire-less relay technologies, and advanced signal process-ing for wireless cooperative communication.

Meixia Tao (S’00-M’04-SM’10) received the B.S.degree in electronic engineering from Fudan Uni-versity, Shanghai, China, in 1999, and the Ph.D.degree in electrical and electronic engineering fromHong Kong University of Science and Technologyin 2003. She is currently an Associate Professor withthe Department of Electronic Engineering, ShanghaiJiao Tong University, China. From August 2003 toAugust 2004, she was a Member of ProfessionalStaff at Hong Kong Applied Science and Tech-nology Research Institute Co. Ltd. From August

2004 to December 2007, she was with the Department of Electrical andComputer Engineering, National University of Singapore, as an AssistantProfessor. Her current research interests include cooperative transmission,physical layer network coding, resource allocation of OFDMnetworks, andMIMO techniques.

Dr. Tao is an Editor for the IEEE TRANSACTIONS ONCOMMUNICATIONS

and the IEEE WIRELESSCOMMUNICATIONS LETTERS. She was on the Edi-torial Board of the IEEE TRANSACTIONS ONWIRELESSCOMMUNICATIONS

from 2007 to 2011 and the IEEE COMMUNICATIONS LETTERSfrom 2009 to2012. She also served as Guest Editor forIEEE Communications Magazinewith feature topic on LTE-Advanced and 4G Wireless Communications in2012, and Guest Editor forEURISAP J WCN with special issue on PhysicalLayer Network Coding for Wireless Cooperative Networks in 2010.

Dr. Tao is the recipient of the IEEE ComSoC Asia-Pacific OutstandingYoung Researcher Award in 2009.

Xiaodong Wang(S’98-M’98-SM’04-F’08) receivedthe Ph.D degree in Electrical Engineering fromPrinceton University.

He is a Professor of Electrical Engineering atColumbia University in New York. Dr. Wang’s re-search interests fall in the general areas of com-puting, signal processing and communications, andhas published extensively in these areas. Among hispublications is a book entitled “Wireless Commu-nication Systems: Advanced Techniques for SignalReception”, published by Prentice Hall in 2003. His

current research interests include wireless communications, statistical signalprocessing, and genomic signal processing.

Dr. Wang received the 1999 NSF CAREER Award, the 2001 IEEECommunications Society and Information Theory Society Joint Paper Award,and the 2011 IEEE Communication Society Award for Outstanding Paper onNew Communication Topics. He has served as an Associate Editor for theIEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONSON WIRELESSCOMMUNICATIONS, the IEEE TRANSACTIONS ON SIGNAL

PROCESSING, and the IEEE TRANSACTIONS ON INFORMATION THEORY.He is a Fellow of the IEEE and listed as an ISI Highly-cited Author.

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Reviewer(s)' Comments to Author:Reviewer: 1Comments to the Corresponding AuthorThe authors have carefully revised their paper according to the reviewer comments. The paper contains a solid contribution, it is clearly written, comprehensive, and could be accepted.

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