3850 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 8, AUGUST 2011

A Unified Analysis of Max-Min Weighted SINR forMIMO Downlink System

Desmond W. H. Cai, Student Member, IEEE, Tony Q. S. Quek, Member, IEEE, and Chee Wei Tan, Member, IEEE

Abstract—This paper studies the max-min weighted signal-to-in-terference-plus-noise ratio (SINR) problem in the multiple-input-multiple-output (MIMO) downlink, where multipleusers are weighted according to priority and are subject toa weighted-sum-power constraint. First, we study the mul-tiple-input-single-output (MISO) and single-input-multiple-output (SIMO) problems using nonlinear Perron–Frobeniustheory. As a by-product, we solve the open problem of conver-gence for a previously proposed MISO algorithm by Wiesel, Eldar,and Shamai in 2006. Furthermore, we unify our analysis withrespect to the previous alternate optimization algorithm proposedby Tan, Chiang, and Srikant in 2009, by showing that our MISOresult can, in fact, be derived from their algorithm. Next, wecombine our MISO and SIMO results into an algorithm for theMIMO problem. We show that our proposed algorithm is optimalwhen the channels are rank-one, or when the network is operatingin the low signal-to-noise ratio (SNR) region. Finally, we provethe parametric continuity of the MIMO problem in the powerconstraint, and we use this insight to propose a heuristic initial-ization strategy for improving the performance of our (generally)suboptimal MIMO algorithm. The proposed initialization strategyexhibits improved performance over random initialization.

Index Terms—Beamforming, multiple-input–multiple-output(MIMO), uplink–downlink duality.

I. INTRODUCTION

I N bandwidth-limited wireless networks, the spatial diver-sity of networks arising from the use of antenna arrays

at both the transmitter and the receivers can be exploited tomitigate the interference between multiple users and increasethe overall performance. Beamforming is a spatial diversitytechnique used at both the transmitter and the receiver to

Manuscript received July 06, 2010; revised December 08, 2010; acceptedApril 08, 2011. Date of publication May 05, 2011; date of current version July13, 2011. The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Ali Ghrayeb. This work was partially sup-ported by the Research Grants Council of Hong Kong under Project No. RGCCityU 112909, CityU 7200183 and CityU 7008087. The material in this paperhas been presented in part at the IEEE International Symposium for InformationTheory (ISIT), Austin, Texas, June 2010, and at the Ninth International Sympo-sium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks(WiOpt), Princeton, NJ, May 2011.

D. W. H. Cai was with the Institute for Infocomm Research. He is now withthe Department of Electrical Engineering, California Institute of Technology,MC 305-16, 1200 E. California Blvd., Pasadena, CA 91125 USA (e-mail:wccai@caltech.edu).

T. Q. S. Quek is with the Institute for Infocomm Research, A*STAR, #21-01Connexis, Singapore 138632 (e-mail: qsquek@i2r.a-star.edu.sg).

C. W. Tan is with the Department of Computer Science, City University ofHong Kong, Kowloon, Hong Kong (e-mail: cheewtan@cityu.edu.hk).

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

Digital Object Identifier 10.1109/TSP.2011.2150218

increase the signal-to-interference-plus-noise ratio (SINR)of each user when multiple users share a common band-width. We consider the joint optimization of transmit power,linear transmit beamformers at the base station, and linearreceive beamformers at the mobile users in a wireless mul-tiple-input-multiple-output (MIMO) downlink system.

Two important performance metrics in a wireless networkare the minimum weighted SINR and the total transmit power.Clearly, the total transmit power is directly related to the re-source cost of the network. The minimum weighted SINR, atfirst glance, appears to characterize only the worst performinguser relative to a set of user priorities. However, maximizingthe minimum weighted SINR actually amounts to equalizingthe weighted SINR performance of all users (where the weightsreflect the user priorities); and in the case when all users are as-signed the same priority, amounts to equalizing the SINR per-formance. Hence, maximizing the minimum weighted SINRis a strategy for enforcing the desired level of fairness in thenetwork.

Thus, two optimization problems of interest are the max-minweighted SINR problem subject to a given total power constraintand the total power minimization problem subject to given min-imum SINR constraints. Whichever optimization goal is chosendepends on the priorities of the system designer. Moreover, bothoptimization goals are closely related. It was proved that formultiple-input-single-output (MISO) downlink systems, thesetwo optimization problems are inverse problems [1]. This meansthat, suppose we know the optimal minimum weighted SINRin the max-min weighted SINR problem, then by plugging inthis optimal minimum weighted SINR as the weighted SINRconstraint in the total power minimization problem, the optimaltotal transmit power will be equal to the total transmit powerconstraint in the max-min weighted SINR problem.

Despite the close relationship between the two problems,there are significantly more results on the total power minimiza-tion problem. Two of the earliest and most significant resultswere the distributed power control (DPC) algorithm [2] and thestandard interference function framework for analyzing itera-tive power control systems [3]. The DPC algorithm, togetherwith uplink-downlink duality [4], was extensively applied inalternate optimization approaches to derive optimal algorithmsfor the MISO problem [5]–[7], and suboptimal algorithms forthe MIMO problem [8], [9]. In a completely different approach,the authors in [1] showed that the standard interference func-tion framework can also be used to solve the MISO problem.By reformulating the MISO problem as a second-order coneprogram (SOCP), the authors derived a fixed-point equation inthe dual uplink power, and applied the standard interference

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CAI et al.: A UNIFIED ANALYSIS OF MAX-MIN WEIGHTED SINR FOR MIMO DOWNLINK SYSTEM 3851

function properties to prove the convergence of the fixed-pointiteration. The authors also derived a similar fixed-point itera-tion for the MISO max-min weighted SINR problem withouta convergence proof.

To our knowledge, there are no successful attempts toprovide a complete analytical and algorithmic analysis to theMIMO max-min weighted SINR problem. The earliest strategyproposed to solve the max-min weighted SINR power controlproblem was the extended coupling matrix approach [10]. Un-fortunately, this approach, when extended to the MISO/MIMOproblems via alternate optimization [8], [11]–[15], required acentralized power update involving an eigenvector computa-tion. Recently, the authors in [16] and [17] derived a differentalgorithm for the max-min weighted SINR power controlproblem. The key insight was that the problem is equivalentto solving an affine eigenvalue equation, in which the optimalvalue is associated with the eigenvalue and the optimal power isgiven by the eigenvector. By applying nonlinear Perron–Frobe-nius theory, a DPC-like power control algorithm was derivedwhich avoided the eigenvector computation and normalizationnecessary at each iterative step in the extended-coupling matrixapproach.

In this paper, we extend the recent results in [16] to theMIMO max-min weighted SINR problem with linear beam-forming. First, we analyze the MISO and single-input-mul-tiple-output (SIMO) problems using nonlinear Perron–Frobe-nius theory. Specifically, we derive a fixed-point equation foreach problem and apply nonlinear Perron–Frobenius theoryto prove the convergence of the fixed-point iterations. As aby-product, we solve the open problem of convergence for thepreviously proposed MISO algorithm in [1]. We also showthat our fixed-point iteration can, in fact, be decoupled andinterpreted as an alternate optimization of power, via nonlinearPerron–Frobenius theory [16], and beamformers, via min-imum-variance distortionless response (MVDR) beamforming(cf. Fig. 1). Finally, we apply nonnegative matrix theory to gainfurther insight into the structure of the optimal solutions of bothproblems.

Next, we combine our MISO and SIMO results into an al-gorithm for the MIMO problem via an alternate optimizationprocedure. The nonconvexity of the MIMO problem imply thatalternate optimization algorithms are in general suboptimal. Wefurther our analysis by proving that our proposed algorithm isoptimal when the channels are rank-one, or when the network isoperating in the low signal-to-noise ratio (SNR) region. Basedon the latter property, and by studying the continuity of the op-timal solutions in the power constraint, we propose a heuristicstrategy for initializing our alternate optimization algorithm. Wecall this the parametric search initialization algorithm. Numer-ical examples show that our proposed initialization strategy per-forms significantly better than random initialization. Finally, webriefly describe how a similar strategy can be employed for theMIMO total power minimization problem.

This paper is organized as follows. We begin in Section IIby introducing the problem formulation. Next, in Section III,we analyze the MISO and SIMO problems using nonlinearPerron–Frobenius theory. In Section IV, we combine our MISO

Fig. 1. These two diagrams illustrate how nonlinear Perron–Frobenius theoryunifies key approaches for solving the MISO and SIMO max-min weightedSINR problems. For the MISO problem, uplink-downlink duality (UL-DLDuality) is used to convert the problems into corresponding problems in thedual uplink variables. (a) MISO Max-Min Weighted SINR. (b) SIMO Max-MinWeighted SINR.

and SIMO results into an algorithm for the MIMO problem. Wealso prove the parametric continuity of the optimal solutionsin the power constraint. Then, we exploit this property topropose the parametric search initialization algorithm, whichimproves the performance of the original MIMO algorithm.In Section V, we evaluate the performance of the proposedalgorithms. Finally, Section VI concludes the paper.

The following notations are used.• Boldface upper-case letters denote matrices, boldface low-

ercase letters denote column vectors, and italics denotescalars.

• For denotes fordenotes for , and denotesand .

• denotes the Perron–Frobenius eigenvalue of a non-negative matrix .

• and denote the Perron (right) and left eigenvec-tors of associated with .

• denotes the diagonal matrix formed by the com-ponents of the vector .

• and denote transpose and complex conjugatetranspose, respectively.

• denotes the set of nonnegative real numbers, de-notes the set of positive real numbers, and denotes theset of complex numbers.

• denotes the norm.

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II. PROBLEM FORMULATION

We consider a MIMO downlink system where independentdata streams are transmitted over a common frequency band.The transmitter is equipped with antennas and the receiver ofthe th stream, denoted by , is equipped with antennas. Thedownlink channel can be modeled as a vector Gaussian broad-cast channel given by

where is the received signal vector atis the channel matrix between the transmitter

and is the transmitted signal vector, andis the circularly symmetric Gaussian noise

vector with covariance at for some .We assume linear transmit and receive beamforming. The

transmitted signal vector can be written as, where is the signal intended for the

th stream, is the normalized transmit beamformer,and and are respectively the information signaland transmit power for that stream. The th stream is decodedusing a normalized receive beamformer . Definethe power vector , the tuples of beamformingmatrices and , the noisevector , and the matrix suchthat the entry . We assume that the receivebeamformers are chosen such that all the effective links arecoupled so for all and . The SINR of the th streamin the downlink can be expressed as

(1)

We are interested in the max-min weighted SINR problem in thedownlink given by

where is the given power constraint,is the priority vector such that is

the priority assigned by the network to the th stream, andis the weight vector such that

is the weight associated with in the power constraint. Therestriction that the weights be positive is for technical purposesand is not a stringent limitation on the model. In cases where acertain stream must be excluded from a power constraint, theweight on that stream in the power constraint can simply bechosen to be extremely small. Since for all and ,we have that at the optimal solution to , all the weightedSINR’s are equal, that is, are equal for all. Moreover, the power constraint must be active.

For ease of notation, we define the (cross channel interfer-ence) matrix and the normalized priority vector

with the following entries:

ifif

Fig. 2. Multiuser MIMO with one data stream per user.

Hence, can be rewritten as

III. ANALYSIS BY NONLINEAR PERRON–FROBENIUS THEORY

In this section, we apply nonlinear Perron–Frobenius theoryto analyze the max-min weighted SINR problem in the MISOand SIMO scenarios. First, in Section III-A, we apply nonlinearPerron–Frobenius theory to propose an algorithm for solving theMISO problem. Then, in Section III-B, we show that nonlinearPerron–Frobenius theory can resolve the open problem of con-vergence of the MISO max-min SINR algorithm proposed in[1]. Next, in Section III-C, we show that our new MISO resultcan, in fact, be derived from the alternate optimization algorithmproposed in [16]. In Section III-D, we exploit the connectionbetween Perron–Frobenius theory and the Friedland–Karlin in-equalities in [18] to draw further insights on the optimal solu-tions. Finally, in Section III-E, we present a similar analysis ofthe SIMO problem.

A. MISO Optimization

In this section, we fix the receive beamformers in andconsider the MISO max-min weighted SINR problem given by

CAI et al.: A UNIFIED ANALYSIS OF MAX-MIN WEIGHTED SINR FOR MIMO DOWNLINK SYSTEM 3853

For notational brevity, we omitted the dependence ofon . First, consider the following problem obtained by fixingthe transmit beamformers in :

(2)

It is well-known that (2) can be reformulated as a geometric pro-gram by introducing an auxiliary variable . Making the loga-rithmic change of variables and , we arriveat the following equivalent convex problem:

(3)The Lagrangian associated with (3) is

where and are the nonnegative Lagrange dual variables.It is easy to check that the convex problem given by (3) satis-

fies Slater’s condition. Hence, the Karush–Kuhn–Tucker (KKT)conditions are necessary and sufficient conditions for optimalityof (3). Recall that, at the optimal solution to (3), all the inequalityconstraints must be active. Hence, we can replace the inequalityconstraints (in the feasibility conditions) with equality, and dropthe complementary slackness conditions. Making a change ofvariables back into and , we arrive at the following neces-sary and sufficient conditions for optimality of (2):

(4)

(5)

(6)

(7)

(8)

(9)

(10)

where and are the optimal primal and dualvariables, respectively. Here, (6) was obtained from and(9)–(10) follows from the fact that and must be strictlypositive in order for and to be boundedfrom below. Note that the auxiliary variable is strictlypositive.

From (6), we conclude that is the optimal dual uplinkpower and is the noise vector in the dual uplink network [4],

[9], [19]. To obtain the equivalent power constraint in the dualuplink network, we first rewrite (6) in vector form as

Then, taking the inner product of this equation with and sub-stituting for and from (4) and (5), we have

(11)

Note that (4)–(8) are also necessary conditions for the op-timal solution of . To obtain necessary conditions forthe optimal transmit beamformers , first apply the constraint

for all and rewrite (6) as

(12)

Let the mapping be defined by

Since , we have that is positive definite for anyand hence it is invertible. Define the normalized minimum

variance distortionless response (MVDR) beamformers in thedual uplink network [20]:

By applying the fact that the optimal transmit beamformersin the dual uplink network are the normalized MVDR beam-formers, i.e., for , we eliminate

from (12) and obtain the following fixed-point equation:

(13)

where the mapping is defined by

We now state the key result for solving .Theorem 1: Define the norm on as fol-

lows: . The conditional eigen-value problem given by andhas a unique solution , and

. Moreover, the normalized fixed-point iterationconverges to

geometrically fast.

3854 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 8, AUGUST 2011

Proof: Refer to Appendix A.Theorem 1 implies that the optimal dual uplink power for

is unique. With the optimal dual uplink power, canbe computed using the normalized MVDR beamformers, andcan be computed using the max-min weighted SINR power con-trol algorithm [16], [19]. In fact, since the fixed-point iterationfor does not depend on and (cf. steps 1 and 2 of Algorithm1 below), the latter two quantities can be updated in parallel with

. The following algorithm converges to the optimal solution of. It can also be shown that the algorithm converges

with monotonically increasing minimum weighted SINR.

Algorithm 1: MISO Max-Min Weighted SINR

• Initialize: arbitrary , andfor , such that, and for all .

1) Compute dual auxiliary variables:

2) Update dual uplink powers:

3) Compute auxiliary variables:

4) Update downlink powers:

5) Compute normalized MVDR transmit beamformers:

B. Resolving a Previously Open Problem on Convergence

Next, we briefly address an open problem regarding the con-vergence of an algorithm proposed for under the spe-cial case when all data streams have equal priority ,the network has a total power constraint , and all userssuffer from the same noise covariance . Under this con-ditions, can be rewritten as

(14)

In [1], it was proved that if and satisfy

(15)

where the mapping is defined by

then is the optimal dual uplink power and is the optimalvalue of (14). Moreover, the authors proposed the followingfixed-point iteration to solve for the optimal dual uplink powers:

(16)

(17)

By comparing (15) with (13), it is easy to modify the proofof Theorem 1 to prove the geometric convergence of the fixed-point iteration (16)–(17). The complete algorithm for solving(14) is obtained by replacing steps 1–2 of Algorithm 1 with(16)–(17). This algorithm has the advantage that the denomi-nator of (16) can be computed by estimating the covariance ma-trix at the base station in the uplink direction. However, this al-gorithm is limited to the special case when all data streams havethe same priority.

C. Connection With Alternate Optimization

In this section, we provide more insight into Theorem 1 by an-alyzing the relationship between the fixed-point iteration and thealternate optimization algorithm proposed in [16] forfor the special case when . There are two differ-ences between Algorithm 1 and the algorithm in [16]. First, thealgorithm in [16] only treated the specific case of a total powerconstraint while Algorithm 1 handles a general weighted-sumpower constraint. Second, the algorithm in [16] was derivedby applying nonlinear Perron–Frobenius theory to the downlinkpower control problem and then exploiting uplink-downlink du-ality to update the transmit beamformers. On the other hand, Al-gorithm 1 was derived by applying nonlinear Perron–Frobeniustheory to the dual uplink fixed-point equation (13) and then ex-ploiting uplink-downlink duality to update the downlink powerand transmit beamformers. Nevertheless, we show that for thecase when , Theorem 1 can, in fact, be derived fromthe results in [16].

Given the dual uplink power at some time step, considerusing MVDR beamforming to optimize . The uplink SINRof the th stream after MVDR beamforming is given by

(18)

where

CAI et al.: A UNIFIED ANALYSIS OF MAX-MIN WEIGHTED SINR FOR MIMO DOWNLINK SYSTEM 3855

is the SINR of the th stream in the dual uplink network. Using(18), we can decouple the fixed-point iteration in Theorem 1 intothe following steps:

(19)

(20)

(21)

For notational brevity, we omitted the dependence of on.Hence, the fixed-point iteration in Theorem 1 can be viewed

as an alternate optimization of and . Observe that when, (20)–(21) is exactly the max-min weighted SINR

power control algorithm (applied in the uplink) from [16], whichwas derived by applying nonlinear Perron–Frobenius theory tothe power control problem. Furthermore, since (19)–(21) do notdepend on , the power control algorithm from [16] can also beused to update in parallel with (19)–(21). The complete al-gorithm is exactly the algorithm proposed in [16] for the MISOproblem. Hence, the fixed-point iteration in Theorem 1 can alsobe derived from the algorithm proposed in [16] by combiningthe updates for and .

We illustrate the above analysis in Fig. 1(a). Each block inthe left and right portions represent a subproblem of .The left portion shows nonlinear Perron–Frobenius theorybeing applied to the downlink power control problem via thedual uplink power control problem. The right portion showsMVDR beamforming being used to solve the transmit beam-forming problem by using uplink–downlink duality to convertinto a receive beamforming problem. The center portion showsnonlinear Perron–Frobenius theory being applied to the MISOproblem directly via (13). This direct approach gives the samealgorithm as that obtained by studying the two subproblems of

separately and then alternately optimizing and .

D. Characterization via Friedland–Karlin Spectral RadiusMinimax Theorem

In this section, we apply nonnegative matrix theory to gainfurther insight into the solution of . Define the non-negative vector as

Suppose is the unique optimal dual uplink power ofand is the optimal value. Since and satisfy

(11) and (13), it follows that

Hence, at optimality, (normal-ized with respect to ) and .

Applying the Friedland–Karlin spectral radius minimax char-acterization [21, Lemma 2] to ,it can be observed that are dual pairs with re-spect to . Moreover, from (18), it can be seen that

is the optimal SINR of the thstream. Hence, it can be deduced that at optimality, the SINRallocation in is a weighted geometric mean of theindividual SINRs, where the weights are the dual uplink powersnormalized with respect to , that is,

(22)

E. Analysis of SIMO Max-Min Weighted SINR

It is easy to see that all the analysis in Sections III-A–III-Dcan be modified for the SIMO max-min weighted SINR problemgiven by

which is obtained by fixing the transmit beamformers in. For notational brevity, we omitted the dependence of

on . We will briefly state the results for .These results, in particular the fixed-point iteration, are also rel-evant toward understanding Algorithm 2 later in Section IV-A.

Let and be the optimal downlink power and receivebeamformers of , and let be the optimal value.Since the receive beamformers are already decoupled in thedownlink network, we have that and satisfy the fixed-pointequation given by

(23)

where the mappings andare defined by

for , and . From (23),we can obtain a fixed-point iteration (independent of ) whichconverges to . This fixed-point iteration can be decoupled intothe following steps:

(24)

(25)

(26)

3856 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 8, AUGUST 2011

TABLE ITHE FIXED POINT EQUATIONS CHARACTERIZED THROUGH THE PERRON–FROBENIUS THEOREM AND THE FRIEDLAND–KARLIN SPECTRAL

RADIUS MINIMAX THEOREM. EXCEPT FOR THE FIXED QUANTITIES, ALL OTHER QUANTITIES DENOTE OPTIMAL VALUES OF �� �� �

where

(27)

are defined as the normalized MVDR beamformers in thedownlink network. We illustrate in Fig. 1(b) the correspon-dence between the algorithms obtained by applying nonlinearPerron–Frobenius theory to the power control problem and thatobtained by applying it to the SIMO problem directly.

Define the nonnegative vector as follows:

Then, we have that

and are dual pairs with respect to . Moreover,the optimal SINR allocation satisfies

We summarize the connections between the fixed-point equa-tions, the Perron–Frobenius theorem, and the Friedland–Karlinspectral radius minimax characterization in Table I. For com-pleteness, we also list the Friedland–Karlin spectral radius min-imax characterizations for the power control problem, i.e., with

and fixed. Notice that while form a dualpair in the power control problem, and arethe dual pairs in the MISO and SIMO problems, respectively.

IV. MIMO OPTIMIZATION

In this section, we study the MIMO max-min weighted SINRproblem . In general, global optimization of is anopen problem because the nonconvexity and the mutual cou-pling of the transmit and receive beamformers make it difficultto jointly optimize the beamformers. We begin in Section IV-Aby proposing a suboptimal algorithm for based on alter-nately implementing our MISO and SIMO algorithms. In addi-tion, we show that our proposed algorithm is optimal when thechannel matrices are rank-one, or when the network is operatingin the low SNR region. Next, in Section IV-B, we study the con-tinuity of the optimal solutions to in the parameter , andwe propose a heuristic approach for tackling the suboptimalityof Algorithm 2. Finally, in Section IV-C, we briefly describe

how the results in Sections IV-A and IV-B can be applied to thetotal power minimization problem.

A. Alternate Optimization Algorithm

In this section, we propose an alternate optimization al-gorithm for based on our fixed-point iterations for theMISO and SIMO problems in Section III. The main idea isto alternate between fixing and . Recall that, whenis fixed, becomes . Hence, (24)–(26) can beused to update and . Similarly, when is fixed,becomes , so (19)–(21) can be used to update and

. Since the fixed-point iteration for is in the dualuplink power , prior to updating , uplink–downlink dualitymust be used to compute corresponding to the current

, and . Define the diagonal matrices andof normalized downlink and uplink SINR values,

respectively, i.e., and. Uplink-downlink duality theory states

that if the downlink and dual uplink networks have the sameSINR values, i.e., for all, and , then the following relations hold [4], [9]:

(28)

(29)

Since the fixed-point iterations converge with monotonicallyincreasing minimum-weighted SINR, we can alternate betweenone pass of (24)–(26) in the downlink domain, and one passof (19)–(21) in the dual uplink domain, using (28) and (29) totransition between the downlink and dual uplink updates. Werefer to the resultant algorithm as Algorithm 2.

Algorithm 2: MIMO Max-Min Weighted SINR

• Initialize: arbitrary , andfor , such that ,

and for all .1) Update receive beamformers:

2) Compute auxiliary variables:

for .

CAI et al.: A UNIFIED ANALYSIS OF MAX-MIN WEIGHTED SINR FOR MIMO DOWNLINK SYSTEM 3857

3) Update downlink powers:

4) Compute dual uplink powers:

5) Update transmit beamformers:

6) Compute dual auxiliary variables:

for .7) Compute dual uplink powers:

8) Compute downlink powers:

Remark 1: It is also possible to combine the MISO andSIMO iterations into an algorithm which does not requireuplink–downlink transitions. Recall that the fixed-point iter-ation in Theorem 1 converges to the optimal solution of theMISO problem regardless of the initial value of the dual uplinkpower . Suppose steps 5–7 of Algorithm 2 are looped tillconvergence.1 Then, regardless of the value of passed intothe loop by step 4, steps 5–7 will converge to the optimal (forfixed receive beamformers ). Hence, the downlink-to-up-link conversion in step 4 can be omitted. Similarly, by loopingsteps 1–3 till convergence, the uplink-to-downlink conversionin step 8 can be omitted. A slightly different modificationwhich also enables us to drop the uplink–downlink transitionsis to loop steps 2–3 and 6–7 until convergence. This is becausesteps 2–3 and 6–7 ensure that and always convergeto the optimal and respectively for any fixed beamformers

[16].Convergence of Algorithm 2 is guaranteed because each step

in the Algorithm increases the minimum weighted SINR. Thepower constraint and the unity norm constraint on the beam-formers imply the existence of a limiting value for the SINRso the sequence must converge monotonically. In addition, itis easily seen that Algorithm 2 is coordinate-wise optimal in

and . This means that, if Algorithm 2 converges to, then

1In practice, this can be implemented in a two time-scale algorithm by run-ning steps 5–7 on a significantly faster time-scale such that convergence can beassumed.

Unfortunately, the global optimality of Algorithm 2 is notguaranteed and depends on the initial values .However, we can identify two special cases under which Algo-rithm 2 is optimal. We give these results in Proposition 1 andObservation 2. In particular, Observation 2 is significant as itwill form part of the intuition behind a heuristic strategy forinitializing Algorithm 2, which we propose in the next section.

First, consider the case when for all , andsuppose for some , and

, where and . It is well-knownthat the optimal receive beamformer for the th stream is the nor-malized matched filter (scaled by arbitrary complex phases) cor-responding to . Notice that by substituting the decompositionfor into (27) and applying the Sherman–Morrison–Wood-bury formula, (27) can be rewritten as

up to a scaling factor, and hence the MVDR beamformers arealways optimal. It follows that Algorithm 2 is optimal when allthe channels are rank-one. We state this result formally in thefollowing proposition.2

Proposition 1: If for all , then Algorithm 2converges to the optimal solution of .

Second, observe that as , the interference betweendifferent users can be neglected. Hence, the optimization overand can be decoupled into independent single-user MIMObeamforming problems. It follows that for the th stream, theoptimal and the optimal , where andare the left and right singular vectors associated with the largestsingular value of . Next, notice that for small , steps 1 and5 of Algorithm 2 can be approximated as

Hence, if and are not orthogonal to and respec-tively, then as , steps 1 and 5 of Algorithm 2 will convergeto the optimal receive and transmit beamformers respectively of

[22, Ch. 8]. This insight is significant as it forms part ofthe basis for the heuristic algorithm which we will propose inthe following section. Hence, we summarize this analysis in thefollowing remark.

Remark 2: Let and be the left and right singularvectors associated with the largest singular value of . If

and for all , then as , Al-gorithm 2 converges to the optimal solution of . In otherwords, Algorithm 2 is optimal in the asymptotically low SNRregion.

2Notice that although Algorithm 2 updates � repeatedly, every update givesa global optimal� so, in fact,� need only be updated once for a single channelrealization.

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B. Parametric Search Initialization Algorithm

In this section, we propose a method for tackling the sub-optimality of Algorithm 2, which we call parametric searchinitialization. Due to the existence of multiple coordinate-wiseoptimal points, the performance of Algorithm 2 depends onthe initial values . Our proposed strategy runsmultiple instances of Algorithm 2 for different values of . It isbased on the following two ideas:

1) optimality of Algorithm 2 in the asymptotically low SNRregion (cf. Remark 2);

2) parametric continuity of the optimal solutions ofwith respect to (cf. Theorem 2 below).

Theorem 2: Let denote the optimal value of ,and let denote the optimal solution.Then is continuous over andis upper semicontinuous3 over .

Proof: Refer to Appendix B.From Theorem 2, it is reasonable to make the following hy-

pothesis: If and are sufficiently close to each other, thenthe global optimal solution for is a good choice of ini-tial power and beamformers for Algorithm 2 to use in solving

. We can exploit this insight repeatedly to obtain an in-telligent strategy for initializing Algorithm 2. Suppose we havethe optimal solution of for some , then we can obtaingood solutions of for any by using Algorithm 2to solve over fine increments of . At each value for ,Algorithm 2 is initialized with the solution it returned at the pre-vious power constraint. Clearly, this strategy requires an optimalsolution to begin with. Fortunately, Algorithm 2 is optimal in thelow SNR region. We summarize our proposed approach, whichwe refer to as parametric search initialization, as Algorithm 3.

Algorithm 3: Parametric Search Initialization AlgorithmInput:

• Desired power constraint .• Fixed step-size .• Fixed tolerance .• A total power constraint such that the network is

operating in the low SNR region.• Arbitrary , and

for , such that, and for all .

• Iteration index .Iterate:

1. Initialize Algorithm 2 with .Use a total power constraint of and run Algorithm2 till convergence to within a tolerance . Store thesolution in .

2. If , exit the iteration. Otherwise, update thepower constraint as follows:

3. Update . Repeat from step 1.Output: .

At first glance, the above linear search strategy appears toincur significant computational cost. Theoretically, must

3Semicontinuity reduces to the usual notion of continuity if �� �� � has aunique optimal solution.

be infinitesimally small in order for the proposed strategy tosuccessfully exploit the parametric continuity of the solutions.However, empirical tests reveal that does not necessarilyhave to be small in order to obtain performance gains overrandom initialization.

C. Application to Total Power Minimization

In this section, we consider the weighted-sum-power mini-mization problem subject to given SINR constraints:

We will briefly explain how our results on can be extendedto , focusing only on an algorithm for proposed in [9]which we refer to as Algorithm E.

Observe that Algorithm E is largely similarly to Algorithm 2,the only difference being that Algorithm 2 has additional stepsto normalize the power (steps 3 and 7). It is easy to see that theanalysis leading to Proposition 1 and Remark 2 can be modi-fied to show that Algorithm E converges to the optimal solutionof when the channels are rank-one, and also when the net-work is operating in the asymptotically low SNR region, as longas and for all . It is also possible toprove a similar parametric continuity result for in the pa-rameter , analogous to Theorem 2. Therefore, a similar para-metric search strategy can be employed for .

Recall that and are inverse problems so a solu-tion for is also a solution for . Hence, the parametricsearch initialization algorithms can be used to solve using

and vice versa.4 For instance, Algorithm 3 can be termi-nated when the minimum weighted SINR exceeds and thepower constraint at termination will be an approximation of theoptimal value of . A similar analysis can be made the otherway round.

Furthermore, the inverse relationship between andalso implies that a common framework can be used to

compare the performance of Algorithm 2 and Algorithm E.Suppose Algorithm 2 achieves a minimum weighted SINR(which could be suboptimal) for . Then, we can compare

against the weighted-sum power achieved by Algorithm Efor . The better algorithm is the one that yields a smallerweighted sum power. This performance is relevant because ofthe suboptimality of both algorithms. We will apply this frame-work in Section V to compare the performance of Algorithm 2and Algorithm E, where we plot their performance on the samegraph.

V. NUMERICAL EXAMPLES

In this section, we evaluate and compare the performance ofthe various algorithms in a single-cell network with four users

4A bisection search method was proposed to use an algorithm for ����� tosolve �� �� �. However, this method requires the algorithm to return the optimalsolution, and hence it is only applicable to the MISO and SIMO scenarios.

CAI et al.: A UNIFIED ANALYSIS OF MAX-MIN WEIGHTED SINR FOR MIMO DOWNLINK SYSTEM 3859

Fig. 3. Performance comparison of max-min weighted SINR and min poweralgorithms.

at km from the base station, which trans-mits one independent stream to each user. The performance isevaluated in terms of the minimum weighted SINR and the totalpower cost. It will be assumed for simplicity that every datastream has the same priority , all users suffer from thesame noise variance , and the network is subject to atotal power constraint . In all simulations, the base sta-tion is equipped with antennas, and each user is equippedwith antennas. The noise power spectral density is set to

162 dBm/Hz. Each user communicates with the base stationover independent MIMO Rayleigh fading channels, with a pathloss of dB, where is the distancein kilometers. We assume an antenna gain of 15 dB.

A. Performance

We plot the performance of Algorithm 2 and Algorithm E(with random initialization and parametric search initializa-tion) on the same graph in Fig. 3. For the max-min weightedSINR problem, the horizontal axis represents the sum powerconstraint while the vertical axis represents the performancein terms of minimum SINR. For the total power minimizationproblem, the vertical axis represents the SINR thresholdwhile the horizontal axis represents the performance in termsof total transmit power. In the parametric search initializationalgorithm, the steps are chosen by dividing the range of and

into 20 equally spaced values on the dB scale.As stated in Remark 2, both Algorithm 2 and Algorithm

E attain optimal performance for small or small . Asor increases, the performance of the algorithms deteriorate,with Algorithm 2 performing slightly better than Algorithm E.Hence, optimizing the cellular network according to a fixed de-sired total power provides approximately the same performanceas optimizing according to a fixed desired SINR threshold.In the low to medium range of or , both algorithms withrandom initialization or parametric search initialization achievethe same performance. However, as or increases further,parametric search initialization performs significantly betterthan random initialization. Moreover, the performance achieved

Fig. 4. Convergence plot of max-min weighted SINR algorithm for �� �

2.6367 W.

by the parametric initialization is only about 2–3 dB away fromthe interference-free bound (which is obtained by ignoringthe interference between users while computing the SINR),demonstrating that parametric search initialization is effectiveat searching for good initial power and beamformers for Al-gorithm 2 and Algorithm E. There is no visible difference inperformance between the parametric search over (using Al-gorithm 2) or the parametric search over (using Algorithm E).Similar results are obtained for a network in which the basestation serves two users and transmits two independent streamsto each user.

B. Convergence

We assume the same network configuration as in Fig. 3 andstudy the convergence of Algorithm 2 and Algorithm E withrandom initialization. Fig. 4 shows the average minimum SINRof Algorithm 2 as a function of iteration index for2.6367 W while Fig. 5 shows the average total transmit powerof Algorithm E as a function of iteration index for an SINRthreshold of 24.2105 dB. Both algorithms converge fairlyquickly. However, Algorithm E has a dominant spike in thetotal transmit power during the initial iterations. Further empir-ical tests attribute the spike to poor initial beamformers, whichforces Algorithm E to increase the transmit power significantlyin order to satisfy the SINR thresholds. Subsequently, when theSINR thresholds are satisfied, the total transmit power decreasesmonotonically (as proven in [9]). In contrast, Algorithm 2 main-tains the power constraint at every iteration and converges witha monotonically increasing minimum weighted SINR.

C. Varying Step Size on Parametric Search Initialization

We assume the same network configuration as in Fig. 3 andinvestigate the effects of varying the step size on the parametricsearch initialization. For Algorithm 2, we divide the range ofbetween and W into 2, 4, or 6 equally-spaced values(on the dB scale). For Algorithm E, we divide the range ofbetween 30 and 40 dB into 2, 4, or 6 equally spaced values

3860 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 8, AUGUST 2011

Fig. 5. Convergence plot of total power minimization algorithm for �� �

24.2105 dB.

Fig. 6. Performance of parametric search initialization algorithm for variousstep sizes.

(on the decibel scale). The results are given in Fig. 6. The per-formance of Algorithm 2 and Algorithm E are similar and thereis no visible increase in performance beyond using 4 steps. The2-step procedure deserves further comment. Notice that in the2-step procedure, the initial beamformers in the second step arethe optimal beamformers in the first step, which are approx-imately the singular vectors corresponding to the largest sin-gular values of the channels. Thus, although singular-vector-de-composition (SVD) initialization does not perform better thana parametric search over steps, it still performs signifi-cantly better than random initialization (cf. Fig. 3). Hence, al-though parametric search initialization, in theory, requires an ex-tremely small step-size to search for good initializations, Fig. 6shows that this might not be necessary in practice. It is possibleto achieve significant gains in performance even with a largestep-size.

VI. CONCLUSION

We began by analyzing the MISO and SIMO max-minweighted SINR problems using nonlinear Perron–Frobeniustheory. Nonlinear Perron–Frobenius theory was key in thederivation of the DPC-like max-min weighted SINR powercontrol algorithm in [16]. We extended the application ofnonlinear Perron–Frobenius theory to the MISO and SIMOproblems, and we showed that our result can be interpretedas an alternate optimization of power and beamformers. As aby-product, we resolved the open problem of convergence forthe previously proposed MISO algorithm in [1]. In summary,we demonstrated that nonlinear Perron–Frobenius theory pro-vides an analytical framework for the max-min weighted SINRproblem, analogous to the relationship of the standard inter-ference function framework with the total power minimizationproblem. Recently, the nonlinear Perron–Frobenius theory hasalso been used to analyze other power control problems relatedto the max–min weighted SINR problem [23].

The nonconvexity of the MIMO max-min weighted SINRproblem and the total power minimization problem, and themutual coupling between the transmit and receive beam-formers, imply that both problems are, in general, difficult tosolve. Hence, most previous work are based on the approachof alternate optimization. In general, these approaches are sub-optimal as they fail to tackle the nonconvexity of the problem.Nevertheless, we believe that an in-depth understanding of theoptimality characteristics of the problems and their algorithmscan provide insights into possible strategies for obtaining bettersolutions to the nonconvex problems.

Hence, in our study of the MIMO max-min weighted SINRproblem, we conducted an in-depth analysis of the optimalitycharacteristics. Specifically, we identified two scenarios underwhich this nonconvex problem can be optimally solved: (i) whenthe channels are rank-one, and (ii) when the network is oper-ating in the low SNR region. We also proposed an algorithmwhich converges to the optimal solution under both scenarios.To improve the performance of our (generally) suboptimal al-gorithm, we proposed a method for initializing our algorithm,which is motivated by the parametric continuity of the problemin the power constraint. The proposed parametric search initial-ization algorithm exhibits better performance than random ini-tialization, and can be easily extended to the MIMO total powerminimization problem.

APPENDIX

A. Proof of Theorem 1

Proof: First, we show that is a concave mapping. Itwas shown in [1] that for a positive semidefinite matrix anda vector in the range of , we have

where the matrix inverse denotes the Moore–Penrose pseudoin-verse. Recall that is invertible so it has full rank. Hence,

CAI et al.: A UNIFIED ANALYSIS OF MAX-MIN WEIGHTED SINR FOR MIMO DOWNLINK SYSTEM 3861

is in the range of . Choosing and, we have that

Since is an affine mapping of , and the point-wiseminimum of affine mappings is concave, we have thatis a concave mapping of . Since (because )and , we have that for all . The restof the proof follows from applying Theorem 1 in [24]. That thefixed-point iteration converges geometrically fast follows fromthe remark after Theorem 1 in [24].

B. Proof of Theorem 2

Proof: First, we need the following result from [25]:Theorem 3: Let be a continuous function and

be a compact-valued, continuous correspon-dence, where denotes the power set of . Letand be defined by

(30)

(31)

Then is a continuous function on , and is a compact-valued, upper-semi-continuous correspondence on .

We apply Theorem 3 to prove our desired result. Let, and

, i.e., is the Cartesian product of with com-plex spaces of dimension and complex spaces ofdimension . Define andas follows:

Replacing in Theorem 3 with , we see that (30) and (31)give respectively the optimal value and solution of . Tosatisfy the requirement of being compact, we relaxed theoriginal constraint into . This relaxation is validbecause we always have at optimality of (otherwisesome stream will have zero SINR).

It is easy to see that is continuous. Hence, tocomplete the proof, it suffices to show that is continuous. Acorrespondence is said to be continuous at

if is both upper-semicontinuous (u.s.c) and lower-semicontinuous (l.s.c) at [25]. First, we prove that is u.s.c.at any . Define the correspondences

and as follows:

Hence, can be rewritten as the Cartesian product .Notice that is a constant mapping so is both u.s.c. andl.s.c. Since the Cartesian product of a finite number of u.s.c.mappings is also u.s.c., it remains to show that is u.s.c. [26,Ch. 6]. Suppose is an open set such that . Since

is compact, there exist some such that. Choose such that .

Then for all , we have that .Hence, is u.s.c.

Next, we prove that is l.s.c. at any . The cartesianproduct of a finite number of l.s.c. mappings is also l.s.c. so itremains to show that is l.s.c. Suppose is an open set suchthat . Then, contains some open subset of sothere exist some , where , such that

. Choose such that .Then for all , we have that

so . Hence,is l.s.c.

ACKNOWLEDGMENT

The authors would like to thank Prof. Steven Low of Caltechfor helpful discussions, and the anonymous reviewers for theirhelpful and insightful comments.

REFERENCES

[1] A. Wiesel, Y. C. Eldar, and S. Shamai, “Linear precoding via conicoptimization for fixed MIMO receivers,” IEEE Trans. Signal Process.,vol. 54, no. 1, pp. 161–176, Jan. 2006.

[2] G. J. Foschini and Z. Miljanic, “A simple distributed autonomouspower control algorithm and its convergence,” IEEE Trans. Veh.Technol., vol. 42, no. 4, pp. 641–646, Nov. 1993.

[3] R. D. Yates, “A framework for uplink power control in cellular radiosystems,” IEEE J. Sel. Areas Commun., vol. 13, no. 7, pp. 1341–1347,Sep. 1995.

[4] P. Viswanath and D. N. C. Tse, “Sum capacity of the vector Gaussianbroadcast channel and uplink-downlink duality,” IEEE Trans. Inf.Theory, vol. 49, no. 8, pp. 1912–1921, Aug. 2003.

[5] F. Rashid-Farrokhi, K. J. R. Liu, and L. Tassiulas, “Transmit beam-forming and power control for cellular wireless systems,” IEEE J. Sel.Areas Commun., vol. 16, no. 8, pp. 1437–1450, Oct. 1998.

[6] F. Rashid-Farrokhi, L. Tassiulas, and K. J. R. Liu, “Joint optimal powercontrol and beamforming in wireless networks using antenna arrays,”IEEE Trans. Commun., vol. 46, no. 10, pp. 1313–1324, Oct. 1998.

[7] E. Visotsky and U. Madhow, “Optimum beamforming using transmitantenna arrays,” in Proc. IEEE Veh. Technol. Conf., Jul. 1999, vol. 1,pp. 851–856.

[8] J.-H. Chang, L. Tassiulas, and F. Rashid-Farrokhi, “Joint transmitterreceiver diversity for efficient space division multiaccess,” IEEE Trans.Wireless Commun., vol. 1, no. 1, pp. 16–27, Jan. 2002.

[9] B. Song, R. L. Cruz, and B. D. Rao, “Network duality for mul-tiuser MIMO beamforming networks and applications,” IEEE Trans.Commun., vol. 55, no. 3, pp. 618–630, Mar. 2007.

[10] W. Yang and G. Xu, “Optimal downlink power assignment for smartantenna systems,” in Proc. IEEE Int. Conf. Acoust., Speech, SignalProcess., May 1998, vol. 6, pp. 3337–3340.

[11] M. Schubert and H. Boche, “Solution of the multiuser downlink beam-forming problem with individual SINR constraints,” IEEE Trans. Veh.Technol., vol. 53, no. 1, pp. 18–28, Jan. 2004.

[12] D. Wajcer, S. Shamai, and A. Wiesel, “On superposition coding andbeamforming for the multi-antenna Gaussian broadcast channel,” pre-sented at the IEEE Inf. Theory Appl. Workshop, San Diego, CA, Feb.2006.

[13] P. Wrycza, M. Bengtsson, and B. Ottersten, “On convergence proper-ties of joint optimal power control and transmit-receive beamforming inmulti-user MIMO systems,” in Proc. IEEE Workshop Signal Process.Adv. Wireless Commun., Jul. 2006, pp. 1–5.

[14] S. Shi, M. Schubert, and H. Boche, “Downlink MMSE transceiveroptimization for multiuser MIMO systems: MMSE balancing,” IEEETrans. Signal Process., vol. 56, no. 8, pp. 3702–3712, Aug. 2008.

3862 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 8, AUGUST 2011

[15] H. T. Do and S.-Y. Chung, “Linear beamforming and superpositioncoding with common information for the Gaussian MIMO broadcastchannel,” IEEE Trans. Commun., vol. 57, no. 8, pp. 2484–2494, Aug.2009.

[16] C. W. Tan, M. Chiang, and R. Srikant, “Maximizing sum rate and min-imizing MSE on multiuser downlink: Optimality, fast algorithms andequivalence via max-min SIR,” in Proc. IEEE Int. Symp. Inf. Theory,Jun. 2009, pp. 2669–2673.

[17] C. W. Tan, M. Chiang, and R. Srikant, “Fast algorithms and perfor-mance bounds for sum rate maximization in wireless networks,” inProc. IEEE INFOCOM Conf., Rio de Janeiro, Brazil, Apr. 2009, pp.1350–1358.

[18] C. W. Tan, M. Chiang, and R. Srikant, “Optimal power controland beamforming in multiuser downlink systems,” presented at theAsilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Nov.2009.

[19] D. W. H. Cai, T. Q. S. Quek, and C. W. Tan, “Coordinated max-minweighted SIR optimization in multicell downlink—Duality and algo-rithm,” in IEEE Int. Conf. Comm., 2011, to be published.

[20] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.Cambridge, U.K.: Cambridge Univ. Press, Jun. 2005.

[21] S. Friedland and S. Karlin, “Some inequalities for the spectral radiusof non-negative matrices and applications,” Duke Math. J., vol. 42, no.3, pp. 459–490, Sep. 1975.

[22] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore,MD: The Johns Hopkins Univ. Press, Oct. 1996.

[23] C. W. Tan, “Optimal power control in Rayleigh-Fading heterogeneousnetworks,” presented at the IEEE INFOCOM Conf., Shanghai, China,May 2011.

[24] U. Krause, “Concave Perron–Frobenius theory and applications,” Non-linear Anal., vol. 47, pp. 1457–1466, Aug. 2001.

[25] R. K. Sundaram, A First Course in Optimization Theory. Cambridge,U.K.: Cambridge Univ. Press, Jun. 1996.

[26] C. Berge, Topological Spaces: Including a Treatment of Multi-ValuedFunctions, Vector Spaces and Convexity. Cambridge, U.K.: Cam-bridge Univ. Press, Sep. 2010.

Desmond W. H. Cai (S’10) received the B.Sc.degree in electrical and computer engineering (withhighest distinction) from Cornell University, Ithaca,NY, in 2009. He is currently working towardsthe Ph.D. degree in electrical engineering at theCalifornia Institute of Technology, Pasadena.

From July 2009 to August 2010, he was a Re-search Officer with the Institute for InfocommResearch, Singapore. His current research interestsinclude smart grids, communications, optimization,and game theory.

Mr. Cai received the National Science Scholarship from the Agency for Sci-ence, Technology, and Research in 2004. He was awarded the John G. PertschPrize in 2008 and the Sibley Prize in 2009 by Cornell University.

Tony Q. S. Quek (S’98–M’08) received the B.E. andM.E. degrees in electrical and electronics engineeringfrom Tokyo Institute of Technology, Tokyo, Japan, in1998 and 2000, respectively, and the Ph.D. degree inelectrical engineering and computer science from theMassachusetts Institute of Technology (MIT), Cam-bridge, in 2008.

Since 2008, he has been with the Institute for In-focomm Research, A*STAR, where he is currently aPrincipal Investigator and Scientist. He is also an Ad-junct Assistant Professor with the Division of Com-

munication Engineering, Nanyang Technological University. His main researchinterests are the application of mathematical, optimization, and statistical theo-ries to communication, detection, signal processing, information theoretic andresource allocation problems. Specific current research topics include cooper-ative networks, interference networks, heterogeneous networks, green commu-nications, wireless security, smart grid, and cognitive radio.

Dr. Quek has been actively involved in organizing and chairing sessions andhas served as a member of the Technical Program Committee (TPC) in a numberof international conferences. He served as the Technical Program Chair for theServices & Applications Track for the IEEE Wireless Communications andNetworking Conference (WCNC) in 2009, the Cognitive Radio & CooperativeCommunications Track for the IEEE Vehicular Technology Conference (VTC)in Spring 2011, and the Wireless Communications Symposium for the IEEEGlobecom in 2011; as Technical Program Vice-Chair for the IEEE Conferenceon Ultra Wideband in 2011; and as the Workshop Chair for the IEEE Globecom2010 Workshop on Femtocell Networks and the IEEE ICC 2011 Workshop onHeterogeneous Networks. He is currently an Editor for the Wiley Journal onSecurity and Communication Networks. He was Guest Editor for the Journalof Communications and Networks (Special Issue on Heterogeneous Networks)in 2011. He received the Singapore Government Scholarship in 1993, TokyuFoundation Fellowship in 1998, and the A*STAR National Science Scholar-ship in 2002. He was honored with the 2008 Philip Yeo Prize for OutstandingAchievement in Research, the IEEE Globecom 2010 Best Paper Award, and the2011 JSPS Invited Fellow for Research in Japan.

Chee Wei Tan (M’08) received the M.A. and Ph.D.degree in electrical engineering from Princeton Uni-versity, Princeton, NJ, in 2006 and 2008, respectively.

He is an Assistant Professor at City UniversityHong Kong. Previously, he was a PostdoctoralScholar at the California Institute of Technology(Caltech), Pasadena, CA. His research interests arein wireless and broadband communications, signalprocessing and nonlinear optimization.

Dr. Tan was the recipient of the 2008 PrincetonUniversity’s Gordon Wu Prize for Excellence.