1746 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, …

1746 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 4, APRIL 2011

Multiuser MIMO Downlink Beamforming DesignBased on Group Maximum SINR Filtering

Yu-Han Yang, Student Member, IEEE, Shih-Chun Lin, Member, IEEE, and Hsuan-Jung Su, Member, IEEE

Abstract—In this paper, we aim to solve the multiusermulti-input multi-output (MIMO) downlink beamformingproblem where one multi-antenna base station broadcasts data tomany users. Each user is assigned multiple data streams and hasmultiple antennas at its receiver. Efficient solutions to the jointtransmit-receive beamforming and power allocation problembased on iterative methods are proposed. We adopt the groupmaximum signal-to-interference-plus-noise-ratio (SINR) filterbank (GSINR-FB) as our beamformer which exploits receiverdiversity through cooperation between the data streams of a user.The data streams for each user are subject to an average SINRconstraint, which has many important applications in wirelesscommunication systems and serves as a good metric to measurethe quality of service (QoS). The GSINR-FB also optimizes theaverage SINR of its output. Based on the GSINR-FB beamformer,we find an SINR balancing structure for optimal power alloca-tion which simplifies the complicated power allocation problemto a linear one. Simulation results verify the superiority of theproposed algorithms over previous works with approximately thesame complexity.

Index Terms—Beamforming, broadcast channel, iterative algo-rithms, linear precoding, MIMO, multiuser.

I. INTRODUCTION

I N this paper, the joint beamforming and power alloca-tion optimization problem for the multiuser multi-input

multi-output (MIMO) downlink channel is considered. In thissystem, transmit and receive beamformings are used to sup-press the multiuser interference and exploit the multi-antenna

Manuscriptreceived November 07, 2009; revised March 15, 2010, November24, 2010; accepted November 29, 2010. Date of publication December 13, 2010;date of current version March 09, 2011. The associate editor coordinating thereview of this manuscript and approving it for publication was Dr. Geert Leus.This work was supported by the National Science Council, Taiwan, R.O.C.,under Grants NSC 97-2219-E-002-012, 95-2219-E-002-005, and the IndustrialTechnology Research Institute (ITRI) under Grant 97-EC-17-A-99-R6-0626.The material in this paper was presented in part at IEEE International Confer-ence on Communications (ICC), Dresden, Germany, 2009, and the InternationalConference on Communications (ICC), Beijing, China, 2008.

Y.-H. Yang was with the Department of Electrical Engineering and Grad-uate Institute of Communication Engineering, National Taiwan University. Heis now with the Department of Electrical and Computer Engineering, Universityof Maryland, College Park USA (e-mail: [email protected]).

S.-C. Lin is with the Institute of Communications Engineering, Na-tional Tsing Hua University, Hsinchu, Taiwan, 30013 R.O.C. (e-mail:[email protected]).

H.-J. Su is with the Department of Electrical Engineering and GraduateInstitute of Communication Engineering, National Taiwan University, Taipei,Taiwan, 10617 R.O.C. (e-mail: [email protected]).

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.2010.2099221

diversity. Power allocation at the transmitter is performed toefficiently utilize the available transmission power. Such a jointbeamforming and power allocation problem has been studiedby many researchers [1]–[5]. In [2] and [3], block diagonal-ization (BD) was proposed to block-diagonalize the overallchannel so that the multiuser interference at each receiver isthoroughly eliminated. Such a zero-forcing approach suffersfrom the noise enhancement problem, because it removesthe multiuser interference by ignoring the noise. Hence, theperformance can be improved if the balance between mul-tiuser interference suppression and noise enhancement can befound [4], [5].

Under individual signal-to-interference-plus-noise-ratio(SINR) constraints for users, Schubert and Boche studied thesituation where each user has only one data stream and singlereceive antenna [4]. It was shown that the optimal solution canbe efficiently found by iterative algorithms. Khachan et al. [5]generalized the scheme in [4] to allow several transmissionbeams to be grouped to serve a user, and each user has multiplereceiver antennas [5]. However, each data stream is processedseparately. Thus, in addition to the multiuser interference fromthe other users, there is intra-group interference between thedata streams of a user. This drawback motivates our workto use a more sophisticated receiver processing to tackle theintra-group interference.

In this work, we adopt the group maximum SINR filterbank (GSINR-FB) proposed by [6] as the beamformer, whichcollects the desired signal energy in the streams of each user andmaximize the total SINR at its output. That is, the GSINR-FBlets these streams cooperate while the filters in [5] let themcompete. Based on the GSINR-FB beamformer, we consider asystem which uses the average SINR over data streams for a useras a metric to measure the quality-of-service (QoS). This cri-terion is very useful in many communication scenarios [6]–[8]including the celebrated space-time block coded systems. It willbe shown that the GSINR-FB based beamformer does improvethe performance over the scheme in [5]. Moreover, we findthat the SINR balancing structure exists for this beamformingmethod, that is, the optimal power allocation results in the sameSINR to target ratio for all users with the GSINR-FB basedbeamforming. As will be shown later, this property makessolving the complicated power allocation problem much easier.Our work can be seen as a nontrivial generalization of [4] to themulti-antenna setting which also subsumes [5] as a special case(with independent processing of data streams). For simplicity,we will first consider group power allocation which restrictsequal power on the data streams of each user to benefits fromthe low-complexity power allocation schemes similar to those

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YANG et al.: MULTIUSER MIMO DOWNLINK BEAMFORMING DESIGN BASED ON GSINR FILTERING 1747

in [4] and [5]. This restriction is later relaxed by allowing thepower of individual data streams to be adjustable. Besidesthe GSINR-FB based beamforming, this per stream powerallocation scheme is new compared with [4] and [5] and hasbetter performance than the group power allocation. These twotechniques are the key ingredients to make our performancebetter than that in [5]. With approximately the same complexityas [5], our approach exhibits a better performance compared tothe existing methods in [5] and the BD based methods.

We will investigate two optimization problems. One is mini-mizing the total transmitted power while satisfying a set of av-erage SINR targets. The other is maximizing the achieved av-erage SINR to target ratio under a total power constraint. Basedon the uplink–downlink duality [9], our methods iteratively cal-culate the GSINR-FB based beamforming and power alloca-tion matrices. The rest of the paper is organized as follows.The system model and problem formulation are introduced inSection II. We also briefly discuss the basic design concept ofour iterative algorithms in this section. Backgrounds such asthe GSINR-FB based beamformers and the applications of theaverage SINR criterion are provided in Section III. Section IVpresents our power allocation results. The numerical results aregiven in Section V, and the computational complexity issuesare discussed in Section VI. Finally, we give the conclusion inSection VII.

II. PROBLEM FORMULATION AND EFFICIENT

ITERATIVE SOLUTIONS

A. Notations

In this paper, vectors and matrices are denoted in bold-facelower and upper cases, respectively. For vector , meansthat every element of is nonnegative. For matrix ,denotes the trace; and denote the transpose and Hermi-tian operations, respectively. denotes the Frobenius norm,

which is defined as . andare, respectively, the inverse and determinant of a square matrix

. And denotes the identity matrix of dimension . A di-agonal matrix is denoted whose th parameter is the

th diagonal term in the matrix. denotes the expectation op-erator.

B. System Model

Consider the downlink scenario with users, where a basestation is equipped with antennas. The upper part of Fig. 1shows the overall system block diagram for user , who hasreceive antennas and receives data streams, where satis-fies the constraint to make sure effective re-covery of the data streams at the receiver. Thus, the users havea total of receive antennas receiving a total of

grouped data streams. For a given symbol time,the data streams intended for user are denoted by a vector ofsymbols . The data streams areconcatenated in a vector . Without loss ofgenerality, we assume that is zero mean with covariance ma-trix . The precoder processes user ’s datastreams before they are transmitted over the antennas. Theseindividual precoders together form the global transmitter

Fig. 1. MIMO downlink system model for user � and its virtual uplink.

beamforming matrix . The power allo-cation matrix for user is a diagonal matrix

(1)

where is the power allocated to the th data stream of user, and the global power allocation matrix

(2)

is a block diagonal matrix of dimension . The transmitterbroadcasts signals to all of the users.

User receives a length vector , whichcan be expanded as

(3)

where the channel between the transmitter and user is repre-sented by the matrix , the Hermitian of ; rep-resents the zero-mean additive white Gaussian noise (AWGN)at user ’s receive antennas with variance per antenna and thecovariance matrix . The resultingglobal channel matrix is , with .We assume that the transmitter has perfect knowledge of thechannel matrix , and receiver knows its perfectly. Thesecond term on the right-hand-side of (3) is the inter-group mul-tiple user interference for user . To estimate its symbols ,user processes with its receive beamforming ma-trix . The resulting estimated signal vector is

(4)

Without loss of generality, as [6], we assume that the interfer-ence-plus-noise components of the filter bank output in (4) areuncorrelated. For any filter bank that produces correlated com-ponents, one can easily find another filter bank which makes


these component uncorrelated but with the same performance.The details can be found in [6].

Finally, owing to the noncooperative nature between usersin broadcast channels, the global receiver beamformingfilter , formed by collecting the individual receiver fil-ters, is a block diagonal matrix of dimension where

.

C. Problem Formulation

In this paper, we consider the average SINR of user overall its data streams SINR SINR as the per-formance measure, where SINR is the SINR of the th datastream of user . The importance and applications of this designcriterion will be reviewed in detail later in Section III-B. Basedon the average SINR constraints and system model describedin Section II-B, we consider two problems as follows. The firstoptimization problem, which will be referred to as Problem Prin the subsequent sections, follows.

Problem Pr: Given a total power constraint and theSINR target for user , maximize SINR over all

beamformers , , and power allocation matrix , that is,

SINRsubject to (5)

We call SINR the SINR to target ratio for user .If the minimum SINR to target ratio in (5) can be made greater

than or equal to one, then the second optimization problem is tofind the minimum power required such that the SINR targets canbe all satisfied. The mathematical formulation of this problem,which will be referred to as Problem Pp in the subsequent sec-tions, follows.

Problem Pp: Given a constraint on the minimum SINR totarget ratio, minimize the total transmitted power over all beam-formers , , and power allocation matrix as

subject toSINR

and (6)

D. Iterative Methods Based on Uplink–Downlink Duality

We briefly review the uplink–downlink duality, which playsan important role in finding efficient solutions based on itera-tive methods for our problems. In [9]–[12], it was shown that itis always possible to find a virtual uplink system for the down-link system. We plot the virtual uplink for user in the lowerpart of Fig. 1, where is the corresponding power alloca-tion matrix in the virtual uplink defined similarly as . To bemore specific, with fixed beamforming filters , , SINR tar-gets , and the same sum power constraint forboth the downlink and the virtual uplink, the downlink and its

TABLE IBASIC STEPS OF THE �TH ITERATION

virtual uplink system have the same SINR to target ratio withoptimal and .

With the aids of the uplink–downlink duality, the optimiza-tion problems Pr and Pp in Section II-C can be solved effi-ciently with iterative algorithms. Now we introduce the basicconcepts of these algorithms, as summarized in Table I. For sim-plicity, we use Problem Pr as an example. From Table I, for itera-tion , with the downlink transmitter and receiver beamformers

and fixed, we can obtain a new power allocationmatrix to increase the minimum SINR to target ratio

SINR . Note that the downlink power allocation are ex-

ecuted two times (Step 1 and 3) for the th iteration, as shownin Table I. To simplify notations in the following sections, weuse and to represent the new power allocationmatrices for the first and second downlink power allocations re-spectively. With fixed and , we can obtain a newdownlink receiver beamformer to increase SINRfor all users. The minimum ratio SINR is further opti-

mized using the new power allocation matrix computedfrom and . Then we turn to the virtual uplink toupdate . Similarly, fixing uplink transmitter beamformer

and receiver beamformer , we obtain a new up-link power allocation matrix . After power allocation,the SINR to target ratios of the downlink and virtual uplink areequal. Then we can find based on and .After that, is updated according to the new and

, and so on.Note that all the iterations are done at the transmitter, and

the transmitter does not need to feed forward the optimized re-ceive filters to the receivers during the iterations. The receivercan compute the final filter by itself after the iterative algorithmstops. This procedure is the same as [13, Sec. II-B], and webriefly describe it here. First, as in the “common training” phasein [13] and [14], each receiver can estimate its own channel

by using the known training sequence. After receiver feedsback to the transmitter, the transmitter can iteratively com-pute transmit and receive beamforming filters, as well as powerallocation matrices in Table I according to . After the itera-tive algorithm stops, the “dedicated training” phase as in [13] isperformed to let the receivers compute the final receiver filter.In this phase, the transmitter will broadcast orthogonal training


sequences to the receivers as in [13], and each receiver can es-timate the final equivalent channel formed by , the transmitfilters, and power allocation matrices to calculate its final re-ceive beamformer. We will first show how to calculate the beam-forming filters in the next section, and then show how to usethese filters to determine power allocation in Sections IV-B andIV-C.

III. GROUP MAXIMUM SINR FILTER BANK FOR THE

AVERAGE SINR CONSTRAINT

In this section, we introduce the key motivation of our paper,that is, the use of GSINR-FB in [6] as the beamfomer to solve(5) and (6). This filter bank is a nontrivial generalization of theone used in [5]. It uses the dimensions provided by the multiplereceive antennas at each user more efficiently than [5]. Specif-ically, the streams of each user (or group) cooperate with oneanother in our scheme, rather than interfere with one another asin [5]. Since this filter bank maximizes the total SINR of thestreams of each user, it also maximizes the average SINR crite-rion adopted in this paper. We will also review the applicationsof the average SINR criterion at the end of this section.

A. Group Maximum SINR Filter Bank

To solve (5) and (6), the GSINR-FB is adopted for our trans-mitter beamformer and receiver beamformer to maximizethe average SINR. Moreover, as will be shown in Proposition 1,the optimal SINR balancing structure based on the GSINR-FBbeamforming will make the corresponding power allocationproblem trackable. Let us first focus on Step 2 in Table I, thatis, given and , finding filter to maximize

SINR , ( times of the average SINR), whereSINR is the SINR of the th stream of user in this step.For brevity, we shall omit the iteration index in most of thefollowing equations. Following [6], the optimization problembecomes

subject to

(7)where , while

and

(8)

are the signal covariance matrix and the interference-plus-noisecovariance matrix for user , respectively. It is now evident thatwe must let since the number of eigen-vectors is limited by the dimension of . The optimizationproblem in (7) was shown to be equivalent to solving the gener-alized eigenvalue problems [6] as

(9)

with

SINR (10)

Then can be computed easily. The receive beamformingfilter designed for the downlink can be carried over to thetransmit beamforming filter for uplink, and vice versa. Thus,the receive beamforming filter for the virtual uplinksystem in Step 4 in Table I can be computed similarly.

Now we show why the GSNIR-FB performs better than thosein [4] and [5]. In [5], all streams interfere with one another and

satisfies

where is the maximum generalized eigenvalue of;

and

(11)

are the signal covariance matrix and the interference-plus-noisecovariance matrix for stream of user , respectively, and

. Comparing (11) with (8), one caneasily see that, in [5], the streams of the same user interferewith one another and there is additional intra-group interferencein (the first term of ) compared with in (8).The GSINR-FB beamforming exploits additional dimensionsfrom the multiple receiver antennas, which are not providedin [4] (where ), much more efficiently, by letting thestreams of each user cooperate rather than compete as in [5].

B. Average SINR Criterion and Its Applications

The average SINR criterion SINR is very useful in manycommunication systems [6]–[8] and can serve as a good metricfor the QoS. Here we briefly review some of its applications.Note that in these applications, it is the total SINR SINRwhich serves as the performance metric, which equals totimes the average SINR. However, as will be discussed inSection V, to have a fair comparison with the results in [5]where the per stream SINR is considered, the average SINR isused in the comparison.

Approximation of Maximum Achievable Rate at Low SINR[7]: The maximum achievable rate for user is

SINR SINR

SINR (12)

where is the SNR gap to capacity [15, p. 432], [16, Ch. 7] dueto suboptimal channel coding schemes and the limitation of cir-cuit implementation in practical systems. According to [15, p.432], the gap is huge (8.8 dB) for uncoded PAM or QAM oper-ating at bit error rate. This approximation is also useful insystems with large numbers of users where the total interferencepower in (4) is large.

Receiver SINR [6], [8]: Assuming that the maximum ratiocombining (MRC) is applied to in (4), the receiver SINR


at the output of the MRC is the sum of individual SINRs asSINR . This metric is very useful when space-time coding is

applied and contains the space-time coded symbols. In thiscase, the decoding is based on the MRC results [6].

Minimization of the Pairwise Error Probability [7]: When aspace-time block code (STBC) is applied and contains theSTBC symbols. Assuming that the channel is slow fading andremains constant during the transmission of a codeword, andthat the maximum-likelihood detector is used at the receiver, onecan approximately transform the minimization of the pairwisecodeword error probability to the maximization of SINRfollowing the steps in [7]. This approximation applies to boththe orthogonal and quasi-orthogonal STBCs.

IV. POWER ALLOCATION

Now we focus on the optimal power allocation strategy forthe Step 3 in Table I, where the maximum SINR beamformingfilter banks , and a set of SINR targetsare given. The optimization problem corresponding to ProblemPr (5) is

SINRsubject to (13)

The other one corresponding to Problem Pp (6) which min-imizes the total transmitted power, such that each individualSINR target can be achieved, is

subject toSINR

and

(14)

We will first explore the structure of the optimal solutionsfor these problems in Section IV-A. However, even with thisstructure which significantly simplifies the problems, the twoper-steam power allocation problems are very complicated andthe solutions in [4] and [5] do not apply. Thus, we first in-tensionally introduce some restrictions to the power allocationstrategies to simplify the problems and benefit from the simplepower allocation schemes similar to those in [4] and [5]. InSection V, the simulation results show that even without the newper stream power allocation, the performance of [4] and [5] canbe enhanced by simply applying the GSINR-FB as the beam-formers. This verifies our motivation to use the GSINR-FB. Theresults for the simple “grouped” power allocation are presentedin Section IV-B. We then remove the restrictions and presentthe general per-stream power allocation results in Section IV-C.The insights to why the proposed algorithms perform better thanthose in [4] and [5] are given in Section IV-D.

A. Optimal SINR Balancing Structure UnderGSINR-FB Beamforming

By carefully rearranging the complicated SINR to a sim-pler equivalent form and using the properties of the GSINR-FB,we prove the following structure for the optimal power allo-cation which makes solving the complicated power allocationproblems (13), (14) possible.

Proposition 1: For the optimization problem (13), the op-timal solution makes all users achieve the same SINR to targetratio, that is, SINR , for all . Here is theSINR balanced level.

Proof: The vector norms of the beamforming filters ,, can be adjusted such that:

1) is a scaled identity matrix [6];2) .

When the above two conditions are satisfied, the average SINRof user in the downlink scenario can be expressed as

SINR SINR (15)

Expanding and ,

SINR

(16)Since [17], the terms canbe written as

(17)

where and denotes theth diagonal element of . Therefore, the average SINR of

user is

SINR (18)

Observing (18), we know that the maximizer of the optimiza-tion problem (13) satisfies

SINR(19)

The reason is as the following. Since , , each

SINR is strictly monotonically increasing in and mono-tonically decreasing in for . Thus, all users must havethe same SINR to target ratio . Otherwise, the users withhigher SINR to target ratios can give some of their power to theuser with the lowest ratio to increase it, which contradicts theoptimality.


Following the same steps of the above proof, the SINR bal-ancing structure also exists for Problem Pp in (14). Now we cansolve power allocation problems (13) and (14) with the aid ofProposition 1 which makes these problem trackable as shownin the following.

B. Simplified Solution: Group Power Allocation

For clarity, we present the simple group power allocation firstthen the general per-stream power allocation in the next sub-section. The group power allocation intentionally restricts thepower allocation strategy to make the complicated power allo-cation problem with multiple receiver antennas similar to thesimple one in [4] and [18], where . Thus, the grouppower allocation takes the advantage of the spatial diversity pro-vided by the GSINR-FB based beamforming to improve the per-formance, while keeping the complexity moderate.

To be more specific, the allocated power for a user using thegroup power allocation is evenly distributed over all streams ofthat user as

(20)

Let the power allocated to user be . Consequently, the diag-onal power allocation matrix for user can be written as ascaled identity matrix, that is,

(21)

We also define a vector to replace matrixin the optimization problems. Substituting into

(15), the average SINR in problems (13) and (14) is

SINR (22)

With the “grouped” constraint on the power allocation strategy(21), the simplified average SINR (22) for has the samestructure as that in [4] and [18] where . Thus, the solu-tions of this simplified group power allocation for Problems Prand Pp can be easily obtained. These solutions are briefly pre-sented in the following subsections. The overall optimizationalgorithms are also summarized at the end of each subsection.

Group Power Allocation for Problem Pr: With the SINRbalancing structure from the GSINR-FB beamforming in Propo-sition 1, the group power allocation for Problem Pr (13) can besolved by a simple eigensystem as

(23)

where the extended coupling matrix and the extended powervector are defined as

and (24)

TABLE IIITERATIVE ALGORITHM FOR PROBLEM PR WITH GROUP POWER ALLOCATION

respectively, where

(25)

and the th element of the matrix is zero whenor when .

By using Proposition 1 and the simplified average SINR (22)in (13), the rest of the proof of the previous results is similar tothose in [4] and [18] and omitted. With (9) and (23), we summa-rize the final optimization algorithm for Problem Pr in Table II,which iteratively calculates the optimal beamforming filter andpower allocation vector between the downlink and the uplink,where eig means the generalized eigenvalue solver. Due to theuplink–downlink duality described in Section II-D, it is guaran-teed that the uplink balanced level equals to the downlinkbalanced level .

Group Power Allocation for Problem Pp: Again, withProposition 1, the minimizer of (14) satisfies

SINR (26)

Substituting (26) into (22), the resulting power allocation vectoris

(27)

The optimal for the virtual uplink can be obtained similarly.The overall algorithm for Problem Pp is summarized in Table IIIwhich iteratively finds the optimal solution minimizing the re-quired power.

Note that (27) does not necessarily have a solution with non-negative elements. When there exists at least one nonnegativepower allocation satisfying the target SINR constraints and total


TABLE IIIITERATIVE ALGORITHM FOR PROBLEM PP WITH GROUP POWER ALLOCATION

power constraint in (14), we call the system feasible. De-pending on the channel conditions, the total power required toachieve the target SINRs could be quite large and exceed .For the purpose of studying the effects of the algorithms on thesystem feasibility, we use the sum power allocation algorithmin Table II with a large (43 dBm) to check the feasibilityas in [4] and [5]. In checking the feasibility, as soon as the bal-anced level becomes larger than 1 (which means that a feasiblesolution can be obtained), the algorithm switches to the powerminimization steps. On the other hand, if the balanced level re-mains below 1 when the feasibility testing stage ends, the feasi-bility test fails and the power minimization algorithm stops. Inpractical applications, when the system is infeasible, one mustrelax the constraints by reducing the number of users or de-creasing the target SINR.

C. General Solution—Per Stream Power Allocation

Now, we remove the restriction of evenly distributing powerin a group in Section IV-B. The performance is expected to befurther improved since the group power allocation is a subset ofthe per stream power allocation. The general power allocationsolutions presented in this subsection are much more compli-cated than the results in [4] and [5]. The overall optimizationalgorithms for Problems Pp and Pr are also summarized at theend of each subsection.

Per Stream Power Allocation for Problem Pp: The powerminimization problem using the result of Proposition 1 becomes

and

SINR (28)

With the equivalent SINR expression in (18), we will show that(28) can be elegantly recast as a well-known linear-program-ming problem. We first recall that the average SINR of user(18) is

SINR (29)

Substituting (29) into (28), the original power minimizationproblem turns into a linear programming problem, that is,

and (30)

where represents the vector comprising the diagonal elementsof as in Section IV-B. It is known that a linear programmingproblem can be solved in polynomial time using, for example,the ellipsoid method or the interior point method [19].

Table IV summarizes the proposed iterative algorithm withgroup maximum SINR beamforming and per stream powerallocation. The virtual uplink power allocation problemcan be similarly solved as (30) with replaced by

. Like the group power min-imization algorithm in Table III, the feasibility of this algorithmshould also be checked using the per stream sum power alloca-tion which is described in the next subsection.

Per Stream Power Allocation for Problem Pr: With afixed beamforming matrix , a fixed receive filter , and atotal power constraint, the optimization problem obtained byapplying Proposition 1 in (13) is

SINRand

(31)

where SINR is rearranged in form (29).The optimal power allocation vector for this complicated

problem is difficult to obtain, thus we consider a suboptimalsolution which can be found by simple iterative algorithms.First, using the concept of waterfilling, we fix the proportionof the power of data streams in each group according to theequivalent channel gains. That is, let

for (32)

Therefore, the variables can be reduced to onevariable such that for each ,and . The SINR for user in Equation (29) canbe rewritten as


TABLE IVITERATIVE ALGORITHM FOR PROBLEM PP WITH PER

STREAM POWER ALLOCATION

SINR

(33)

(34)

with .We then solve problem (31) with concepts similar to the sum

power iterative water-filling algorithm proposed in [20]. The thiteration of the algorithm is described in the following. Note thatthis problem has a similar form as (13), thus the balanced levels,defined as SINR , of all users must be equal according toProposition 1. At each iteration step, we generate a new effectivelevel gain for each user based on the power of other users fromthe previous step , as

(35)

TABLE VITERATIVE ALGORITHM FOR PROBLEM PR WITH PER

STREAM POWER ALLOCATION

for . The power variables s are simultane-ously updated subject to a sum power constraint. In order tomaintain an equal level, we allocate the new power proportion-ally to the inverse of the level gain of each user as

(36)

Note that when updating , the power variables of other usersare treated as constants and .

Similarly, for the virtual uplink, we denote the power variablefor user as and the effective level gain for user as . Theproposed algorithm for the overall problem Pr is summarized inTable V.

D. Insights to the Performance Advantage of theProposed Approaches

The insights to why the proposed approaches outperformthose in [4] and [5] are discussed as follows. First, under thesame power allocation matrix in (5) and (6), the GSINR-FBwill perform better than the beamformers in [5]. This is becausethe streams of each user cooperate with one another in ourscheme rather than interfere with one another as in [5]. Themathematical validation was given in Section III-A. Indeed, asshown in [6, Sec. III], the GSINR-FB includes the minimummean-squared error (MMSE) filter used in [4], [5] as a special


case (without cooperation). Thus, the GSINR-FB should havea better performance. As for the power allocation part, notethat our suboptimal group power allocation has a formulationsimilar to that of the power allocation methods in [4] and [5].Thus, they should be similar in terms of optimality. Our morecomplicated per-stream power allocation includes the grouppower allocation as special case. Therefore, it should performbetter than the group power allocation and the power allocationmethods in [4] and [5].

Finally, we note that we have no proof whether our iterativealgorithms converge to the global optimum or merely local op-tima. However, as shown by the simulation in the next section,the local optima still result in much better performance than [4]and [5].

V. SIMULATION RESULTS

In this section, we provide some numerical results to illus-trate the advantages of the proposed algorithms over [5] andthe simple BD methods [3]. The design concept of the BDtransmit beamformer is to remove the inter-user interferencein (3) completely. A BD beamformer can be found when

. To solve Problems Pr and Pp in (5)and (6), respectively, and to maximize the average SINR ofthe worst user, we also apply the GSINR-FB as the receivebeamformers for the BD cases. Note that this paper focuses onthe QoS of individual users, where the average SINR servesas a metric of QoS. For the BD cases, the conventional BDreceive beamformer design is more for the purpose of sumrate maximization (with waterfilling power allocation), whichusually does not maximize the SINR of the worst user. Thanksto Proposition 1, the corresponding power allocations can bederived similarly to those in Section IV-B and the details areomitted here. We also consider both the group and per streampower allocation strategies for BD, named “group BD” and“per stream BD,” respectively.

For the system simulation parameters, the channel matrixis assumed flat Rayleigh faded with independent and identicallydistributed (i.i.d.) complex Gaussian elements with zero meanand unit variance. The noise is white Gaussian with variance1 W. The transmitter is assumed to have perfect knowledge ofthe channel matrix , and each user knows its own equivalentchannels as discussed in Section II-D. Since typically the trans-mitter has more antennas than the receivers, we set the numberof streams equal to the number of receive antennas foruser . Without loss of generality, we assume a common SINRconstraint for all users, i.e., for all . In the followingsimulation, we generate 1000 channel realizations and averagethe performance. The convergence criterion of the iterative al-gorithms is set to .

Fig. 2 shows the simulation results of the balanced levelversus total power for Problem Pr, where is definedas in Proposition 1. The two proposed algorithms in Table II andTable V are compared with the method proposed in [5] and BD.Note that in [5], the data streams are processed separately andthe balanced levels are the same for all streams. With a commonSINR constraint to be satisfied by all streams, the per-streambalanced level defined in [5] gives the same value as the bal-anced level defined in Proposition 1. So the comparison

Fig. 2. Comparison with [5] and BD for Problem Pr.� � �,� � �,� � �,��.

of is fair in Fig. 2. The simulation parameters are:users, transmit antennas, each user has 2 receive an-tennas and 2 streams , and the SINR con-straint 1 dB. For each channel realization, all the three algo-rithms run until convergence but for at most 50 iterations. Forfair comparison, only the cases where all the three algorithmshave converged within 50 iterations are considered in averagingthe performance. We will discuss the convergence probabilitieslater. It can be seen that the proposed group power allocationachieves higher balanced levels than the method in [5] at thepositive SINR region. The proposed per stream power alloca-tion further outperforms group power allocation. Similarly, theper stream BD achieves higher balanced levels than the groupBD since the group BD is a special case of the per stream BD.Note that the BD schemes perform better when the total avail-able power is high and perform worse when the availablepower is low, since BD is a zero-forcing method which suffersfrom the noise enhancement problem at low . When ex-tremely large power is available, BD will perform close to theproposed methods. However, the operating region where thisphenomenon is obvious needs a much higher power than oursetting in Fig. 2. We do not show the simulation results in thisregion since it is less practical.

In Fig. 3, we plot the minimum total required powerversus SINR constraint for Problem Pp. Simulation parame-ters are users, transmit antennas, and each userhas receive antennas and streams. Again, forthe method in [5], a common SINR target has to be achievedby all streams. Thus, it has the same average SINR targetfor each user as the other algorithms. For each channel real-ization, all algorithms first perform feasibility test using a large

43 dBm. The feasibility test for the method in [5] canbe done similarly as the proposed algorithms. As soon as thefeasibility test passes, the corresponding algorithm switches tothe power minimization steps and runs until convergence butfor at most 50 iterations. The feasibility test for BD can be donetrivially. Only the cases where all the algorithms have passed


Fig. 3. Comparison with [5] and BD for Problem Pp.� � �,� � �,� �

�, ��.

the feasibility test, and converged within 50 iterations, are con-sidered in averaging the performance. Again, we will defer thediscussions for the infeasible cases and the convergence issueslater. As shown in the figure, the proposed group power allo-cation performs better than the method in [5] at high SINR.However, at low SINR, it requires more power. This is becausegroup power allocation suffers for the fact that it cannot adjustthe power within a group as the method in [5]. At low SINR, theinterference is larger and the method in [5] can adjust the powerwithin a group to better deal with the interference. On the otherhand, the proposed per stream power allocation performs betterthan the other algorithms at both high and low SINR. Similar toFig. 2, the BD methods perform better than the method in [5] athigh SINR, but are worse than the proposed methods in all casespresented. The results at extremely high power, where the per-formances of BD and the proposed methods are close, are notshown due to the same reason discussed before.

We also present the sum rate comparison in Fig. 4 where thebalanced levels of all users are the same (as in Fig. 2) as an in-dication of the QoS guaranteed and the fairness achieved. Thesimulation parameters are the same as in Fig. 2. Note that thesum rates of both BD methods are worse than the method in[5], while their balanced levels cross over that of [5] in Fig. 2.This is because under the same average SINR, the method in [5]will make all streams of a user have equal SINR and achieve thehighest sum rate due to the concavity of the log function. Thus,when a scheme’s balanced level advantage over the method in[5] is not significant enough (e.g., the BD schemes), its sum ratemay be lower than that of [5]. We emphasis again that our algo-rithms focus on the QoS (average SINR) of individual users. Ourproblem formulations are fundamentally different from thosefocusing on sum rate optimization and not guaranteeing theQoS.

Now we show the feasibility and convergence properties ofthe proposed algorithms. In the above simulation setting, thenumber of transmit antennas is equal to the total numberof data streams of all users (also equal to the total

Fig. 4. Sum rate comparison with [5] and BD for Problem Pr.� � �,� � �,� � �, ��.

number of receive antennas ). We further consider thecases where by increasing the number of users

. That is, for Problem Pp, , , and; while for Problem Pr, , ,. We name these cases as Case 2 and the settings for Figs. 2

and 3 as Case 1. Note that typically the system will performscheduling [21] when , that is, it uses time-division multiple access (TDMA) to schedule a number of userssuch that each time. Thus, the simulation resultsof Case 1 represent the performance of fully loaded systems andthose of Case 2 well represent the performance of overloadedsystems.

First, we discuss the feasibility issues. From the simulationsof Case 1, we observed that the proposed algorithms and themethod in [5] passed the feasibility test for almost all channelrealizations. Intuitively, group power allocation is more fea-sible than the method in [5] because the average, instead of perstream, SINR constraints are easier to be achieved, and theymake (27) better conditioned than the corresponding equation in[5]. Thus, nonnegative solutions of (27) are easier to be found.In addition, since the group power allocation is a special case ofthe per stream power allocation with equal power distributionamong the streams of a user, the per stream power allocationmethod should be even more feasible. As an example, when ahigh target SINR 12 dB is desired, simulation shows thatthe probabilities of feasibility for the group power allocation,the per stream power allocation, the method in [5], group BDand per stream BD are 99%, 100%, 67%, 100%, and 100%, re-spectively. For Case 2, the system can only support lower targetSINR and the probabilities of feasibility for the above five al-gorithms when 3 dB are 100%, 100%, 0%, 0%, and 0%,respectively. Note that for Case 2, the BD based methods cannot be applied since is not large enough.

As for the insights to the convergence behavior, typically eachoptimization step improves its objective function as outlined inSection II-D. The beamforming step maximizes each user’s sum


Fig. 5. Convergence behaviors of the proposed algorithms for Problem Pr inCase 1 (fully loaded) and Case 2 (over loaded). The arrows point at the numbersof iterations where the algorithms meet the convergence criteria.

SINR of the data streams and the power allocation step opti-mizes the balanced level. As an example, in Fig. 5, we plot thebalanced levels versus iteration times of the two proposed al-gorithms for Problem Pr under the same channel conditions.The total power constraint is set to 15 dBm, and the SINR con-straint . The arrows point at the numbers of iterationswhere the algorithms meet the convergence criteria. From thefigure, the two proposed algorithms typically do not oscillateoften and exhibit smooth transient behaviors. We also observethat the convergence behavior of the group power allocation isslightly better than that of the per stream power allocation, i.e.,the per stream power allocation is not as smooth as the grouppower allocation and needs more iterations to approach the bal-anced level. The reason why the per stream power allocation hasworse convergence behavior is that after a power allocation step,the noise whitening property obtained by the previous maximumSINR filter bank may no longer be valid, that is,may no longer be a scaled identity matrix. This effect may de-crease the balanced level. However, in most cases this negativeeffect has a small impact on the eventual performance. Table VIlists the iteration times needed to converge for both problems.From these results, we can see that all three methods need moreiterations to converge in Case 2. Note that for Problem Pp, thetarget SINRs for Case 2 are smaller than those of Case 1 sincethe method in [5] is not feasible for 3 dB. Also, the methodin [5] needs significantly more iterations when 2 dB.

Fig. 6 shows the probabilities of the proposed algorithms andthe method in [5] converging within 50 iterations given thatthey have passed the feasibility test, for the power minimiza-tion problem in Case 1 (settings of Fig. 3). We can see that thegroup power allocation and the method in [5] both exhibit goodconvergence probabilities while the per stream power allocationconverges better at low SINR than at high SINR. The reason forthe lower convergence probability of the per stream power allo-cation is that the linear programming makes the algorithm proneto oscillation between feasible solutions from iteration to itera-tion.

TABLE VIAVERAGE NUMBERS OF ITERATIONS NEEDED FOR CONVERGENCE

Fig. 6. Comparison of the convergence probabilities for Problem Pp in Case 1(fully loaded): � � �,� � �, � � �, ��.

In practice, as long as the solution is a nonnegative powervector, the SINR constraints are achieved, no matter the algo-rithm oscillates or not. Moreover, even when the per streampower allocation oscillates at the final iterations, typically theSINRs are still higher than that of the group power allocation. Soone can simply pick the solution at the final iteration and still ob-tain a better performance. The other way is to avoid oscillationby switching to the group power allocation whenever the perstream power allocation algorithm oscillates. The performanceof this combined algorithm should be between the performanceof the per stream power allocation and the group power alloca-tion. Fig. 7 shows the convergence probability for Problem Ppin Case 2. Since the method in [5] is not feasible when SINRconstraint for 3 dB, we only plot for 3 dB. The group


Fig. 7. Comparison of the convergence probabilities for Problem Pp in Case 2(over loaded): � � �,� � �, � � �, ��.

TABLE VIICOMPLEXITIES OF THE OPTIMIZATION STEPS IN ONE ITERATION

power allocation still converges almost surely in this overloadedcase.

VI. COMPUTATIONAL COMPLEXITY

In Table VII, we compare the computational complexityin one iteration based on the number of complex multiplica-tions. From this table, one can see that there is no single stepwhich dominates the complexity for each algorithm, so welist all of them for comparison. For each optimization step,the complexity of group power allocation (Table II) or powerminimization (Table III) is lower than that of the method in[5], and we have shown in Section V that the performancesof the proposed methods are also superior. The reason for thecomplexity saving of the group power allocation method is dueto the fact that the method in [5] processes the streams sepa-rately (matrix dimension ), while the group power allocationprocesses the streams of a user jointly (matrix dimension ,

). For the per stream algorithms (Tables V and IV), thecomplexity of power allocation is at most slightly higher thanthat of the method in [5], but the performance is much better.

In addition to the computational complexity in one iteration,the average number of iterations needed for convergence alsoaffects the system complexity. The average numbers of itera-tions for the three algorithms in the simulation settings of Fig. 3

and Fig. 2 are shown in Table VI (for the power minimiza-tion Problem Pp, the average number of iterations needed bythe feasibility test is included). This table shows that the grouppower allocation method has the fastest convergence among thethree algorithms, while the per stream power allocation has theslowest convergence. Compared to the method in [5], the grouppower allocation has a lower computational complexity, con-verges faster and performs better. If more complicated compu-tation is allowed, the per stream power allocation exhibits evenbetter performance.

As for the BD algorithms used in this paper, the computationof the zero-forcing transmit beamformers has approximately thesame complexity as that of the “uplink beamforming” step ofGroup (Pr) in Table VII; while the complexity for receiver beam-formers is approximately the same as that of “downlink beam-forming” step of Group (Pr). The complexity of the power al-location steps is negligible compared with those of the beam-formers. Since the BD algorithms do not need iterations, theyare not listed in the comparisons in Tables VII and VI (nor inFig. 6).

VII. CONCLUSION

Efficient solutions to the joint transmit-receive beamformingand power allocation under average SINR constraints in themulti-user MIMO downlink systems were proposed. The beam-forming filter is a GSINR-FB which exploits the intra-groupcooperation of grouped data streams. Due to this selection, theSINR balancing structure of optimal power allocation holds andsimplifies the computation. Based on the uplink–downlink du-ality, we formulated the dual problem in the virtual uplink, anditeratively solved the optimal beamforming filters and powerallocation matrices. The proposed algorithms are generaliza-tions of the one in [4] to the scenario with multiple receiveantennas per user, and exploit the receiver diversity more ef-fectively than [5]. Simulation results demonstrated the superi-ority of the proposed algorithms over methods based on inde-pendent data stream processing [5] and BD in terms of perfor-mance. Moreover, the computational complexities of the pro-posed methods are comparable with that of [5].

REFERENCES

[1] 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, 1998.

[2] A. Bourdoux and N. Khaled, “Joint Tx-Rx optimisation for MIMO-SDMA based on a null-space constraint,” in Proc. EEE Vehicular Tech-nology Conf. (VTC 2002 Fall), Sep. 2002, pp. 171–174.

[3] L.-U. Choi and R. D. Murch, “A transmit preprocessing techniquefor multiuser MIMO systems using a decomposition approach,” IEEETrans. Wireless Commun., vol. 3, no. 1, pp. 2–24, Jan. 2004.

[4] 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.

[5] A. Khachan, A. Tenenbaum, and R. S. Adve, “Linear processing forthe downlink in multiuser MIMO systems with multiple data streams,”in Proc. IEEE Int. Conf. Commun., Jun. 2006, vol. 9, pp. 4113–4118.

[6] H.-J. Su and E. Geraniotis, “Maximum signal-to-noise array processingfor space-time coded systems,” IEEE Trans. Commun., vol. 50, no. 8,pp. 1419–1422, Sep. 2002.

[7] J. Wang and D. P. Palomar, “Worst-case robust MIMO transmissionwith imperfect channel knowledge,” IEEE Trans. Signal Process., vol.57, pp. 3086–3100, Aug. 2009.


[8] A. Abdel-Samad, T. Davidson, and A. Gershman, “Robust transmiteigen beamforming based on imperfect channel state information,”IEEE Trans. Signal Process., vol. 54, no. 5, pp. 1596–1609, May 2006.

[9] D. Tse and P. Viswanath, “Downlink-uplink duality and effective band-widths,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jul. 2002, p. 52.

[10] H. Boche and M. Schubert, “Optimal multi-user interference bal-ancing using transmit beamforming,” in Proc. Wireless PersonalComm. (WPC), 2003, vol. 26, no. 4, pp. 305–324.

[11] M. Schubert and H. Boche, “A unifying theory for uplink and downlinkmulti-user beamforming,” in Proc. IEEE Int. Zurich Seminar, Jul. 2002,pp. 27-1–27-6.

[12] N. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality ofGaussian multiple-access and broadcast channels,” IEEE Trans. Inf.Theory, vol. 50, no. 5, pp. 768–783, May 2004.

[13] G. Caire, N. Jindal, M. Kobayashi, and N. Ravindran, “MultiuserMIMO achievable rates with downlink training and channel statefeedback,” IEEE Trans. Inf. Theory, vol. 56, no. 6, pp. 2845–2866,Jun. 2010.

[14] M. Biguesh and A. Gershman, “Training-based MIMO channel esti-mation: A study of estimator tradeoffs and optimal training signals,”IEEE Trans. Signal Process., vol. 54, no. 3, pp. 884–893, Aug. 2006.

[15] S. Haykin, Communication Systems, 4th ed. New York: Wiley, 2001.[16] J. Proakis, Digital Communications, 4th ed. New York: McGraw-

Hill, 2000.[17] R. Horn and C. Johnson, Matrix Analysis. Cambridge, U.K.: Cam-

bridge Univ. Press, 1990.[18] W. Yang and G. Xu, “Optimal downlink power assignment for smart

antenna systems,” in Proc. IEEE Int. Conf. Acoust., Speech, SignalProcess. (ICASSP), May 1998, vol. 6, pp. 3337–3340.

[19] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge,U.K.: Cambridge Univ. Press, 2003.

[20] N. Jindal, W. Rhee, S. Vishwanath, and S. A. Jafar, “Sum power it-erative water-filling for multi-antenna Gaussian broadcast channels,”IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1570–1580, Apr. 2005.

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

Yu-Han Yang (S’06) received the B.S. degree inelectrical engineering in 2004 and two M.S. degreesin computer science and communication engineeringin 2007, from the National Taiwan University, Taipei,Taiwan, R.O.C. He is currently working towards thePh.D. degree at the University of Maryland, CollegePark.

His research interests include wireless communi-cation and signal processing.

Shih-Chun Lin (M’08) received the B.S. and Ph.D.degrees in electrical engineering from the NationalTaiwan University, Taipei, Taiwan, R.O.C., in 2000and 2007, respectively.

He was a Visiting Student at the Ohio StateUniversity, Columbus, in 2007. After serving hismilitary duty in 2008, he is a Postdoctoral ResearchAssociate at the Institute of Communications En-gineering, National Tsing-Hua University, Hsinchu,Taiwan, R.O.C. His research interests includecoding/information theory, communications, and

signal processing.Dr. Lin also served as a TPC member of the IEEE Vehicular Technology

Committee (VTC) 2010-Spring.

Hsuan-Jung Su (M’00) received the B.S. degreein electronics engineering from the National ChiaoTung University, Taiwan, R.O.C., in 1992, and theM.S. and Ph.D. degrees in Electrical Engineeringfrom the University of Maryland, College Park, in1996 and 1999, respectively.

From 1999 to 2000, he was a Postdoctoral Re-search Associate with the Institute for SystemsResearch, University of Maryland. From 2000 to2003, he was with the Bell Laboratories, LucentTechnologies, Holmdel, NJ, where he was involved

in the design and performance evaluation of adaptive coding/modulation, fasthybrid-ARQ, scheduling, and Radio Link Control protocol for 3G wirelessnetworks. In 2003, he joined the Department of Electrical Engineering, Na-tional Taiwan University, where he is currently an Associate Professor. Hisresearch interests cover coding, modulation, signal processing, power control,synchronization, and MAC protocols of wideband wireless communicationsystems.

Dr. Su has served on the organizing committees and TPCs of many interna-tional conferences. He served as the Finance Chair of the IEEE InternationalConference on Acoustics, Speech and Signal Processing (ICASSP) 2009 andthe Finance Co-Chair and a TPC Track Chair of the IEEE Vehicular TechnologyConference (VTC) 2010 Spring.

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