THE JOURNAL OF CHINA UNIVERSITIES OF POSTS AND TELECOMMUNICATIONS Volume 14, Issue 2, June 2007

ZENG Yi, ZHANG Zu-fan

Joint transmit beamforming and power control in multi-user

MIMO downlink using a game theoretic approach CLC number TN911.7 Document A

Abstract This study addresses the problem of jointly optimizing the transmit beamformers and power control in multi-user multiple-input multiple-output (MIMO) downlink. The objective is minimizing the total transmission power while satisfying the signal-to-noise plus interference ratio (SINR) requirement of each user. Before power control, it uses the maximum ratio transmission (MRT) scheme to determine the beamformers due to its attractive properties and the simplicity of handling. For power control it introduces a supermodular game approach and proposes an iterated strict dominance elimination algorithm. The algorithm is proved to converge to the Nash equilibrium. Simulation results indicate that this joint optimization method assures the improvement of performance.

Keywords multi-user MIMO, power control, transmit beamforming, MRT, supermodular game

1 Introductlon

MIMO communication techniques have stimulated great interest in recent years because of its potential for high capacity, increased diversity, and interference suppression. It is well known that in a MIMO system with nT transmit and nR

receive antennas, capacity grows linearly with min (nT, nR). For applications, such as wireless LANs and cellular telephony, MIMO systems will probably be deployed in environments where a single base must communicate with many users within the same time and frequency slots. In this multi-user MIMO system, it is usually assumed there is no coordination between the users since cooperating is too costly. Therefore, although classic multi-user detection techniques can be used in the uplink, capacity analysis and transmit precoding are more challenge tasks in the downlink.

In next-generation wireless communication system, users may have different bandwidth and/or SINR requirements.

Received date: 2006-11-29 ZENC Yi (m), ZHANG Zu-fan Special Research Center for Optical Internet and Wireless Information Networks, Chongqing University of Posts and Telecommunications, Chongqing 400065, China E-mail: yee.zengOyahoo,corn.cn

Article ID 1005-8885 (2007) 02-0014-05

Thus, power control and transmit beamforming should be jointly optimized to support the heterogeneous services. The power control problem in signal antenna scenario has been widely studied [ 1-31. Despite the obvious similarities, the multi-antenna case considered here has a more complicated analytical structure, which is due to the interaction between powers and beamformers. Single antenna power control can be seen as a special case of the more general problem addressed here.

In this article, such an optimization problem is addressed which can be depicted as minimizing the total transmission power while fulfilling the SINR target of each user. In Ref. [4], a joint optimization algorithm was proposed by exploiting network duality, and this method is much suitable for distributed network. In Ref. [5 ] , the authors proposed an improved approach that was based on traditional iterative algorithm, but it was initialized by applying a block- diagonalization algorithm for improving convergence speed. In Ref. [6], a scheme based on the MRT was put forward; however, it formulated the power control problem as outage probability computation for each mobile station.

Different from the above-mentioned methods, the power control problem in this study was modeled as a supermodular game, which can largely facilitate the implementation. Game theory-aided analysis has been widely used in code division multiple access (CDMA), Ad-hoc, and cognitive radio networks [7, 81. It has been proved that this method is much more efficient. Here, the beamfonners were determined by the MRT scheme. Then the supermodular game approach was applied to the power allocation problem, and an iterative algorithm was used to find the optimal power level for each user.

As in most published reports, the perfect knowledge of channel state information (CSI) was assumed at transmitter. The most common method for obtaining CSI at the transmitter is through the use of training or pilot data in the uplink (e.g., for time-division duplex systems) or via feedback of the receiver’s channel estimate found using downlink training data (e.g., for frequency-division duplex transmission) [9].

The remaining article is organized as follows. Section 2

No. 2 ZENG Yi. et al.: Joint transmit beamformine and M 3wer control in multi-user MIMO downlink using.. . 15

mathematically presents the joint optimization problem. The MRT scheme is briefly introduced in Sect. 3. Section 4 introduces a game theory framework as well as the supermodular game. Section 5 models the optimization problem as a supermodular game and proposes an iterative algorithm. Numerical results are given in Sect. 6. Finally, Sect. 7 concludes this article and brings up the further study.

The system model used in this article is similar to that in Ref. [lo], which can be formulated as follows.

Figure 1 illustrates a typical multi-user MIMO downlink system. The base station had M transmit antennas and K mobile stations each with N,, l<i<K antennas. The information symbol s,, l d i d K was first mapped to the M antennas by the transmitter beamformer ti E CM* before

transmitting. The channel was flat fading and quasi-static, thus the MIMO channel for user i could be represented by H , E @ N , x M . At the receiver of user i , the received signal

vector yts CNrx' was then decoded by the receiving

vector rt E @IxNc to generate the final output information

symbol S, . Hence, the received information symbol for user i

could be expressed as: K

Hitpi + H,tkSk +ni k = l , k # i

Base station u

Mobile station

Fig. 1 A typical multi-user MIMO downlink system

The first term in the bracket is the desired signal, and the second term reflects the multiple access interference (MAI) introduced by other users since spatially orthogonal signaling was not assumed. n, denotes the background noise with the same average power o2 .

Without loss of generality, for user i, l d i d K the beamformer t , is assumed to be unitary lltiI12 = 1 , and the

transmission power level is denoted by pi. Thus it is straightforward to obtain the SINR of user i

h=i .k#i

where, 'H denotes Hermitian transpose. Therefore, the joint optimization problem can be formulated as

minimize pi

s.t. RsINRl ( t , P ) a n (3)

where,

transmission power constraint.

is the SINR target of user i and Pm is the total

The MRT scheme is a simple and efficient precoding method which maximizes the individual signal-to-noise ratio (SNR) by choosing appropriate ti. This method can be formulated to K independent optimization problems as

t,'H,!H,tipi 0 2

maximize &M1, = (4) -

The optimization problem is the well-known Rayleigh quotient problem. Its solution, denoted by thp, is the eigenvector corresponding to the largest eigenvalue of

Although the solution of Eq. (4) ignores the MA1 terms, it maximizes the desired signal power. The MRT beamforming is asymptotically optimal for the low SINR region and the Corresponding optimal transmit power vector is element-wise minimal [4]. The MRT scheme was adopted in this study due to its attractive property and simplicity. it The numerical results indicate that the users can benefit much from the MRT scheme. It is known that increasing one's signal power can enhance its SINR, but meanwhile deteriorate the others'. Therefore, this power allocation problem as a game and this will indeed facilitate analysis.

H $ H , [lo].

4 Supermodular game8

Game theory represents a set of mathematical tools developed for the purpose of analyzing players' interactions in decision processes. It can be used to predict the outcome of these interactions and to identify optimal strategies for the players. Consider a pure strategic noncooperative game expressed as

where, K is the set of players (decision makers who select a particular power level to transmit), Pi is the set of strategies associated with player i (power levels which could be selected by i). If there is no coupling amongst players' strategies, the strategy space can be defined as the Cartesian product of the strategy set of each player, P = xe . And u, : P H R is the

set of payoff functions that the players associate with their strategies. For every player i in game , the payoff function

16 The Journal of CHWT 2007

ui is a function of pi , p-i that is the choice of player i and the

choices of all other players. Each player's objective is maximizing his own payoff function regardless of the others'.

Definition 1 A strategy profile for the players, P = [ p , , pz, . . ., p K ] , is a Nash equilibrium (NE) if and only if

~ ~ ( P ) 2 u , ( p ~ ' , p - ~ ) ; ViE K,p, 'e 4 (5)

Nash equilibrium identifies the steady-state condition wherein no players would unilaterally deviate from their chosen action since this would diminish their payoff.

The sufficient condition for the existence of pure strategy Nash equilibrium of general games is that the strategy space must be convex and compact. However, in power control problems the global strategy set is generally not convex Ell]. More recently, a class of games called potential game is widely used in power control problem, but they seem to be awkward in handling such problems with coupled strategy sets like those in this study. Therefore, the authors appeal to making use of supermodular games to address the problem in Eq. (3) .

Supermodularity was introduced into the game theory published report by Topkis in 1979. These types of noncooperative game have been shown to possess certain attractive properties, for example, a Nash equilibrium exists, best response strategies are monotone in other players' strategies. Furthermore, the simplicity of supermodular games makes convexity and differentiability assumptions unnecessary. A formal definition of supermodular games can be found in Ref. [12], with single dimension strategy sets, which are of interest in this study. The definition could be simplified as follows:

Definition 2 A function ui : P H R has increasing differences (or supermodularity) if for d p : 3 p i , p i > p ,

Ui (Pl, P I ) - ui ( P i , P J ) 3 U i (Pl, Pj) - M i ( Pi$ , ) ( 6 )

In case of ui is twice differentiable, then supermodularity is

D e m ~ o n 3 The gamer={ K,{ e}kK , { u ~ } ~ ~ } is a supermod~lar

C 1. Pi is a compact set of R ; C2. ui has supermodularity in (pi, p-i ); C3. ui is upper semi-continuous in pi, p-i . Next, an important property which was proved in Ref. [12]

Corollary 1 Suppose r is a supermodular game, let the

equivalent to aui ( p)/apiapj 20 for all j # i .

game if for all i the following conditions hold

is introduced.

best response be BRi(p-i)=arg max u ~ ( P ~ . P - ~ ) ( 7 )

P , . 4 ( P - , )

then, BR, ( p - ' ) has a largest and smallest element denoted by

E, ( p - { ) and 13Ri ( p - i ) , respectively, and if p l i 3 p + , then

- BRi ( pci)2BRi ( p - i ) , m, ( p:i)2BRi ( where, Pi ( p - i )

is the constraint strategy of user i depending on the other users' choices P - ~ . Corollary 1 shows that each player's best

response function is increasing in the actions of other players. Theorem 1 [12] The set of pure strategy Nash equilibria of

a supermodular game are nonempty, and possess greatest and least Nash equilibrium points.

Theorem 1 assures the existence of NE in a supermodular game. The greatest and least vector in a set of vectors refers to the component-wise comparison between vectors in that set.

5 Power contra1 modellng and convergence of the algorithm

Motivated by Yates [13] , an interference function I i@) is defined as

where, tbp,Vic K is determined based on MRT scheme. In

this study, the focus is on the problem which is feasible, that is, the set { p : p 3 Z ( p ) , l l p l ) , d P m } is nonempty. When it is

infeasible, there is a call admission problem in finding a subset of users that can obtain acceptable connections, which is beyond the scope of this study.

To make use of supermodularity, let the objective function be -pi. Assume p' is a feasible solution and let a constant a31, note that ap' could also be considered to be feasible as long as allp'I1, d P,, . Hence, for the game r the

coupled strategy set of user i is

c: ( P - , > = { P i : Pi 3 1 , (PI 90dPt Wl? IlPll, GPm> (9)

Pi( p-i ) could be treated as a subset of Pi.

meorem 2 The power allocation game T = { K , { ~ ( P _ ~ ) } ~ ~ ,

{ -p i } , , 1 is a supermodular game.

Proof It is straightforward to verify r' satisfies C1 and C2 of Definition 3, hence there is a need to prove the objective function -pi is upper semi-continuous.

Let f ( p, ) = -p , , obviously f (.) is decreasing and linear

in pi , for Vpi ,p ;€ q ( p - , ) and E > O , there exists a

6 > 0 , when Ipi - < 6 , the following inequality holds,

f( p i ) < f( p ; ) + ~ . Hence the objective function -p , is

upper semi-continuous. An iterated strict dominance elimination algorithm called

Algorithm 1 is introduced to find the Nash equilibrium points. Let pi" be the smallest element in q . In each iterative

No. 2 ZENG Yi, et al.: Joint transmit beamforming and power control in multi-user MIMO downlink using.. . 17

process checking whether p,k>Ii(pk) for every user, if no,

updating p i by p,? = BR, ( p t , ) and renewing the strategy

set. The iteration ends until p 3 I ( p ) . Theorem 3 Algorithm 1 converges to the unique pure

strategy Nash equilibrium of r' . Proof The proof of uniqueness can be found in~Ref. [13],

recalling that a fixed point in best response correspondences implies a Nash equilibrium. Here we will show that algorithm 1 converges to the NE.

= 4 , p o = (pp ,p: , . . . , p i ) the smallest element in

4 and p,! =BRi(pOi)3pro . Note that, for game the

best response is unique. Let 4' ={pi : pi E qo, p,>p:} , if

pi E 4' , for example, pi < p,! , then it is dominated and

eliminated, since by supermodularity (10)

Let

u, (Pf a,) - u, P-, 12% ( Ptl? POi) -ui (Pi9POi) > 0

Iterating the argument, and define

Pf+l = BR, ( P k , )

q k + ' = { p, : p , E 4 k , pi >p,!+'}

Now, if ~ , ~ > p f - ' , then according to monotonicity of best

response (Corollary l), there is p,k+l = BR, ( ~! , )>BR, (p!;;') = p,k

By induction, p,! is a nondecreasing sequence for each i.

Define p, = 1imk+- pZk , since p , is bounded by a p : , the limit exists. = (PI, p, , . . . , jTK ) is the Nash equilibrium.

For all i, p, , u, ( p,"", pkt)>u, ( p , , p!, ) . Taking limits as

k -+ m , u, (j?, , Zi_, )>u, ( p , , Ti_, ) , therefore j j is the strategy

profile of the Nash equilibrium.

6 Numerlcalrerulk

In the simulation, the behavior of the proposed algorithm is illustrated and the two different schemes are compared, one was based on the MRT scheme and the other used random beamformers. The channel model used here was a quasi-static flat Rayleigh fading channel. The evaluation of the complete system performance was not included in this study, since it required additional assumptions, such as the multiple access scheme, modulation, and coding.

Three users each with single receive antenna and the same SINR targets (e.g., - 10 dBm) were assumed to illustrate the convergence behavior of the Algorithm 1. Simulation results are given in Fig. 2. It shows that five iterative processes have been experienced until the SINRs reach the targets. The number of the iterative processes may be smaller if the requirement for

the precision of SINR is not rigorous. In addition, the SINRs are enhanced after each iterative process, which is mainly due to the monotonicity of Algorithm 1. Therefore, the SINR of each user could be improved using Algorithm 1 even if the power control problem is infeasible.

Fig. 2 Convergence behavior of the Algorithm I

One major motivation of this study is demonstrating the benefit of using the MRT scheme before power control, and the results are shown in Fig. 3. It was assumed that there were 3 users each with 2 antennas and the same SINR target in the multi-user MIMO system and the base station had 5 antennas, which is usually denoted by [5 2 2 21. At each receiver, the average power of the background noise was considered to be - 20 dE3m. For comparison, three different beamformers were used, they were the one's based on the MRT scheme, the one

with equal elements (e.g., t , =[I/& ,..., I / A ~ , v ~ € K ),

and the one with random elements. It can be observed that the total required transmission power based on the MRT scheme is approximately 10 dB less than the other two schemes. Hence it reveals that the performance of the power control scheme is generally better than the others without precoding by the MRT. Besides, as the SINR target is -2 dB, it does not converge while using equal beamformers, which stands for the infeasible case.

-16 -14 -12 -10 -8 -6 -4 -2 SINR targetldB

Performance comparison between the different schemes Fig. 3

18 The Journal of CHUPT 2007

Figure 4 describes the minimal power level that is required to achieve the same targets for various numbers of users. The simulation parameters are the same as those in the above-described situation except that the channel matrices may be different for various numbers of users.

Fig. 4 Minimum transmission power required to achieve the different SINR targets for various numbers of users

7 Condudon8 and further study

In this study, the transmit beamforming and power control are jointly optimized in multi-user MIMO downlink to minimize the transmission power while fulfilling the SINR target of each user. Due to simplicity and efficiency, the MRT scheme was used to determine the transmit beamformers, and then a supermodular game theoretic approach was used for power control. An iterated strict dominance elimination algorithm was introduced and its convergence was proved. Simulation results show that the convergence to the unique Nash equilibrium is assured as long as the power control problem is feasible. In addition, it also illustrates using MRT scheme before power allocation could obtain better performance. However, it is known that the perfect CSI is generally difficult to obtain, hence the impact of partial CSI on the proposed method should be further investigated in further studies. Besides, it is necessary to carefully analyze the complexity of the proposed algorithm and make possible modifications to improve it. How to handle the infeasible case also needs to be taken into .,onsideration.

Acknowledgements This work is supported by the National Natural Science Foundation of China (60602057) and the Natural Science Foundation of' Chongqing Science and Technology Commission (CSTC, 2006BB2360).

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Biographies: ZENG Yi, from Chongqing, M. S. Candidate in Chongqing University of Posts and Telecommunications, interested in wireless communications and adaptive resource allocation.

ZHANG Zu-fan, from Hubei Province, Ph. D., associate professor of Chongqing University of Posts and Telecommunications, interested in smart mobile cellular systems and wireless resource management.