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SPATIAL SCHEDU LING ALGORITHM S FOR WIRELESS SYSTEM SDiego Bartolome, Antonio Pascual-Iserte, Ana I. Perez-Neira *

{diego, tonip, anuska} @gps.tsc.upc.esUniversitat Politbcnica de Catalunya (UPC)c/ Jordi Giron a 1-3,Mhdul D5, Campus Nord UPC

08034 Barcelona (Spain)

A B S T R A C TThis paper a ddresses the problem of the spatial scheduling of usersin a cell for simultaneous downlink transmission from a Base Sta-tion (BS) having multiple antennas under a perspective of jointPhysical and M edium Access C ontrol (PHY -MAC) design. First ofall, wec ompu te the transmit beamvectors for each group accordingto a Zero Forcing (ZF) criterion, which gives a simple closed-formsolution. We show first that it is equivalent to the minimization ofthe maximum B it Error Rate (BER). In this paper, the main con-tribution lies on the resolution of the NP-complete combinatorialproblem tha t comes up as cost function if w e want to minimize thetotal transmit power. The solution of the NP-complete problem isperformed by the stochastic technique Simulated Annealing (SA).Additionally, we present two heuristic algorithms that may enablea real-time implementation of this scheduling approach.

1. I N T R O D U C T I O NMultiple Element Antenna systems may provide a large increase incapacity for future wireless communications standards. However,in realistic scenarios and systems, the number of antennas mightbe limited by the high cost of the RF ront-ends. T herefore, wefocus on the use of a small set of antennas at the BS, or at theAccess Point (AP) in Wireless Local Area Networks (WLANs).

In this paper, we address mainly the problem of clustering a setof users for simultaneous transmission in th e downlink of a T DM Asystem. We assume that the BS has to allocate the K users in thecell in groups of Q , which is the number of transmit antennas.After solving this combinatonal problem, G groups are formedand the scheduler at the BS will choose how to allocate them indifferent time slots, but this is out of the scope of this paper.To begin with, we have to choose a transmit beamforming de-

sign. Optima l downlink beamforming is addressed in [ I ] or in[2]. In the former, the transmit power is minimized while assur-ing a certain Signal to Interference and Noise Ratio (SINR) at thereceivers. However, this kind of techniques are difficult to imple-ment on a real system. As practical issues play an important role,we have selected the minimization of the transmit power subjectto a ZF criterion. This criterion is suboptim al, but the key point isthat it provides a simple closed-form solution and guarantees thatthere is no interference among users belonging to the same group.

'This work is partially supported by the Telecommunications Tech-nological Center of Catalonia (CTTC), by the Spanish govemment FIT-070000-2000-649 (Medea+ A105 UniLan). TlC99-0849and the IntegratedAction HF2WI-0055 and by the Catalan govemment thanks to grants2001SGR 00268.2003FI 00190 and 2001FI 00714.

0-7803-7663-3/03/$17.00 2003 IEEE

Centre Tecnolhgic de Telecomunicacions de Catalunya (CT TC)c/ Gran Capita 2-4,Edifici Nexus 1, Planta 208034 Barcelona (Spain)

As reported in [31, the performance of a Spatial Division Mul-tiple A ccess (SDM A) system is better when each antenna at the BSis assigned to a user rather than using all the antennas for transmis-sion to a single user. On the other hand , several papers in the liter-ature have dealt with the problem of spatial scheduling, but they donot take into account the spatial signature of the users. Therefore,the overall efficiency of the system is reduced because co-linearspatial signatures may be allocated in the same group. In this pa-per, the spatial signatures of the users are taken into account in theoptimal group assignment for the scheduling.In [4], the authors maximize the capacity of a SDMAITDMAsystem. As in our work, their problem a lso resides on the bestcombination of users that optimizes a certain criterion, in their casethe maximization of capacity. They propose a solution based ongraph-theory, which is a NP-complete problem, so they point outthe need for efficient heuristic algorithms.

In our paper, instead of the maximization of the capacity, wehave chosen the minimization of transmit power as a cost function.In fact, we vote for a realistic physical parameter that yields to areduction in the power consumption and that can be easily usedin higher layers. On he other hand, we are taking into consider-ation a spatio-temporal scheduler that could be developed in realtime, so exhaustive search methods or the solution in [4] based ongraph-theo ry have been disregarded. Firstly, we implement SA ,which obtains good near-optimal solutions with reasonable com-puting time. In fact, simulations have shown to obtain the opti-mum when the number of users is low. As a main contribution, wepropose simple heuristic solutions that may enable the real-timeoperation of the scheduler with only a slight power degradation.This paper is organized as follows. First, in Section 2 we de-scribe our problem and find the global cost function to be min-imized, together with the above-mentioned equivalence. Then,Section 3 deals with the proposed solutions to the problem, i.e. SAand the heuristic algorithms. After that, we give som e simulationsresults in Section 4 and the conclusions.

2. P R O B L E M S T A T EM E N TIn the following, boldface capital (lowercase) letters refer to mat%ces (vectors). The operator (.)* denotes conjugation, ( . ) T transpo-sition,and (.)" = ((.)*)=.T hee lement a t r ow i tha ndc o lum nj thof matrix A is denoted by [A],,.We consider the downlink of acommunications system, where the BS is provided with Q transmitantennas, although the notation and solutions presented henceforthcan be directly applied to the uDlink. Let K > Q be the number of.. - .active users in the cell, each having a single antenna. The users arethen distributed into G = [K/Q1 groups. Each group is sched-uled for transmission in a different time slot, whereas the Q users

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in one group are served simultaneously by a SDMA scheme, i.e. adifferent beamformer for each user.Motivated within the context of Orthogonal Frequency Divi-son Multiplexing (OFDM) systems, such as WL AN S', the channelis assumed Rat fading. In those systems, the frequency-selectivechannel in the time domain is converted into a flat fading channelat each of the useful subcaniers, see [5 ] or details.Our problem is the minimization of the total transmit power,subject to a Z F design criterion for each group. ZF implies that atgroup g: 1 5 g 5 G, there is no interference among users and thesame sig nal level is achieved for all of them. Let us first considerthe signal model for this MISO link'. Th e received signal vectorfor the group at time instant n s y (n) :

Qy(n)= xHb i s x ( n ) E C'"'; ( 1 )k = l

where bb is the transmit beamvector for the kth user and H is theQx Q complex flat-fading channel matrix, the ith row (1 5 i 5 Q)of which contains the 1 x Q vector of the channel gains for the ithuser, i.e. hi. At position k , 5 k 5 Q, of vector y(n)we findthe received signal for the user kth. Th e transmitted symbols haveunitary mean energy E { I s c ( n ) / ' } = 1.As stated before, for each user w e want to minimize the powerof the beam vector according to a ZF criterion:

m in bfbk s . t . Hbk = l k , (2 )where the vector 1 k is 1 at position kth and 0 elsewhere, It isstraightforward to obtain the solution to this problem:

bk = HH (HH")-' 1 k E CQx'. (3)Applying an eigenvector decomposition to the hennitian ma-

, l is easy lo verify that the power fromrix HHH = UXUHthe kth transmit beamvector reduces to:

(4)where A, is the mth eigenvalue of matrix HHH and um is thenormalized eigenvector associated with that eigenvalue (the mt hcolumn of matrix U) . f we sum up the powers of the beamvectorsfor the Q users in group, the total power can be expressed as:

Inserting the index g denoting the group, the total transmit powerP, dded up over all the groups is finally expressed as:

In order to minimize the global transmit power P, in Eq . (61,we h ave to find the best distribution of the users in groups (amongall alternatives). Indeed, this is a comb inatorial problem . As itwill be stated. the global transmit power may vary substantiallydepending on how the users are grouped together.

'The main current swdard s are Hiperlad2 and IEEE 802.1 a.'Suictly speaking, we should also add the group index g. But far sim-'E is b e diagonal matrix of the positive eigenvalues and the columnsplicity, this is omitted until Eq. (6).of the unitary matrix U contain the eigenvectors associated to them.

m . . -Fig. 1. System configuration: a BS with Q antennas aiming totransmit simultaneously to groups of Q users (m obile terminals).2.1. Equivalent problemWe show here that the solution to Eq . (6) is equivalent to the min.imization of the maximum BER of all users and groups. First,recall that an additional parameter a b can be inserted in Eq. (2).which denotes the desired signal power a t the receiver4. With thisapproach, the optimum solution is that minimizing the global BERof all the users in the cell, e.g. if the sym bols are BPS K an d thereis no interference among users in the same group',

(7)

where SNRk , , = w,eing the noise power, is the SNRof user kth in group g. This is equivalent to the minimization ofthe I-norm of the B ERs for all users in all groups. As this problemis difficult to solve, we can choose the infinity-norm instead of theI-norm. Therefore, Eq. (7 ) is now transformed into:

This problem is in fact the minimization of the maximum BERamong all the users and groups. Intuitively, let us have an scenariowhere a user has a high SNR (thus very low BER) and the otherusers equally low SNR. Then, we can always increase the SNRsof bad users so that their BER is reduced. In the end, all th e userswill have the same SN R (and BER ), therefore (ckk,g/ ' = a , V k , g ,thus the ZF solution if we set LI = 1. Finally. note that the ZFapproach does not guarantee the minimization of the mean B ER.

3. P R O P O S ED M E T H O D SSeveral options have been evaluated so as to solve the combinato-rial problem of Eq. ( 6 ) . First, exhaustive search in the possibilityspace or the graph-based solution in [4J, re prohibitively expen-sive in terms of complexity when the number of users increases.Instead, we have applied the stochastic method S A , which mayyield to the optimum solution with a reasonable computing time.Moreover, we have developed some low complex heuristic algo-rithms that achieve a low degradation with respect to SA .3.1. Simula t ed annea l ingSA is an iterative algorithm which is capable of finding the globaloptimum of a function even if the problem is not convex. SA hasanalogies with the physical process of annealing, i.e. cooling of a

'The lk vector will have now air at position kth5Th is condition is satisfi ed by the ZF solution.

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Table 1 . Simulated AnnealingI . Choose an initial solution G. This G x Q matrix contains 2user index at each position g, i, i.e. each row contains the 4

users belonging to the same group.2. Compute the global transmitted power P (G ) as in Eq. (6).3. Initially. P,,,*" - (G )and G,i, - .4. Generate a new solotion G' exchanging tw o users' positions irdifferentgroups.5. As in Eq. (6). compute the power P (G') and the differenctwith the previous solution A(P) = P (G ' )- P (G ) .6. Accept 6' with probability min I , e T If G' is ac( "1.cepted. G t '.1. If (G') 5 Pm

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Fig. 2. Example of two users that have to be assigned to groups.M A X - M I N first allocates user j whereas M A X - R A T I O selects useri . If user i is not assigned to group 1 a high degradation occurs,while if user j is not in group 1, he loss in performance is negligi-ble. M A X - R A T I O allocates first the users incurring in a bi g loss ifthey are not in the best group, thus achieving a better performance.

Table4. Spatial distribution of users. Single run, G = 4, Q = 3.M A X - R A T I O proposal: instead of selecting the users according tothe minimum costs, we son them by means of the ratio betweenthe maximum and the minimum costs. Note that other approachesmight also be possible as long as they consider the variation ofthe cost values. Fo r an illustrative example of the behavior of theheuristic algorithms, see Fig. 2.

4. S I M U L A T I O N SN , = 10000 rnns of the simulations have been conducted for anumber of transmit antennas ranging from Q = 2 to Q = 4, an da number of groups' from G = 2 to G = 7. The channel is flatfading, and static users are distributed uniformly between -60'an d 60" (sectored antennas at the BS ). No noise is included in th esimulations. The aim is to see what is the performance in terms ofpower for the techniques that have been previously described.

We have also simulated the random sched uler, i.e. the users arearbitrarily allocated to the groups. We see in Figs. 3 an d 4 that thissolution is not convenient at all as the output power' is at least onemagnitude order worse than the M A X - R A T I O solution. We alsosee that the M A X - M I N approach yields to had performances dueto the fact explained in the example (se e Fig. 2): it does not takeinto account all the possible options far a user, i.e. it may not beassigned to the best-suited group, so this worsens the performancebecause of a high penalty in term s of power.As expected, S A has the hest performance. Note that the totalpower when G is low is greater than in the high range. This isdue to the fact that, if we have very few users and they are closetogether, we cannot cope with their separation. On the other hand,when we increase G, we a lso increase the degrees o f freedom o fthe system. This allows a better separation of the groups. In turn,th e M A X - R A T I O echnique approaches the minimum power of SA.The degradation is mainly between 3 dB and 6 dB. T he key advan-tage is that the the computational lo ad is redwmd drastically payingthe price of a slight increase in the global transmit power.Finally. we show in Table 4 the spatial distribution of usersgiven by a single Tun of the simulation with the M A X - R A T I O SO-lution. It is seen that the users that are closer (in terms of angle

6For simplicity, we assume K to be a multiple of Q.'In those figures,we show the ourage power for a cenain threshold Z .That means that the x% of the obtained powers arebelow the platted value.

'4L i_DI-sFig. 3. Outage Powers (90% an d 95%) fo r Q=2

l . . , , , . , , , l'a 1 IS 1 ,I 8 .I I I, . .I .Fig. 4. Outage Powers (90% an d 95%) fo r Q=4.

of anival) are separated in different groups. If we grouped themtogether, the increase in power would he remarkable.

*---r

5. C O N C L U S I O N SThere are tw o main contributions of this paper. First, w e deal withthe spatial schedulinga setofusers in groups for simultaneous ser-vice and propose a new joint MAC -PHY spatial channel allocation.Second, we focus on the techniques for the resolution of this NP-complete problem. Therefore. we apply the stochastic techniqueSA and propose suboptimal solutions that are shown to have a lowdegradation with respect to the SAperfo rmance wi th much lowercomputational load. As this is a first insight in the joint Physicaland Medium Access Control (MA C-PH Y) design, the next step isto take fairness issues [7 ] into consideration.

6. REFERENCES[ I ] E Rashid-Fmkhi et al., "Transmit Beamforming and Power Controlfar Cellular Wireless Systems:' lEEE JSAC, Oct. 1998.121 R. SVidh and M. Benglssan and B. Ottersten, "System Evaluationof Optimal Downlink Beamforming in W ireless Com munication," in

VTC Fall. 2001.[3] R.W. Heath and M. Airy and A.J. Paulraj. "Multiuser Diversity farMIMO WirelessSystems with Linear Receivers:' inilrilomar, 2001.[4 ] R. Zhang, "Scheduling for Maximum Capacity in SDMA,TDMA Net-works: in ICASSP. 2002.(51 L.J. Cimini, "Analysis and Simula tion of a Digitll Mobile RadioChannel using Orthogonal Frequency Division Multiplexing," IEEETransrcrionr on Communications, July 1985.[61 P.J.M. van Laarhaven and E.H.L. AaRr, Simuiared Annealing: Theory

andApplicatk" Kluwer Academic Publishers, 1987.171 D.N.C. Tse, "Multiuser Diversity in Wireless Networks," WirelessCommunicorions Seminnr Standford Univerriry. April ZOOl.