9
An Efficient Scheduling Algorithm For Downlink Multi-Antenna CDMA Systems Elmahdi Driouch and Wessam Ajib Department of Computer Science Universit6 du Quebec A Montr6al Montr~a1, Quebec, Canada driouch.el-mahdi @courrier.uqam.ca, ajib.wessam@ uqam.ca Abstract We design efficient scheduling algorithms for multi- antenna CDMA downlink systems using zero forcing beam- forming. Our proposition maximizes the system sum rate and keeps the computational complexity low. We make use of a graph theoreticalapproach to represent the system as an undirected weighted graph. As a second step, we for- mulate the scheduling problem as the maximum weight k- colorable subgraph problem. We propose two heuristic so- lutions to find the users to serve in each time slot in an ac- ceptable polynomial time. Finally we evaluate the efficiency of the proposed schemes by mean of simulations and the re- sults shows the near-optimal performance of the proposed schedulers with very low computational complexity com- pared to the optimal exhaustive search over all the possible users combinations. Assuming the use of non-orthogonal spreading codes, our proposed algorithms are shown to still have near-optimal performance even in the case of high val- ues of the orthogonalityfactor~ 1 Introduction Due to the high expansion of wireless services, one of the main concerns in the design of next generation wire- less communication systems is to provide high data rates to the mobile users. Multiple input multiple output (MIMO) technology has gained great research interest over the last decade as it represents a very attractive solution which pro- vides high wireless system capacity without paying any ex- tra bandwidth or power [ 12]. In multiuser MIMO0 systems, the base station and/or the mobile equipments are equipped with multiple antennas which bring about more complexity and challenges in the design of efficient scheduling algo- rithms. In the case of single antenna multiuser systems, the authors in [7] show that the optimal scheduling strategy (the one maximizing the overall system capacity) is to serve op- portunistically the user with the current high channel qual- ity based on a kind of diversity named multiuser diversity. However, when the base station is equipped with multiple transmit antennas, designing the optimal scheduling strat- egy that extract the maximum gain from multiuser diversity becomes more challenging. In fact, MIMO systems, due to the space dimension arising from the use of multiple anten- nas, can support transmissions to multiple users simultane- ously and hence increases the system sum rate. In the case of a MIMO broadcast channel (Base station to users), when the base station has a perfect knowledge of the channel state information between its antennas and the ones of the users to serve in the system, an optimal trans- mit scheme known as Dirty paper coding (DPC) is proven in [2] to achieve the sum capacity of the system. In spite of its optimality, DPC has a very high complexity, due to its iterative encoding and decoding processes, which in- creases dramatically as we add more antennas or users to the system (even for moderate number of users). Therefore, it is almost impossible to implement DPC in practical sys- tems. Linear precoding schemes such as Zero forcing beam- forming (ZEBE) presents a promising alternative to DPC al- though their suboptimality, since they give a good trade-off between system capacity and implementation complexity. Effectively, ZFBF is proven to obtain a good approxima- tion of the capacity region of multiuser NM1MO broadcast channels. Moreover, if the number of users in the system is very large (i.e. goes to infinity), it is shown in [14] that ZFBF approaches asymptotically the sum capacity achieved by DPC. The choice of the users that will be served simulta- neously by the base station is a critical factor that affects the performances of the ZFBF. The optimal set of users which maximizes the sum rate when using ZFBF, can be found by performing an exhaustive search over all the combina- tions of the system users. Although this solution gives the maximum performance, it has a very high computational 978-1 -4244-4439-7/09/$25.00 (P2009 IEEE I Authorized licensed use limited to: Universite du Quebec a Montreal. Downloaded on December 5, 2009 at 19:11 from IEEE Xplore. Restrictions apply.

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An Efficient Scheduling Algorithm For Downlink Multi-Antenna CDMASystems

Elmahdi Driouch and Wessam AjibDepartment of Computer ScienceUniversit6 du Quebec A Montr6al

Montr~a1, Quebec, Canadadriouch.el-mahdi @courrier.uqam.ca, ajib.wessam@ uqam.ca

Abstract

We design efficient scheduling algorithms for multi-antenna CDMA downlink systems using zero forcing beam-forming. Our proposition maximizes the system sum rateand keeps the computational complexity low. We make useof a graph theoretical approach to represent the system asan undirected weighted graph. As a second step, we for-mulate the scheduling problem as the maximum weight k-colorable subgraph problem. We propose two heuristic so-lutions to find the users to serve in each time slot in an ac-ceptable polynomial time. Finally we evaluate the efficiencyof the proposed schemes by mean of simulations and the re-sults shows the near-optimal performance of the proposedschedulers with very low computational complexity com-pared to the optimal exhaustive search over all the possibleusers combinations. Assuming the use of non-orthogonalspreading codes, our proposed algorithms are shown to stillhave near-optimal performance even in the case of high val-ues of the orthogonalityfactor~

1 Introduction

Due to the high expansion of wireless services, one ofthe main concerns in the design of next generation wire-less communication systems is to provide high data rates tothe mobile users. Multiple input multiple output (MIMO)technology has gained great research interest over the lastdecade as it represents a very attractive solution which pro-vides high wireless system capacity without paying any ex-tra bandwidth or power [ 12]. In multiuser MIMO0 systems,the base station and/or the mobile equipments are equippedwith multiple antennas which bring about more complexityand challenges in the design of efficient scheduling algo-rithms. In the case of single antenna multiuser systems, theauthors in [7] show that the optimal scheduling strategy (the

one maximizing the overall system capacity) is to serve op-portunistically the user with the current high channel qual-ity based on a kind of diversity named multiuser diversity.However, when the base station is equipped with multipletransmit antennas, designing the optimal scheduling strat-egy that extract the maximum gain from multiuser diversitybecomes more challenging. In fact, MIMO systems, due tothe space dimension arising from the use of multiple anten-nas, can support transmissions to multiple users simultane-ously and hence increases the system sum rate.

In the case of a MIMO broadcast channel (Base stationto users), when the base station has a perfect knowledge ofthe channel state information between its antennas and theones of the users to serve in the system, an optimal trans-mit scheme known as Dirty paper coding (DPC) is provenin [2] to achieve the sum capacity of the system. In spiteof its optimality, DPC has a very high complexity, due toits iterative encoding and decoding processes, which in-creases dramatically as we add more antennas or users tothe system (even for moderate number of users). Therefore,it is almost impossible to implement DPC in practical sys-tems. Linear precoding schemes such as Zero forcing beam-forming (ZEBE) presents a promising alternative to DPC al-though their suboptimality, since they give a good trade-offbetween system capacity and implementation complexity.Effectively, ZFBF is proven to obtain a good approxima-tion of the capacity region of multiuser NM1MO broadcastchannels. Moreover, if the number of users in the systemis very large (i.e. goes to infinity), it is shown in [14] thatZFBF approaches asymptotically the sum capacity achievedby DPC. The choice of the users that will be served simulta-neously by the base station is a critical factor that affects theperformances of the ZFBF. The optimal set of users whichmaximizes the sum rate when using ZFBF, can be foundby performing an exhaustive search over all the combina-tions of the system users. Although this solution gives themaximum performance, it has a very high computational

978-1 -4244-4439-7/09/$25.00 (P2009 IEEE I

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complexity and hence represents only an upper bound fortheoretical studies. Therefore, the design of efficient heuris-tic scheduling algorithms minimizing the complexity of theexhaustive search solution is considered of great interest.

Recently, communication systems employing multipleantennas (MIMO technology) at the physical layer andcode division multiple access (CDMA) as a multiple accesstechnique at the MAC layer, which are known as MIMO-CDMA systems, has received remarkable research interest.In fact, these systems take advantage from the combina-tion of the robustness of spread spectrum (CDMA) com-munications and the capacity enhancements of MIMO tobecome a promising candidate for next generation wirelesssystems. The authors in [6] generalize of the Bell Labslayer space time (BLAST) codes, which are initially de-signed for MIMO point to point systems, to a multiple ac-cess system using CDMA. They have also shown that theproposed MIMO-CDMA system outperforms conventionalCDMA systems as they increase the number of antennas.Other research works in MIMO-CDMA [4], [3] try to in-crease the number of simultaneously supported users in thesystem by designing different joint and linear detection re-ceivers. However, this type of architectures requires com-plex processing at the receivers side instead of using a pre-coding scheme at the transmitter where more intelligencecan be added more easily.

In this paper, we propose two simple scheduling algo-rithms to solve the problem of choosing users to be servedby the base station of a MIMO-CDMA system with the aimof maximizing the system sum rate. The proposed sched-ulers divide the chosen users into a given number of groups,and serve the users of each group using the same code. In-stead of employing the optimal but complex DPC trans-mit strategy; the base station serves each group of usersby employing ZFBF in order to eliminate the interferencesbetween the users sharing the same code. Though ZFBFhas a very low implementation complexity, it requires agood scheduling algorithm in order to approach DPC per-formances without performing the very complex exhaus-tive search. Therefore, the low complexity proposed algo-rithms are based on a graph theoretical approach. In fact,the MIMO-CDMA system is formulated as an undirectedweighted graph where each node corresponds to an activeuser weighted by its channel gain. The scheduling prob-lem is consequently reduced to a graph coloring problemknown as the maximum weight k-colorable subgraph prob-lem. The users colored with the same color are then sched-uled in the same set and served using the same spreadingcode. Assuming that the base station makes use of non-orthogonal spreading codes, we evaluate the performancesof the proposed algorithms in the presence of different de-grees of inter-set interferences modeled by an orthogonalityfactor [8]. The simulation results prove that our algorithms

Userd ,U UC~ Ce2

Scheduler Zer forcinabeamfoinog

Feedbak CS] (H)

Figure 1. Structure of the proposed MIMO-

CDMA system

still have near optimal performances even for severe inter-set interferences.

This paper is organized as follows. Section UI formu-lates the system model. In Section III, we give a brief in-troduction to the graph theoretical notions used in this pa-per. We introduce in the same section the two proposedheuristic algorithms as a solution to the scheduling prob-lem. Simulation results, provided in Section IV evaluate theperformances of the presented algorithms and prove theirefficiency. Finally, we conclude our paper in Section V andprovide some future work perspectives.

2 System Model

We consider the downlink of a wireless communicationsystem with K single antenna mobile users and a base sta-tion. The base station is assumed to be equipped with Mtransmit antennas. We assume that the mobile terminals aredumb and all the intelligence is located at the base station.This model is practical since most existing mobile equip-ments have single antenna due to their small size. Hence,more transmit antennas and complex processing can be de-ployed easily at the base station side. The basic systemmodel for the downlink communication is shown in Fig. 1.Note that the proposed algorithms in the next section can beeasily adapted to support multi-antenna users.

We consider a time-slotted system where time is dividedinto slots. The base station has to take a new schedulingdecision at each time slot (TS). Let hk the channel coeffi-cients vector (M x 1) between the kth user and the basestation transmit antennas. We assume that the channel co-efficients vectors remain fix during each time slot, but they

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may vary from one time slot to another. Each user tracks thechannel state accurately and provides error-free and instan-taneous feedback (the vector hk) to the base station. Thechannel coefficients are assumed to be independent identi-cally distributed (idi.d.) complex Gaussian variables withunit variance. Furthermore, we assume that the base stationalways have data intended to be transmitted to the chosenusers. Thus, there will be no empty slots caused by lack ofdata, and the scheduler decision will only be affected by thechannel state.

At each time slot the scheduled users are disposed in atmost N independent groups or sets, where N is the num-ber of spreading codes available at the base station. In fact,the BS makes use of Code division multiple access in or-der to serve separately the scheduled sets of users. Thedata to be sent to the users belonging to the same set (de-noted by (,, n =1, ... , N) are multiplied by the samespreading code. The chip-level sampled spreading codeused for the set (,, is denoted by the 1 x C vector c,, whereC is the processing gain. Though orthogonal codes (suchas Waish-Haddamard codes) are shown to be optimal asspreading codes for downlink channels, their orthogonalitycan be damaged by the phenomena of multipath [8]. Hence,we consider c,, as a combination of orthogonal codes andpseudo-random noise (PN) codes to improve the correla-tion properties. Note that the latter assumption causes mu-tual interference between the scheduled sets that have to behandled at the mobile users.

Denote by Y the (K x C) matrix of received signals atthe K mobile stations. It can be written as:

(,. The matrix S,, can be written as the summation of thesignals intended to the users in (,,:

(2)Sn - ~ iWkCnVk

kE("

where Wk is the M x 1 beamforming weight vector, Pk isthe relative power allocated to the kth user and Xk is thetransmitted data symbol for the kth user. Consequently, wecan define the the matrix of transmitted signals S as the sumof the matrices S,,, n = 1, . . . , N and it can be written as:

(3)S S, - /PkWkCnXkn=1 n=1k(

where the notation k E Cnis used to denote that the kth userbelongs to the nth set of users.

The base station is assumed to have a fixed amount ofpower for transmission, denoted by P, during one transmis-sion. Denote by P,, the power portion allocated to the usersbelonging to the set (,, In order to maximize the perfor-mances of the system, the base station takes benefit fromthe perfect knowledge of the CSI to allocate different por-tions of power to each user. Then, P can be written as:

N

n=1

N

ZZ1:Pk.n-1 kEC,.

(4)

Therefore, assuming that the kth user is using the nthspreading code, the received chip-level sampled signal vec-tor for this user is given by:

Yk =hk V~WkCnXk + 1: hk jWjCnXjjE~n, j~k

+ Z k Zv/WjCXj +Zk

where Yk (k = 1, . ..,K) is the 1lx C vector of the re-ceived chip-level sampled signal at the kth user, S denotesthe M x C matrix of the chip-level transmitted signals fromthe M base station antennas and where sm, m =1, . .. ,Mis the 1 x C transmitted signals through the mth antenna ofBS during the current time slot and H represents the K x Mchannel matrix build from the users channel coefficientsvectors hk. Note that Zk is the 1 x C vector representingthe idi.d. additive white complex Gaussian noise with zeromean and unit variance.

The base station spread each user signal using a givenspreading code. The spread signal is then multiplied by aprecoding vector before transmission. We denote by S,' theM x C matrix of the chip-level transmitted signals fromthe base station antennas to the users belonging to the set

Note that the second term in (5) corresponds to the inter-ference among the users sharing the same spreading codeor self-set interference; whereas, the third term in the sameequation corresponds to the interference produced by theusers belonging to the other-sets namely the inter-set inter-ference.

In this paper, since the base station knows perfectly thechannel state information, we assume that it uses zero forc-ing beamforming as a precoding scheme in order to totallyeliminate the self set interference among the users served bythe same spreading code. As said before, we opt for ZFBFfor its simplicity and near optimality. The use of ZFBFlimits the number of users in each set to be smaller thanor equal to the number of antennas, i.e. card((,,) ! Mwhere card(-) denotes the number of elements in the set.

Y1

YK ) SM Z,) +( SM

(1)HS +Z(5)

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Denote by HC,. the matrix formed by the channel coeffi-cients of the users belonging to the set C, The kth lineh(k ) is the channel coefficients vector for the user k EHence, we obtain The beamforming weight matrix of eachset (n, n = ,. N, W~,, from a simple pseudo-inversionof the channel matrix H(.. The kth column of W~n (de-noted w (k)) [9] is given by:

(k) ht (k)- (6)

t1 ~k)1

where hctk is the kth column of the pseudo inverse of H(,..Each served mobile user uses a matched filter to pro-

cess the received signal in order to detect the desired signal.Therefore, the output of the matched filter of the kth usercan be given as:

T-Yk -Cn VP-kh'k WCflCf Xk

+ z: V/~]kwcfcfxiE(n, i9'k

N

+ Z Z p-h7~~~T

+ZkCn T

N

+ZkCn T(7)

where p is the cross correlation among the different PNscrambling codes and a denotes the orthogonality factorrepresenting the degree of imperfection of the orthogonalityamong the orthogonal codes caused by the multipath phe-nomenon. The second term in (5) is no longer present in (7)because of the use of ZFBF that eliminates all the interfer-ence caused by the users belonging to the same set. Hence,the desired signal is corrupted only by the additive noise andthe other sets interfered signals. Also, note that as the mul-tipath fading becomes important, the orthogonality factorapproaches 1 and accordingly the interference produced bythe users belonging to the other sets becomes more severe.

In [11I], it has been shown that ZFBF achieves near opti-mal sum rate region if the channel coefficients of the sched-uled users are orthogonal. Hence, in order to maximize thesum rate of the system the base station constructs a givennumber of sets of near orthogonal users. The maximumachievable sum rate for the overall system is given by:

RT

N2 ZZR (k)

Pk:Z 11Z k IwIIPk<ý!P.n=1lkE(,,(8)

where R (k) denotes the capacity achieved by serving the kthuser belonging to the set (,, and it can be written as:

R~k) = log 2 ~1 + NPk

1 + aep 2 . E Pjlhk',I

(9)The optimal power allocation Pk,,pt for the kth user can

be found through a continuous optimization [10] which addmore complexity to the scheduling algorithm. Hence, weuse in this paper a simple (in terms of complexity) water fill-ing strategy Lo find a near optimal power distribution amongthe scheduled users. The water filling equation is given by:

Pk~pt ~ (k)112 _1+Pkpt L W (n I 1 (10)

where pi is the solution of the equation given as follows:

(11)N-

3 Scheduling

3.1 Graph theory definitions

3.1.1 The vertex coloring problem

Given an unweighted graph G = (V, E) where V is the setof vertices and E is the set of edges, we define the problemof vertex coloring 0 V of the graph G as the problemof assigning to each vertex in V a given color such thatadjacent vertices, i.e. vertices joined by an edge, receivedifferent colors. In other words, coloring a graph with kdifferent colors is equivalent to put the vertices of G intok independent (stable) sets. This is typically given by thefollowing mathematical formulation:

Find CV : V -* Nsuch that if (u, v) E G then Cv (u) =, Cv (v)

One trivial solution to the vertex coloring problem is toassign a different color to each vertex in the graph. How-ever, such a solution is not interesting since the number ofcolors is often limited. In fact, it is proved in [5] that theproblem of vertex coloring, such that the number of usedcolors is minimal, is NP-hard for general graphs. Anyhow,

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near optimal solutions can be found in a reasonable polyno-mial time by applying some heuristic or metaheuristic algo-rithms [1].

3.1.2 The maximum weight k-colorable subgraphproblem

Let us consider a weighted graph G = (V, E) and aninteger number k > 0. Also, every vertex v in V is as-signed a non negative weight av. We define the maximumweight k-colorable subgraph problem as finding a subgraph

G- (V' c V, E') of the graph G, such that there existsa vertex coloring CV, of G' with k colors that maximizesthe value E a, among all the possible subgraphs. Note

VEV'that E' is a subset of E containing the edges formed by thenodes in V'. The announced problem can be formulated as:

Find V' C Vsuch that 3Cv, with kcolors and~ ma E a

It has been proven in [13] that The maximum weight k-colorable subgraph problem is NP-hard for general inputgraphs.

3.2 Graph representation

The main mission of the scheduler in the base stationconsists in finding, at each time slot, the best users to servein order to maximize the system sum rate. Since, the opti-mal solution has a very high computational complexity, weuse in this paper graph theory to reduce this complexity. Inthis subsection we formulate the MIMO-CDMA system asan undirected weighted graph in order to reduce the prob-lem of scheduling into a graph theoretical problem.

The system graph is obtained (similarly to [15]) as fol-lows: The MIMO-CDMA system is formulated as a graphwhere each user present in the system is represented by avertex vi and there is an edge (vi, v3 ) between vertices viand vj if and only if

ei hihj (12)

This means that we draw an edge between two given ver-tices if and only if their channels are not c -orthogonal wheref is a constant called the orthogonality threshold. Also, weassign to each vertex in the graph a non negative weightgiven by its corresponding user channel gain I1hi 112. Thefact that the chosen sets are not perfectly orthogonal willintroduce some rate degradation. The value of the orthog-onality threshold f: will affect directly the amount of thisrate loss. The choice of c~ is subject to two constraints:(i) the limited number of users that can be scheduled ineach set and (ii) the permissible rate penalty for each set.

Table 1. The complete coloring algorithm

In other words, a big value of E will violate the constraint(card((,,) < M) while a small c will provide smaller sets.

3.3 Scheduling algorithms

In the previous subsection, we have seen that our MIMO-CDMA system can be formulated as a weighted graph hav-ing as nodes the users waiting to be served by the basestation. Therefore, Assigning colors to the graph nodes isequivalent to assigning codes to the served users. Conse-quently, the scheduling problem consisting in finding thebest sets of users maximizing the system sum rate is equiva-lent to thc problem of finding a maximum weight subgraphcolored with at most N colors. Furthermore, maximizingthe sum of the weights for the colored nodes in the subgraphis equivalent to maximizing the system sum rate. Since theequivalent problem is proven to be NP hard, we present twolow complexity heuristic scheduling algorithms, namely thecomplete coloring algorithm and the greedy algorithm.

3.3.1 Complete coloring algorithm

The algorithm uses as input the complete channel matrixH between all the base station antennas and the K mobileusers, the orthogonality threshold c as well as the number ofavailable spreading codes N. The algorithm then starts byconstructing the system graph G = (V, F). The set V de-notes the set of vertices corresponding to the users presentin the system and the set E is the set of edges. Rememberthat an edge exists between two nodes in V if and only ifthe channel coefficients of the corresponding users are notE-orthogonal as in the equation (12). When the edges arecompletely built, the second step of the scheduler consists

Input H : the channel matrix,c : the orthogonality threshold,N : the number of codes

1. Build the system graph G =(V, E)using H and f as described in section 3.2.

2. Perform a complete graph coloring of Gusing a minimum coloring algorithm.

3. Sort the resulting groups decreasinglyaccording to their sum weights, The ZFBFsets are given by the N first groups.

(12)

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in performing a complete coloring of G by using an efficientand low complex heuristic coloring algorithm using a min-imum number of colors. This step of the algorithm buildsdistinct groups of nodes (users) colored using different col-ors. Note that depending on the value of c~, users belongingto the same group produce different degrees of interferenceamong each other. For example, when f increases, the num-ber of users in one group may increase and the interferencemay increase also and this may cause a loss in the systemsum rate.

The scheduler sorts the groups, built in the previous step,in a non increasing order according to their sum weights(corresponding to the channel gains of the users formingthese groups which affects directly the sum rate of thegroup). As a third and final step, the scheduler selects onlythe N first sorted groups, i.e. those with the highest sumrate, and assigns one PN/orthogonal spreading code code toeach group. In the case where the algorithm build less thanN groups, the scheduler selects all the available groups.Hence, the BS will not use all of its available spreadingcodes.

Since we have assumed the use of zero-forcing beam-forming, the maximum number of users belonging to eachset cannot exceed M users, which corresponds to the num-ber of antennas at the BS. However, the complete coloringalgorithm may build sets with more than M users. There-fore, the scheduler only chooses the M users having thebest channel gains in each set. Anyhow, an optimal value ofE found by simulation can decrease considerably the proba-bility of having more than M users per group. We summa-rize the complete coloring algorithm in table 1.

3.3.2 Greedy algorithm

This algorithm uses the same input as the previous one: thecomplete channel matrix H and the orthogonality thresholdc in order to build the system graph as well as the numberof available spreading codes N which corresponds to thenumber of available spreading codes.

As a first step, the algorithm builds the system graphG = (V, E). After that, the algorithm sorts decreasinglythe vertices of V according to their weights (correspondingto the user's channel gain).

In this algorithm, the sets of users will be built in agreedy fashion. First, the algorithm builds the first set (1it selects the first vertex in the ordered set V, which corre-sponds to the user having the best channel gain among allthe users in V, delete it from V and put it in (1. As a secondstep, the scheduler selects the user with the highest chan-nel gain from V that is non-adjacent to the users already in(I. The scheduler continues this operation until there is nomore users in V that are non-adjacent to the users already in(1 or when card((,) reaches M. Following, the algorithm

will build the other sets ((2, then (3, etc) using the samesteps but with a modified V at each iteration. Rememberthat each time a user is added to a given group, it shouldbe discarded from the initial set of vertices V. Note thatthe steps of the algorithm are exactly the basic steps of agreedy algorithm. In fact, At each new iteration, the algo-rithm takes the decision of which user to add to the currentgroup. this decision is definitive and the algorithm can notdrop or replace already chosen users in the other iterations.

The algorithm terminates when the set of vertices V be-comes empty or when the number of built sets reaches ex-actly N sets. In the latter case, the remaining users are notserved during the current time slot. Finally, the base stationassigns a different spreading code to each set of users.

The different steps of the greedy algorithm are summra-rized in table 2.

Table 2. The greedy algorithm

Input H : the channel matrix,f : the orthogonality threshold,N : the number of codes

1. Build the system graph G = (V, E)using H and f as described in section 3.2.

2. Sort the vertices of V in a non increasingorder according to their weights.

Initialize: i = 1

3. initialize the set (i by selecting the vertexv from V having the biggest weight.

4. Delete v from V, and put in (i the biggestnon adjacent vertex of all the vertices in (iand delete it from V.

Repeat Step 4 until all vertices in Vare adjacent to vertices in (j (or thereare M vertices in (Q) then go to Step 5.

5. if(i <NandV 7 Otheni --i+ landgo to Step 3.

Else the ZFBF has at most N sets.

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4 Numerical results

We present in this section some simulation results thatshows the near optimal performances of the proposedschedulers detailed in Section III. The performance of thetwo algorithms are analyzed in terms of the sum rateachieved by all the users served in the system. We alsocompare using simulations, the obtained sub optimal sumrates with the optimal ones given by the exact but complexexhaustive search. For all the simulations, the cross corre-

lation value is approximately set to -1where C denotesthe processing gain. Also, note that the value of the sumrate is given by bits per second per channel use since we areconsidering a CDMA system and each code can be spreadover one channel.

26F-

than the optimal value, the scheduler builds bigger sets con-taining users that experience high self set interference andthus decreasing the system sum rate.

Note that for all the following figures, we use the optimalvalue of the orthogonality threshold obtained in Fig. 2.

% - 10 1-2 __14Nun.b. & -e~

Figure 2. Sum rate vs c for a two or four an-tennas/four codes MIMO CDMA system with20 users

We plot in Fig. 2 the system sum rate of the greedyscheduler presented in the previous section when varyingthe orthogonality threshold e from 0 to 1. We simulate aMIMO0-CDMA system where a multiple antenna base sta-tion tries to serve 20 users. The BS makes use of four or-thogonallPN spreading codes and is assumed to be equippedwith two or four transmit antennas. We observe that the sumrates, given by the two curves in the figure (correspondingto M = 2,4), reach different maximum values before de-creasing for high values of c. This maximum value is ob-tained for an optimal value of the orthogonality thresholdwhich depends directly on the number of transmit antennas.Therefore, we notice that when c is below its optimal value,our proposed algorithm forms small sets of users resultingin performance degradation since we are not using all theMIN40O gain. When the orthogonality threshold is higher

-- Greedy:0001gonth. wihlphao - 0-.- Greadyo gorlthmithaphsla.-- Greedy 81900010 wMS alpha = 0.2-4-Greedy a190011m with alpham 0 5

16 18 20

Figure 3. Sum rate of the greedy algorithmvs Number of users for a four antennas/fourcodes MIMO CDMA system

One of the main objectives in the design of efficientscheduling algorithms is to extract the maximum gain fromthe multiuser diversity inherent in the system. Therefore,we prove in Fig. 3 that the proposed greedy algorithm takesbenefit from the multiuser diversity. In fact, Fig. 3 shows thesum rate of the greedy scheduler for different values of theorthogonality factor for a system with four antennas basestation having 10 dB of transmit power. We observe thatthe sum rate increases as we add more users to the systemwithout being saturated. Furthermore, the proposed algo-rithm keeps extracting almost the same degree of multiuserdiversity even for high values of a. However, as we increasethe value of a the sum rate is penalized by the inter-set in-terferences.

Fig. 4 plots the performances in terms of sum ratesof both the proposed greedy algorithm and the exhaustivesearch scheduler when varying the signal to noise ratio(SNR). A base station, equipped with two transmit antennasand using four orthogonallPN spreading codes, tries to servethe eight active users present in the system. We providesimulations for different values of the orthogonality factor(Y = 0, 0. 1, 0.5. It can be observed that the performance ofthe proposed algorithm, in terms of SNR, is approximately0.5 dB lower than the optimal exhaustive search if we as-sume that the signals using different spreading codes arriveperfectly orthogonal to their destinations (i.e. a =0). Thisgap remains almost the same for larger values of ae which

K

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alpha= 0

20

101

alpha = 0.5

2 4 0 8 10 12 14 16Signal to nosrti 05(dB)

18

Figure 4. Sum rate vs SNR for a two an-tennas/four codes MIMO CDMA system witheight users

search when varying the number of users to be served in thesystem. We consider different number of antennas at thebase station, M = 2 (the lower curves) and M = 4 (thehigher curves). We assume an orthogonality factor a = 0.We observe that, as we increase the number of users beyondeight users, the greedy algorithm outperforms clearly thecomplete coloring since it is well adapted for large numberof users. It can also be seen that there is only a very smallgap between the performances of the proposed greedy algo-rithm arnd the exhaustive search. Furthermore, this gap re-mains almost invariant for different number of users whichdemonstrates the near optimality of the proposed algorithm.We also notice that the greedy algorithm extracts the samemultiuser diversity gain than the exhaustive search. Thisgain is achieved with a significant complexity reduction asthe number of users increases. In fact, We were not able tocontinue our simulation for the exhaustive search for highvalues of K(K >l8for M =2and K >14for M-=4)since the complexity becomes extremely high.

5 Conclusionproves that our algorithm performs well even for severeinter-set interference. We can also notice that in spite ofits very low computational complexity the greedy schedulerprovides a sum rate slightly lower than the one provided byan exhaustive search (between -5% and -10%). Note thatfor a high value of a, the sum rate experiences a satura-tion at high SNR values since the power of the interferencescaused by the other sets' users becomes higher.

£9 -.-- Greedy algorithm 2Tx-- Complete coloring 2Tx

5 -.- Exhaustive search 2Tx-s-Greedy algorithm 4Tx

Complete coloring 4Tx-- Exhaustive search 4Tx

5 10 15 2NUmcer of usars

Figure 5. Sum rate vs Number of users for atwo or four antennas/four codes MIMO CDMAsystem

We provide in Fig. 5 a comparison between the sum ratesachieved by the two proposed algorithms and the exhaustive

This paper proposed new scheduling algorithms with theaim of maximizing the sum rate of MIMO-CDMA systemsthat use zero forcing beamforming as a precoding scheme.The proposed algorithms are implemented at the base sta-tion and are responsible of choosing the users to be servedat each time slot. We design the two algorithms based ona graph theoretical approach in order to maximize the sumrate of the system with a reduced computational complexity.

The proposed schedulers, first formulate the MIMO0-CDMA system as a weighted graph and after that they trans-form the scheduling problem into a graph coloring problem.Second, the algorithms build the sets of users to serve bysolving the formulated graph coloring problem. As a fi-nal step, Each set of users is assigned a given spreadingcode. Since the base station makes use of a combinationof orthogonal and pseudo random spreading codes, the sumrate of the system is degraded because of an inter-set in-terference introduced by an orthogonality factor. Anyhow,the proposed algorithms are shown, through simulations, togive near optimal results compared to the optimal but verycomplex exhaustive search scheduler even for high valuesof the orthogonality factor. Furthermore, the simulationresults demonstrate that the proposed schedulers extract amaximum gain from the multiuser diversity.

Future work in this area can be done by extending theproposed algorithms to support some imperfections in thesystem model, such as an outdated or erroneous channelestimation (presented in another work) or correlations be-tween the channel coefficients. Including some practical is-sues as fairness and quality of service constraints may alsobe addressed in future works.

30ý

Gaudy schaduling algontlrni

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