20 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

A Transmit Preprocessing Technique for Multiuser MIMO Systems Usinga Decomposition Approach

Lai-U Choi and Ross D. Murch, Senior Member, IEEE

AbstractWe introduce a transmit preprocessing techniquefor the downlink of multiuser multiple-input multiple-output(MIMO) systems. It decomposes the multiuser MIMO downlinkchannel into multiple parallel independent single-user MIMOdownlink channels. Some key properties are that each equivalentsingle-user MIMO channel has the same properties as a conven-tional single-user MIMO channel, and that increasing the numberof transmit antennas of the multiuser system by one increases thenumber of spatial channels to each user by one. Simulation resultsare also provided and these results demonstrate the potential ofour technique in terms of performance and capacity.

Index TermsMultiple-input multiple-output (MIMO)systems, multiuser decomposition, multiuser multiple-inputmultiple-output (MU-MIMO), transmit single processing.

I. INTRODUCTION

I N RECENT years, wireless multiple-input multiple-output(MIMO) systems with multiple antennas employed at boththe transmitter and receiver have gained attention because oftheir promising improvement in terms of performance and band-width efficiency [1]. In the downlink, several single-user tech-niques have been proposed [2][5] such as vertical Bell Labora-tories layered spacetime (V-BLAST), maximum likelihood de-tection (MLD), and singular value decomposition (SVD)-basedtechniques.

Here we introduce a multiuser MIMO transmit preprocessingtechnique for the downlink of multiuser MIMO systems. Thetechnique is based on decomposing a multiuser MIMO down-link channel into parallel independent single-user MIMO down-link channels. Once the multiuser channels are decomposed, anysingle-user MIMO technique (such as MLD and BLAST) can beapplied in the usual way to each user.

Previously, there has been only limited work on multiuserMIMO systems for the downlink. Examples include[6] and [7],where multiuser MIMO systems are considered and processingat both of the receivers and transmitter is assumed to be linear.The work in [6] attempts to find the antenna weights for thetransmitter and receivers jointly by maximizing the spectralefficiency for a multiuser MIMO time-division multiple-access(TDMA) system and [7] finds the antenna weights jointlyby maximizing the signal-to-interference-plus-noise ratio fora multiuser MIMO code-division multiple-access system.However, the solution for the antenna weights in [6] is not

Manuscript received March 13, 2002; revised November 26, 2002; acceptedNovember 27, 2002. The editor coordinating the review of this paper and ap-proving it for publication is L. Hanzo. This work was supported by the HongKong Research Grant Council (HKUST6024/01E).

The authors are with the Department of Electrical and Electronic Engineering,The Hong Kong University of Science Technology, Clear Water Bay, Kowloon,Hong Kong (e-mail: eeluchoi@ee.ust.hk; eermurch@ee.ust.hk).

Digital Object Identifier 10.1109/TWC.2003.821148

Fig. 1. System configuration of a multiuser MIMO system.

guaranteed to exist since the proposed iterative algorithm isnot guaranteed to converge. On the other hand, the solution forthe antenna weights in [7] is suboptimal. Another issue is thatparticular linear receiver structures are assumed in both of thesystems and these impose certain restrictions on the systems.

Our work is different in that we introduce a transmit pre-processing technique at the base station (BS) for the downlinkof multiuser MIMO systems that decomposes a multiuserMIMO downlink channel into multiple parallel independentsingle-user MIMO downlink channels. Therefore, any tech-nique for single-user MIMO systems, such as V-BLAST,MLD, and joint transmit and receive MIMO processing (e.g.,SVD-based techniques), can be applied for each user of themultiuser MIMO systems. Some key properties of this de-composition include that each equivalent single-user MIMOchannel has the same properties as a conventional single-userMIMO channel, and that increasing the number of transmitantennas of the multiuser system by one increases the numberof spatial channels to each user by one. Simulation results arealso provided and these results demonstrate the potential of ourtechnique in terms of performance and capacity.

The structure of our contribution is as follows. In Section II,the system model of a multiuser MIMO system is introduced,while the problem formulation and the solution are providedin Section III. Then, Section IV gives some discussions of thekey properties and some simulation results are provided in Sec-tion V. Finally, Section VI concludes our work.

II. SYSTEM MODELThe configuration of our proposed multiuser MIMO system

is shown in Fig. 1, where antennas are located at the BS andantennas are located at the th mobile station (MS). In total,

there are MSs or users in the system. At the BS, the dataare processed before transmission, which we refer to as transmit

1536-1276/04$20.00 2004 IEEE

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004 21

preprocessing, and then launched into the MIMO channel. Letrepresent the transmit data symbol vector for user

, where is the number of parallel data symbols transmittedsimultaneously for user . This data symbolvector is passed through a transmit precoder, which is charac-terized by the precoding matrix , a matrix thattakes in nonzero values and outputs terms. Each of the

output terms is transmitted by each of the transmit an-tennas.

We assume that the channel is flat fading and denote theMIMO channel to user as , which is a matrix.Its ( )th element is the complex gain from the th transmitantenna at the BS to the th receive antenna at MS . Also, itselements are independently identically distributed (i.i.d.) zeromean complex Gaussian random variables with unity variance.At the receiver of user , receive antennas are used to receivethe data symbols and the received signals can be written bya vector of length , which is given by

(1)

where the noise is an vector, whose elements arei.i.d. zero mean complex Gaussian random variables with vari-ance .

Throughout this letter, we denote a -user system withtransmit antennas at the BS and antennas at the th MSas a ( ) system, and we will refer to asingle-user system with transmit antennas at the BS andantennas at the MS as a ( ) system.

III. PROBLEM FORMULATION

By using the system model introduced in Section II, our pri-mary objective is to select the nonzero precoding matrices,

, for the users such that at the receiverof each MS there is no interference from the other users.This can be expressed as

.

.

.

(2)

where represents the trace operation and the constraintstates that the transmit power of user is limited . It should benoted that represents the interferenceto user due to other users, and therefore, our primaryobjective function in (2) nulls all interference for each user.

Since are arbitrary data vectors,implies for .

Therefore, it can be shown that the solution to (2) is equivalentto the solution of

.

.

.

.

.

.

for (3)

Letting be the th column of , sinceimplies that is in the null space or kernel of , we cansimplify (3) further as the solution of

(4)

where , denotes the null space or kernelof , and represents the intersection of the subspaces.

Note that the precoding matrix should be a nonzero ma-trix, otherwise, no signal is transmitted. To guarantee the exis-tence of a nonzero precoding matrix, a sufficient condition isthat the number of the transmit antennas is larger than the sumof the number of receive antennas of any users and iswritten as

(5)

Under this sufficient condition, let be an

orthonormal basis of the subspace , where

is the dimension of this subspace. By letting

(6)from (4), we can write a solution to (3) as

(7)where is a nonzero matrix, which can be de-signed alone by some criteria or can be jointly designed withthe structure of the receiver. More explicit details about the de-sign of matrix can be found later in this section. Also, notethat , because is orthonormal and

. The matrix can be computed bySVD

.

.

.

.

.

.

(8)

From (8), we can see that the dimension of is ,where . Under the sufficient

22 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

Fig. 2. MU-MIMO decomposition: Decomposing a multiuser MIMO channelinto parallel single-user MIMO channels.

condition given in (5) and the assumption of i.i.d. channel, wecan obtain that the dimension of the subspace is

(9)

with probability one. By substituting (7) into (1), we can obtain

(10)Note that the multiuser MIMO system denoted by (1) has beendecoupled to parallel single-user MIMO systems. A closeobservation of (10) shows that we can think of the equivalentsingle-user MIMO channel of user as and the equiv-alent transmit processing can be represented as . As shownin Fig. 2, the multiuser MIMO channel is decomposed to par-allel single-user MIMO channels. We refer to the decomposingprocess in (10) as multiuser MIMO decomposition and we referto a multiuser MIMO system applying this decomposition asMU-MIMO system.

Since is the equivalent transmit processing for the equiv-alent single-user MIMO channel of user , , the designof is the same as designing the transmit processing for asingle-user MIMO system. For example, is a scaling iden-tity matrix if the V-BLAST technique is employed for the user,while is consists of the right singular vectors ofif the SVD technique is used.

Finally, we would like to point out that the number of simul-taneous users in the proposed system is restricted by the numberof transmit antennas. If the number of users is large, other ac-cess techniques, such as TDMA, need to be used together withthe proposed technique so that all the users can obtain service.

IV. KEY PROPERTIES

The key properties of the multiuser MIMO decompositiondiscussed in Section III are as follows.

1) A multiuser MIMO downlink channel is decomposed intoparallel independent single-user MIMO channels (see

Fig. 2). Therefore, any technique suitable for the down-link of single-user MIMO systems, such as V-BLAST,MLD, and joint transmit and receive MIMO processing

(e.g., SVD-based techniques), can be applied for eachuser of the multiuser MIMO systems.

2) Each equivalent single-user MIMO channel has the sameproperties as a conventional single-user MIMO channel.As shown in Fig. 2, the equivalent channel of user canbe given by , whose dimension is .Since the elements of are i.i.d. zero mean complexGaussian random variables with unity variance andis orthonormal, the elements of are also i.i.d.zero mean complex Gaussian random variables with unityvariance. Hence, the equivalent system for user afterthe multiuser MIMO decomposition is a system withtransmit antennas and receive antennas.

3) Increasing the number of transmit antennas of the mul-tiuser system by one increases the dimension of the equiv-alent MIMO channel of each user by one. That is, in-creasing the value of by one increases the value of

by one for all users . This can be ob-served from the relationship discussed in Section III that

with probability one.4) The system capacity of this multiuser MIMO system in-

creases linearly with the number of transmit antennas. Forexample, a -user MU-MIMO system with transmitantennas at the BS and two receive antennas at each MSis equivalent to parallel (2,2) systems. Therefore, whenthe number of transmit antennas increases, the number ofusers can increase without affecting the individual sys-tems.

V. SIMULATION RESULTS

In this section, the MU-MIMO system introduced in theprevious sections is investigated by computer simulation. In thesimulation, quadrature-phase-shift keying (QPSK) is utilized.The flat fading MIMO channel, whose elements are i.i.d. zeromean complex Gaussian random variables with variance one,is fixed for 100 symbols and more than 10 000 independentchannels are used to obtain each bit-error-rate simulation.Throughout this section, we consider a -user system withtransmit antennas at the BS and receive antennas at eachMS ( ), and we will refer to itas a ( ) system. In order to satisfy thesufficient condition for the existence of a nonzero precodingmatrix solution, we assume . Also, we assumethat the number of data streams is equal to for each user( ). We denote a single-user systemwith transmit antennas at the BS and receive antennas ateach MS as a ( ) system.

In Fig. 3, we provide sample performance comparison be-tween our MU-MIMO systems and single-user MIMO systems.Two receive antennas ( ) are employed at each MS. Threedifferent cases are compared: with SVD-based technique(i.e., the data stream is transmitted through the channel withthe largest singular value), with minimum mean squareerror (MMSE) receiver at each user, and with maximumlikelihood (ML) receiver at each user. It can be observedthat the performance of our three-user MU-MIMO system,(6,[2,2,2]) configuration, is similar to that of the single-user

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004 23

Fig. 3. Performance comparison between single-user MIMO systems andMU-MIMO systems.

Fig. 4. Performance comparison between single-user MISO systems andMU-MISO systems.

system, (2,2) configuration, for all cases. This is consistentwith the result in Section IV, in which we show that (6,[2,2,2])MU-MIMO system is equivalent to a ( , 2) configurationsingle-user system and from (9). We also deduced inSection IV that a -user MU-MIMO system with transmitantennas at the BS and two receive antennas at each MS isequivalent to parallel (2,2) systems. These results reveal thatthe capacity of the MU-MIMO system increases as the numberof the transmit antennas increases.

Since the multi-input single-output (MISO) system is aspecial case of an MIMO system, our approach is applicable tomultiuser MISO systems (we refer to it as MU-MISO system).In Fig. 4, we provide performance comparison results forMU-MISO systems and single-user MISO systems. We cansee that the performance of (3,[1,1,1]) configuration is similarto that of (1,1) configuration, the performance of (4,[1,1,1])configuration is similar to that of (2,1) configuration, and

Fig. 5. Overall capacity of our MU-MIMO system versus the number of usersfor various values of total transmit power P , when the number of transmitantennas is equal to 15 and the number of receive antennas at each user is two.

the performance of (5,[1,1,1]) configuration is similar to thatof (3,1) configuration. These results are consistent with theanalysis in Section IV that increasing the number of transmitantennas of the multiuser system by one increases the numberof transmit antennas of each single-user by one.

A close observation to Figs. 3 and 4 reveals the flexibility ofour method. Multiple data streams can be transmitted simul-taneously for each user. In Fig. 3, the (6,[2,2,2]) MU-MIMOsystem when outperforms that when . The(7,[2,2,2]) MU-MIMO system outperforms the (6,[2,2,2])MU-MIMO system when . Moreover, the ML receiverprovides better performance than the MMSE receiver when

. In Fig. 4, we can see that the (4,[1,1,1]) MU-MISOsystem when provides better performance than thatwhen , and the (4,[1,1,1]) MU-MISO system outperformsthe (3,[1,1,1]) MU-MISO system when . Therefore,we can trade off between the data rate and the performance.Moreover, we can increase the transmit antennas to improvethe performance and also the structure of the receiver can assistimproving the performance.

In Fig. 5, we provide the overall capacity of our MU-MIMOsystem versus the number of users for various values of thetotal transmit power when the number of transmit antennasis equal to 15 and the number of receive antennas at each useris two. In the simulation, we assume the noise varianceand the total transmit power ranges from 0 to 20 dB. Equalpower is located for each user. The overall capacity is givenby taking the expectation of , (i.e., ), where is thesum of the capacity of all the equivalent parallel single-userMIMO channels, each of which is computed by the standardformula with water filling solution given in [8]. Here, it can beobserved that the capacity may increase when the number ofusers increases. For example, when the number of users ( )equals six and the total transmit power ( ) equals 20 dB, thecapacity is about 60 b/channel-use, which is triple the capacitywhen there is one user and is twice the capacity when there aretwo users. This is because after the multiuser decomposition,

24 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

Fig. 6. Capacity of a single-user MIMO system versus the number of transmitantennas for various number of receive antennas when total transmit powerequals to 10 dB.

the (15,[2,2,2,2,2,2]) channel is equivalent to six parallel (5,2)channels and the (15,[2,2]) channel is equivalent to two parallel(13,2) channels. The capacity of the six parallel (5,2) channelsis larger than both the capacity of the two parallel (13,2) chan-nels and the (15,2) channel. However, note that the capacity de-creases when the number of users ( ) equals seven comparedto that when the number of users ( ) equals six. This is be-cause our MU-MIMO system loses transmit diversity in orderto eliminate the multiple-user interference. The more users, themore transmit diversity it loses for the more multiple-user in-terference. Therefore, the optimum capacity occurs at a certainnumber of users, where it has good tradeoff between the gain ofmultiple-user diversity and the loss of transmit diversity.

Finally, Fig. 6 presents the capacity of a single-user MIMOsystem versus the number of transmit (Tx) antennas for variousnumber of receive (Rx) antennas when total transmit powerequals 10 dB. Similar to Fig. 5, the capacity is given by takingthe expectation of (i.e., ) over the MIMO channel [8].We can observe that the capacity increases as the number oftransmit antennas increases. However, when the number ofreceive antennas is fixed, say two receive antennas, even thoughthere are ten or even more transmit antennas, the capacity issmall. This is because the small number of receive antennaslimits the capacity. By allowing multiusers, each still with asmall number of receive antennas, and using our MU-MIMOdecomposition, the overall capacity increases while each userstill receives comparable capacity to the single-user case. Forexample, the capacity of our MU-MIMO system with four

users, a (15,[2,2,2,2]) channel, can achieve two times as muchas the capacity of a single-user (15,2) channel when the totaltransmit power is 10 dB (see Fig. 5). Similarly, the capacityof our MU-MIMO system with five users, a (15,[2,2,2,2,2])channel, can achieve three times as much as the capacity ofa single-user (15,2) channel when the total transmit power is20 dB. This demonstrates the potential of multiuser operationand our MU-MIMO system, which can provide increasedcapacity when the number of receive antennas at each MS islimited.

VI. CONCLUSION

We have introduced a transmit preprocessing technique forthe downlink of multiuser MIMO systems. It decomposes themultiuser MIMO downlink channel into parallel independentsingle-user MIMO downlink channels. Used together with thisdecomposition technique, all the previous proposed MIMOprocessing techniques, which are suitable for the downlink ofsingle-user MIMO systems, are applicable in the downlink ofmultiuser MIMO systems. Some key properties are provided.It is shown that the capacity of this multiuser MIMO systemincreases linearly with the number of transmit antennas andthis system can provide increased capacity when the number ofreceive antennas at the mobile stations are limited. Simulationresults are also provided and these results reveal the potentialof our technique in terms of performance and capacity.

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