Subspace-based Multiuser Detection forAntenna Array CDMA Systems

Getian Ye, Guoan Bi, and Zhengyu Li

Abstract: In this paper, an adaptive multiuser detector based onsignal subspace estimation is proposed for synchronous CDMA sys-tems with an antenna array. Based on a training sequence trans-mitted by the desired user, we obtain a linear MMSE detectorwhich does not require knowledge of the code sequences and thechannel impulse response. The proposed spatial-temporal adap-tive multiuser detection technique is shown to be robust against thenear-far problem and computationally efficient by using subspacetracking algorithm.

I. INTRODUCTION

Code-division multiple access (CDMA) systems have re-ceived much attention for their application to multiuser systemssuch as mobile cellular and personal communication systems.Since the near-far problem is the main limitation of the conven-tional CDMA systems, a number of near-far resistant multiuserdetectors have been proposed (see [1], and references therein).These techniques can provide significant performance gain overthe conventional detector and substantially increase the capacityof CDMA systems.In recent years, considerable attention has been focused on

the space-time multiuser detection using an antenna array at thereceiver. A number of space-time processing techniques havebeen proposed for enhancing the performance of CDMA sys-tems. In [2], an approach that is the combination of adaptivearray processing and interference cancellation using least meansquared (LMS) algorithm has been presented. The convergenceof this method is shown to be very slow and it requires sev-eral hundred training bits. In [3], the expectation-maximization(EM) algorithm is developed for joint channel estimation anddata detection. The convergence of this detector can be acceler-ated by using an antenna array at the receiver and only a shorttraining sequence is required. However, it is computationallyexpensive and only applicable to systems with a few users. Ad-ditionally, some spatial-temporal blind multiuser detectors havebeen proposed, e.g., [4] and [5]. These methods require knowl-edge of the signature sequence and the estimates of channelparameters and array response of the desired user. In authors’opinion, the blind channel estimation is not an easy task.In this paper, we propose a subspace-based spatial-temporal

Manuscript received March 25, 2000; approved for publication by Dong InKim, Division II Editor, May 3, 2001.G. Ye is with School of Electrical Engineering, University College, the Uni-

versity of New South Wales, Australian Defence Force Academy, Australia, e-mail: g.ye@adfa.edu.au.G. Bi and Z. Li are with School of Electrical and Electronic Engineer-

ing, Nanyang Technological University, Singapore, e-mail: EGBI@ntu.edu.sg,zyli@ntu.edu.sg.

1229-2370/00/$10.00 c� 2001 KICS

multiuser detector which does not require knowledge of the codesequences of all users and the channel impulse response. Theproposed approach only needs a training sequence for deter-mining the weight vector of linear minimum-mean-square-error(MMSE) detector and offers lower computational complexity byusing the modern signal subspace tracking algorithm.The rest of this paper is organized as follows. The discrete-

time system model is briefly derived in Section II. In SectionIII, we develop two subspace-based linear multiuser detectors,namely, the linear zero-forcing detector and the linear MMSEdetector. Section IV presents some simulation results and fol-lowed by conclusions in Section V.

II. SYSTEM MODEL

We consider a synchronous CDMA system with K activeusers over flat fading channels. The baseband signal transmittedby user k is given by

sk�t� �pPk

�Xi���

bk�i�uk�t� iT �� (1)

where Pk is the transmitted power, bk�i� � f�����g is theith transmitted symbol, T is the symbol duration, and uk�t� �PN��

n��uk�n��Tc�t � nTc� is the spreading waveform where

uk�n� � f�����g is a signature sequence, and �Tc�t� denotesa unit rectangular pulse that has chip duration Tc � T�N .We suppose that an antenna array with M elements is em-

ployed at the receiver. The vector impulse response of the chan-nel between the transmitter of user k and the antenna array canbe modeled as

fk�t� � �k�t�a��k�t����t�� (2)

where �k�t� denotes the channel fading attenuation and a��k�t��is theM -vector of array response from user k. For a linear array,themth component of the array response vector a��k�t�� is givenby

akm � exp

�j�d

�

�m�

M � �

�sin��k�t��

�� (3)

where d is the inter-sensor distance, � is the wavelength and�k�t� is the direction of arrival (DOA) of the wavefront of userk.Since K users transmit synchronously over flat fading chan-

nels, the received signal at the mth array element can be repre-

sented as

rm�t� �

KX

k��

pPk�kakm

�X

i���

bk�i�uk�t� iT � � wm�t�

�

KX

k��

hkm

�X

i���

bk�i�uk�t� iT � � wm�t�� (4)

where hkm �pPk�kakm denotes the channel impulse re-

sponse from the transmitter of user k to the mth element of theantenna array and wm�t� is assumed to be complex white Gaus-sian noise with zero mean and covariance � �. It is also assumedthat the channel fading process �k�t� and DOA �k�t� are fixedover the observation interval.The discrete version of the received signal rm�j� can be

formed by sampling the output of a chip matched filter at thechip rate. By buffering rm�j� into blocks of length N duringthe ith symbol duration, the received signal vector rm�i� can bewritten as

rm�i� �

KX

k��

hkmbk�i�uk �wm�i� �

KX

k��

bk�i�gkm �wm�i��

(5)

where uk � ���pN��uk��� uk��� � � �uk�N � ���� is the nor-

malized signature waveform vector of user k and gkm � hkmukis defined as the impulse response vector. The noise vectorwm�i� is a Gaussian random vector with zero-mean and covari-ance matrix ��IN , where IN denotes anN �N identity matrix.Therefore, the received signal vector from the antenna array isgiven by

r�i� �KX

k��

bk�i�gk �w�i�� (6)

where r�i� � �r�� �i� r�� �i� � � � r�M �i���� gk � �g�k� g�

k� � � �g�kM

��� andw�i� � �w�� �i� w�� �i� � � �w�M �i��� is a Gaussianrandom vector with zero-mean and covariance matrix � �IMN .

III. SUBSPACE-BASED LINEAR MULTIUSERDETECTION

A. Subspace Decomposition

We assume that the vectors fgkgKk�� of the K users are lin-early independent. By performing an eigendecomposition of theautocorrelation matrix, we have

Rrr � Efr�i�rH�i�g � Xs�sXH

s�Xn�nX

H

n� (7)

where �s � diag���� ��� � � � � �K� � CK�K contains theK largest eigenvalues of Rrr in descending order and Xs �CMN�K is the signal subspace; �n � ��IMN�K and Xn �CMN��MN�K� is the noise subspace. It is easy to see that

range�G� � range�Xs� whereG � �g�� � � � �gK �.We assume that user 1 is the desired user. A linear multiuser

detector demodulating the data bit of user 1 in (6) has the form

b��i� � sgn���cHr�i���� (8)

where ���� denotes the real part and c � CMN�� is a complex

weight vector. Next we derive the expressions for the linearzero-forcing detector and the linear MMSE detector in terms ofthe signal subspace parameters.

B. Linear Zero-Forcing Detector

The linear zero-forcing detector for detecting the ith bit ofuser 1 has the form of (8) with the weight vector c � cd suchthat the MAI is completely eliminated at the expense of enhanc-ing the background noise.

Preposition 1: A linear zero-forcing detector for detectingthe data bit b��i� of user 1 from the received signal vector r�i�is given by

cd � Xs��s � ��IK���XH

sg�� (9)

Proof: The autocorrelation matrix can be written as

Rrr � E�r�i�rH�i�� �

KX

k��

gkgH

k � ��IMN � (10)

From (7) to (10) we have

GGH � Xs��s � ��IK�XH

s� Xs��X

H

s� (11)

where�� � �s���IK . Denote �k aK-vector with all entrieszeros except for the kth entry, which is one. In the absence ofnoise, we have

cHd r�i� �

KX

k��

gH� Xs��s � ��IK���XH

s gkbk�i�

�

KX

k��

�H� GH�GGH�yG�kbk�i�

�KX

k��

�H� �GHGHy

��GyG��kbk�i�

�

KX

k��

�H� �kbk�i� � b��i�� (12)

where ���y denotes the Moore-Penrose generalized matrix in-verse [6], the second equality follows from (11), and the fourthequality follows from GHGH

y

� GyG � IK . Therefore, cdis indeed a zero-forcing filter which completely eliminates theMAI. �

C. Linear MMSE Detector

The linear MMSE detector for detecting the ith bit of user 1has the form of (8) with the weight vector c � cm where cmminimizes the output mean-square error (MSE) defined as

MSE�cm� � Efk b��i�� cHmr�i� k�g� (13)

Preposition 2: A linear MMSE detector for detecting the databit b��i� of user 1 from the received signal vector r�i� is givenby

cm � Xs���s XH

s g�� (14)

Proof: Using (6), we obtain

MSE�cm�

� cHmEfr�i�rH�i�gcm � ���cHmEfb��i�r�i�g� �

� cHmRrrcm � ���cH

mg�� � �

(15)

where we have used Efb��i�r�i�g � g�. Since Rrr is posi-tive definite, the linear MMSE detector is obtained by solvingrMSE�cm� � for cm, i.e.,

cm � Rrr

��g�� Xs�

��

sXH

sg� � ����XnX

H

n�g�

� Xs���

sXH

sg�� (16)

where the second equality follows from the eigendecomposition(7) ofRrr, and the third equality follows from the fact that g� �range�Xs� is orthogonal to the noise subspace, i.e.,XH

ng� � .

�

D. Subspace-based Estimation of the Channel Impulse Re-sponse Vector

In practice, the autocorrelation matrix Rrr is unknown andcan be estimated by using

�Rrr �

L

LX

i��

r�i�rH�i�� (17)

where L is the sample size. The consistent estimates ofXs and�s are found in the eigendecomposition of �Rrr

�Rrr � �Xs��s

�XH

s� �Xn

��n�XH

n� (18)

From the previous discussion, we can see that the linear MMSEdetector can be obtained if g� is estimated. We next consider theproblem of estimating the vector g� for user 1, given L trainingdata bits fb��i�gLi��, where L � K.

Preposition 3: Let b� � �b��� � � � b��L��� be the training

data bits of user 1, and � � �r�� � � � r�L�� be the matrix ofL received signal vectors during the training stage. Then theestimate �g� of the desired signal vector g� can be expressed as[7]

�g� � �Xs��s� �X

H

s��H �Xs�

�� �XH

s�b�� (19)

It is noted that �g� is the consistent estimate of g�. That is, ifL � �, then �Xs � Xs, ��s � �s, ��L���

H � Rrr and��L��b� � g� and from (19) we have

�g� �Xs�s�XH

sRrrXs�

��XH

sg�

� Xs�s���

s XH

s g� � XsXH

s g�

� g��

(20)

By replacing Xs, �s, and g� in (14) by the correspondingestimates, we have

cm � �Xs����

s�XH

s�g�

Table 1. The PASTd algorithm [8] for tracking the rank and signal

subspace components of the received signal vector r�i�.

Updating the eigenvalues and eigenvectors of signal subspace f�k , vkgKk��x��i� � r�i�FOR k � � � Ki�� DO

yk�i� � vH

k�i� ��xk�i�

�k�i� � ��k�i� �� � jyk�i�j�

vk�i� � vk�i� �� � �xk�i� � vk�i� ��yk�i�� yH

k�i���k�i�

xi���t� � xk�i�� vk�i�yk�i�END���i� � ����i� ��� k xki�����i� k

� ��N �Ki���

Updating the rank of the signal subspace KiFOR k � � � Ki DO

��k� � �PN

i�k���i�t���N �K����

QN

i�k���i�t��

�

N�K

AIC�k� � �N � k�ln��k���� � �� � k�N � k�ENDKi � arg min

��k�N��AIC�k� � �

IF Ki � Ki�� THEN

remove f�k�t��vk�i�gKi��

k�Ki��

ELSE IF Ki Ki�� THENvKi

�i� � xKi�����i�� k xKi�����i� k

�Ki �i� � ���i�END

� �Xs����

s�XH

s�Xs

��s� �XH

s��H �Xs�

�� �XH

s�b�

� �Xs� �XH

s��H �Xs�

�� �XH

s�b�� (21)

It is seen from (21) that the linear MMSE detector can be ob-tained as long as the signal subspace components are identifiedand it does not require knowledge of the code sequence and thechannel impulse response vector of the desired user. Note that ifthe length of the training sequence L is smaller than the numberof users K, i.e., L � K, the matrix � �XH

s��H �Xs� in (21) is

not full rank. Therefore, the weight vector cm of our proposeddetector can’t be obtained if L � K.Subspace tracking algorithms are modern approaches to the

subspace estimation. They are recursive and update the sub-space in a sample-by-sample fashion. In this paper, the PASTdalgorithm [8] is employed for the adaptive multiuser detectionapplication. The advantages of this algorithm include almostsure global convergence to the signal eigenvalues and eigenvec-tors and low computational complexity O�MNK�. By usinginformation theoretic criteria such as the Akaike informationcriterion (AIC), the rank of the signal subspace or the numberof active users in the channels can also be estimated [9]. Thealgorithm for tracking the rank and signal subspace componentsof the received signal vector r�i� is summarized in Table 1.

IV. NUMERICAL RESULTS

In this section, we provide some simulation examples todemonstrate the performance of the proposed training-basedMMSE multiuser detector. In our simulation, the simulated sys-tem is a synchronous CDMA system over flat Rayleigh fadingchannels. All users are assigned Gold code sequences of chiplength N � � and user 1 is the desired user. We assume thatthe DOAs are uniformly distributed in �� o� o�. It is also as-sumed that a 200-bit data sequence is transmitted by each userand the channel gain as well as array response are fixed overthe data block transmission. In each of Monte Carlo runs, thechannel fading processes and the DOAs are independently and

0 1 2 3 4 5 6 7 8 9 1010

−5

10−4

10−3

10−2

10−1

100

Synchronous CDMA with 10 Users over Flat Rayleigh Fading Channels

Signal−to−Noise Ratio (dB)

Bit

Err

or R

ate

of th

e D

esire

d U

ser

Single antennaThree−element antenna array Conventional detectorProposed detectorLinear zero−forcing detector

Fig. 1. BER of three detectors in flat Rayleigh fading channels as afunction of SNR when the number of users is 10, the length of trainingsequence is 15 bits and the near-far ratio is 10 dB.

randomly generated. The PASTd algorithm [8] that is capableof the on-line rank estimation is used in each of the simulationexamples. The forgetting factor used in this algorithm is �����.In the first example, we compare the performance of three

detectors, i.e., the conventional detector, the proposed training-based MMSE detector, and the linear zero-forcing detector withperfect knowledge of the channel gain and antenna array re-sponse. A CDMA system with single antenna or a three-elementantenna array is considered. It is assumed that there are �� activeusers in the system and the length of training sequence is �� bits.All the interferers are assumed to have equal transmitted pow-ers. The power of each interferer is 10 dB above the power ofthe desired user, i.e., the near-far ratio is 10 dB. The simulationresults in this example are obtained from 10,000 Monte Carloruns. Fig. 1 corresponds to bit error rate (BER) as a functionof signal-to-noise ratio (SNR). It is seen that the conventionaldetector is near-far limited, whereas the performance of the pro-posed detector is much better than that of the conventional detec-tor without knowledge of channel gain and array response. Theperformance of the linear zero-forcing detector is better than thatof the proposed detector. However, the linear zero-forcing de-tector is obtained by assuming perfect knowledge of the channelgain and array response of the desired user.We next consider BER as a function of the near-far ratio. A

CDMA system with single antenna or a three-element antennaarray is considered. In this example, the SNR is fixed to be 10dB and the near-far ratio is varied from 0 dB to 20 dB. Othersimulation parameters are the same as those in the previous ex-ample. Fig. 2 shows the BER as a function of near-far ratio. Wecan see that the proposed detector is near-far resistant. It is animprovement over the conventional detector. The zero-forcingdetector is also robust against the near-far problem and performsbetter than the proposed detector with perfect knowledge of thechannel gain and antenna array response.In the third example, we investigate BER of the proposed de-

tector when the length of training sequence is varied from 10 to

0 2 4 6 8 10 12 14 16 18 2010

−4

10−3

10−2

10−1

100

Synchronous CDMA with 10 Users over Flat Rayleigh Fading Channels

Near−far Ratio (dB)

Bit

Err

or R

ate

of th

e D

esire

d U

ser

Single antennaThree−element antenna array Conventional detectorProposed detectorLinear zero−forcing detector

Fig. 2. BER of three detectors in flat Rayleigh fading channels as afunction of near-far ratio when the number of users is 10, the lengthof training sequence is 15 bits and SNR is 10 dB.

10 15 20 25 30 35 40 45 5010

−5

10−4

10−3

10−2

10−1

100

The Length of Training Sequence Transmitted by the Desired User (bits)

Bit

Err

or R

ate

of th

e D

esire

d U

ser

The Proposed Detector with Different Length of Training Sequence

Single antennaA three−element antenna array

Fig. 3. BER of the proposed detector in flat Rayleigh fading channelsas a function of the length of training sequence when the number ofusers is 10, SNR is 10 dB and the near-far ratio is 10 dB.

50 bits. We assume that there are 10 active users in a CDMAsystem with single or a three-element antenna array. All the in-terferers are assumed to have equal transmitted powers and thenear-far ratio is 10 dB. The SNR is 10 dB and 10,000 MonteCarlo runs are used. It is shown in Fig. 3 that the performanceof the proposed detector can be improved if the length of train-ing sequence is increased.Finally, We consider the performance of the proposed de-

tector in dynamic multiple-access channels, where the interfer-ing users may enter the channels. The performance measureadopted is the output signal-to-interference ratio (SIR), which isdefined as [9]

SIR �E�fcHmr�i�g

VarfcHmr�i�g� (22)

0 200 400 600 800 1000 1200 1400 1600 1800 20002

4

6

8

10

12

14

16

18

20

Synchronous CDMA over Flat Rayleigh Fading Channels, SNR = 20 dB

The number of iterations

Tim

e A

vera

ged

SIR

(dB

)

Single antennaThree−element antenna array

Fig. 4. Performance of the proposed multiuser detector in dynamic mul-tiple access channels where one 20 dB interfering user enters thechannels.

where Varf�g denotes the variance of the random variable andthe expectation is taken with respect to the data bits of the in-terfering users and the noise after the training period. By com-bining the PASTd algorithm with the rank tracking described inTable 1, it is possible to track the number of active users in thechannels and update the rank of the signal subspace. We assumethat single antenna or three-element antenna array is employedat the receiver in a CDMA system. In this example, at t � �,there are six 10 dB interfering users in the channels; at t � ����,a 20 dB interfering user enters the channels. The SNR is fixedto be 20 dB. The time averaged output SIR versus the number ofiterations is shown in Fig. 4. It is seen that the proposed detectorcan adapt rapidly to the dynamic channels when single antennaor three-element antenna array is used at the receiver.

V. CONCLUSIONS

In this paper, we have developed a multiuser detection tech-nique based on the signal subspace estimation for antenna ar-ray synchronous CDMA system over flat fading channels. Withthe aid of a training sequence transmitted by the desired user,the proposed detector does not require knowledge of the codesequences of all users and the channel impulse response. Theadaptive implementation is based on a computationally efficientsignal subspace tracking algorithm. Simulation results show thatthe proposed scheme is near-far resistant and offers significantperformance gains compared to the conventional detector.

REFERENCES[1] S. Verd�u, Multiuser Detection, Cambridge, UK: Cambridge Univ. Press,

1998.[2] R. Kohno et al., “Combination of an adaptive array antenna and a can-

celler of interference for direct-sequence spread-spectrum multiple-accesssystem,” IEEE J. Select. Areas Commun., vol. 4, pp. 675–681, May 1990.

[3] R. Wang and S. Blostein, “Spatial-temporal CDMA receiver structures forRayleigh fading channels,” in Proc. ICC’99, 1999, pp. 549–554.

[4] X. Wang and H. Poor, “Space-time multiuser detection in multipathCDMA channels,” IEEETrans. Signal Processing, vol. 47, pp. 2356–2374,Sept. 1999.

[5] A. Chkeif et al., “Spatio-temporal blind adaptive multiuser detection,”IEEE Trans. commun., vol. 48, pp. 729–732, May 2000.

[6] G. Gloub and C. V. Loan, Matrix Computations, The Johns Hopkins Uni-versity Press, 2nd ed., 1991.

[7] X. Wang and H. Poor, “Robust adaptive array for wireless communica-tions,” IEEE J. Select. Areas Commun., vol. 16, pp. 1352–1366, Oct. 1998.

[8] B. Yang, “An extension of the PASTd algorithm to both rank and subspacetracking,” IEEE Trans. Signal Processing Lett., vol. 2, pp. 179–182, Sept.1995.

[9] X. Wang and H. Poor, “Blind Multiuser detection: A subspace approach,”IEEE Trans. Inform. Theory, vol. 44, pp. 677–690, Mar. 1998.

Getian Ye received the B.S. degree from Northwest-ern Polytechnical University, Xian, PR. China, in1995 and the M.S. degree in electrical engineeringfrom Nanyang Technological University, Singapore,in 1999. He is currently pursuing the Ph.D. degreewith the School of Electrical Engineering at Univer-sity College, The University of New South Wales,ACT, Australia. His research interests include digitalcommunications, image processing and coding, andmultisensor data fusion.

Guoan Bi received a B.Sc. degree in Radio com-munications, Dalian University of Technology, PRC,1982, M.Sc. degree in Telecommunication Systemsand Ph.D. degree in Electronics Systems, Essex Uni-versity, UK, 1985 and 1988, respectively. Since 1991,he has been with the school of Electrical and Elec-tronic Engineering, Nanyang Technological Univer-sity, Singapore. His current research interests includeDSP algorithms and hardware structures and digitalsignal processing for communications.

Zhengyu Li received the B.S. and M. Eng. degreesfrom Northwestern Polytechnical Unviersity, Xian,PR. China in 1996 and 1999, respectively. Since Aug1999, he has been pursuing another M. Eng. in elec-trical and electronic engineering from Nanyang Tech-nological University, Singapore. His research inter-ests include wireless communication and digital signalprocessing.