group-blind intersymbol multiuser detection for downlink cdma with multipath

434 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 2, MARCH 2005

Group-Blind Intersymbol Multiuser Detection forDownlink CDMA With Multipath

Gaonan Zhang, Guoan Bi, Senior Member, IEEE, and Liren Zhang

Abstract—Group-blind multiuser detectors for uplink code-di-vision multiple-access (CDMA) were recently developed by Wangand Høst-Madsen. These detectors make use of the spreadingsequences of known users to construct a group constraint to sup-press the intracell interference. However, such techniques demandthe estimation of the multipath channels and the delays of theknown users. In this paper, several improved blind linear detec-tors are developed for CDMA in fading multipath channels. Theproposed detectors utilize the correlation information betweenconsecutively received signals to generate the corresponding groupconstraint. It is shown that by incorporating this group constraint,the proposed detectors can provide different performance gainsin both uplink and downlink environments. Compared with thepreviously reported group-blind detectors, our new methods onlyneed to estimate the multipath channel of the desired user anddo not require the channel estimation of other users. Simulationresults demonstrate that the proposed detectors outperform theconventional blind linear multiuser detectors.

Index Terms—Cross-correlation matrix, group constraint,group-blind multiuser detection, intersymbol information, multi-path.

I. INTRODUCTION

MULTIUSER detection for code-division multiple-access(CDMA) systems has been proposed to mitigate mul-

tiple access interference (MAI) and enhance channel capacity.The concept of multiuser detection for CDMA systems wasfirst started with the work of [1], where an optimum multiuserdetector for multiple-access Gaussian channels was obtained.However, the optimum detection has never become the main-stream because of its unmanageable computational complexity.Therefore, a number of suboptimum detectors with good per-formance/complexity tradeoffs were developed during the pastdecade. In [2] and [3], the adaptive linear decorrelating, or zero-forcing, detector was reported by using training sequences. Thelinear minimum mean square error (MMSE) detector was an-other popular suboptimum detector, as presented and analyzedin [4] and [5]. In [6], the basic techniques for multiuser detec-tion were summarized, depicting the evolution of this disciplineover the past several years.

Some of the previously mentioned detectors require theinformation of active users, including the spreading waveform,the multipath channel, and the delay of each user. Unfortu-nately, such requirements are practically difficult to meet.

Manuscript received August 22, 2002; revised April 21, 2003; accepted De-cember 20, 2003. The editor coordinating the review of this paper and approvingit for publication is X. Wang.

The authors are with the School of Electrical and Electronic Engi-neering, Nanyang Technological University, 529892 Singapore (e-mail:[email protected]; [email protected]; [email protected]).

Digital Object Identifier 10.1109/TWC.2004.842987

Hence, blind multiuser detection, which only requires the infor-mation of timing and spreading waveform of the desired user,has received much attention since it was first reported in [7].Without considering multipath, a blind minimum output energy(MOE) detector was presented in [7] and then the canonicalsubspace representation of the decorrelating detector and theMMSE detector were reported [8]. In [9], a reduced-rank MOEdetector was proposed by using array processing techniques.However, these detectors cannot work properly when the inter-symbol interference (ISI) cannot be ignored and the multipathchannel is not known. In [10]–[12], several improved subspaceapproaches were developed based on the channel estimation. Areduced computational constrained optimization solution wasproposed in [13] and [14] with a little sacrifice of the perfor-mance (i.e., inferior to the performance of MMSE detector).

In [15], several types of group-blind linear detectors weredeveloped to provide substantial performance gains over theblind linear multiuser detection methods for uplink channels.The basic idea behind the group-blind detectors is to suppressthe interference from the known users by using their spreadingsequences to construct a useful group constraint and to suppressthe interference from unknown users based on subspace-basedblind methods. Since these detectors need to estimate the mul-tipath channels and the time delays of all known users to ob-tain the group constraint, computational complexity has to beincreased for the implementation of these detectors. In addition,errors of channel estimation for each known user may deterio-rate the performance of these detectors.

The main focus of this paper is to develop two improved blindlinear detectors for CDMA systems in fading multipath chan-nels. It is noted that due to the existence of ISI, especially inasynchronous CDMA systems, the current received signal hasindeed certain relationships with its preceeding and succeedingones. Hence, proper utilization of ISI would be beneficial forblind multiuser detection. The key idea of the proposed de-tectors is to construct a cross-correlation matrix by exploitingthe correlation between the consecutive received signals. Byutilizing this cross-correlation matrix, we generate an optimalconstraint similar to the group constraint in [15] and then de-velop two improved blind linear detectors. Unlike the detectorsin [15], our methods only require the channel estimation of theinterested user. Although we mainly focus on the asynchronousCDMA systems in this paper, it will be shown that for both asyn-chronous (uplink) and synchronous (downlink) cases, the pro-posed detectors offer superior performance to some extent overthe conventional blind multiuser detection methods that do notutilize the optimal constraint. The performance improvement ofthe proposed detectors are justified analytically.

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ZHANG et al.: GROUP-BLIND INTERSYMBOL MULTIUSER DETECTION FOR DOWNLINK CDMA WITH MULTIPATH 435

Throughout this paper, scalars are represented in lower casecharacters, vectors in bold lower case, and matrices in boldupper case. The following symbols are also used in our presen-tation:

transpose operator;hermitian transpose operator;

diag diagonal matrix;sgn signum operator;

statistical expectation.

This paper is organized as follows. Section II presents thesignal model of asynchronous CDMA systems with multipath.Section III discusses the conventional blind multiuser detectionmethods and their subspace-based implementations. Section IVproposes two improved blind linear multiuser detectors. In Sec-tion V, simulation examples are provided to demonstrate theperformance of the proposed detectors. Section VI contains theconclusion.

II. SIGNAL MODEL

We consider a DS-CDMA system with users and a normal-ized spreading factor of chips per symbol. The transmittedsignal for the th user is given by

(1)

where is the symbol duration, and andare, respectively, the amplitude and symbol stream of the thuser. The spreading waveform is of the form

(2)

where is the spreading sequenceallocated to the th user and is the normalized chip wave-form with a duration . The discrete-time signal foruser , generated by matched filtering and sampling at the chiprate, is given by

(3)

where for . By propagatingthrough the asynchronous multipath channel that is assumed tohave a maximum length of in terms of chip dura-tion, the received discrete-time signal for user is givenby [16]

(4)

where is the th complex channelgain for user and is the delay of user in chipperiods. Based on (3) and (4), we obtain

(5)

(6)

where is the signature pulse of user , which is a distortedversion of the code due to the multipath gain . Fi-nally, the received signal is the superposition of the signalsfrom all users plus additive complex white Gaussian noise

which has a zero mean and a variance of , i.e.,

(7)

From (6) and denotingas the signature sequences of user , we obtain

, where

.... . .. . .

.... . .

...

(8)

In (8), is an matrix formed from theproduct of the spreading sequences and the amplitude of thuser, and is the th user’s multipath channel vector. From(5)–(8), we denote

... ...

...

It is easy to obtain

(9)

where is the th received signal vector of user . Then, thetotal received user’s signal vector

is given by

(10)

wheresignature matrix of the current bits ofall users including MAI;signature matrix of the previous bits ofall users including ISI;signature matrix of all users;

current bits ofall users;


previousbits of all users;

bits of all users;

independent white Gaussian noisevector.

In asynchronous (uplink) CDMA systems, it is often assumedthat and are mutually independent, and is a tall ma-trix with full column rank as the existance of the delaysof different users [15], [13]. However, for synchronous (down-link) CDMA communication, the full-rank condition on ishardly satisfied as argued in [17]. It can be explained that in syn-chronous CDMA communication ( for ),the nonzero elements in are limited by the number of activeusers and the multipath length , i.e., only an sub-matrix in is nonzero. Hence, the signature matrix will notbe of full column rank unless is less than .

III. CONVENTIONAL BLIND LINEAR MULTIUSER DETECTORS

The canonical form of a linear detector for user can be rep-resented by a correlator vector followed by the re-ceived signal , such that the decision on user is

sgn (11)

Based on (11), we next introduce the two most popular linearmultiuser detectors.

A. Linear Decorrelating Detector

The linear decorrelating detector is known as the linear zero-forcing detector. The decorrelating detector is designed to elim-inate the MAI and ISI completely at the expense of enhancingthe ambient noise. It has the form of (11) with the weight vector

, which is given by

subject to: (12)

where is defined in (10) and is the signature vector of thedesired user.

B. Linear MMSE Detector

The linear MMSE detector has the form of (11) with theweight vector , obtained by calculating the minimumoutput mean squared error, i.e.,

(13)

The two detectors can be resolved by subspace methods. Denotethe autocorrelation matrix of the received signal in (10) as

(14)

By performing the eignedecomposition of the matrix , we ob-tain

(15)

where diag contains the largest eigen-values of and contains the corresponding orthonormaleigenvectors of the signal subspace spanned by . Both

and are, respectively, the eigenvalues andthe orthonormal eigenvectors of the noise subspace. The numer-ical value of depends on the column rank of , which has themaximum value of on the condition that is of full columnrank. By using the eigendecomposition in (15), the detectors in(12) and (13) can be expressed in terms of the subspace formsas described in [8], i.e.,

(16)

(17)

The signal subspace components and can be estimatedfrom the eigendecomposition of the autocorrelation matrix ofthe received signal samples. For blind multiuser detection, thedetector has only the prior knowledge of the spreading sequenceof the desired user. Then, the detectors in (16) and (17) need toestimate and . By computing the average of min-imum eigenvalues of can be easily obtained. Meanwhile,when the desired user is synchronized, its signature vectorcan be estimated in terms of the received signal and in (8),[18], i.e.,

Max-eigenvector (18)

(19)

where is composed of the first rows of . Table Isummarizes the algorithms of the blind decorrelating detectorand blind MMSE detector.

IV. IMPROVED BLIND LINEAR MULTIUSER DETECTORS

The conventional blind multiuser detectors discussed inSection III only know the spreading sequence of the desireduser. However, some performance gains may be obtained ifthe detector has the knowledge of the spreading sequencesof a group of users. Such an expectation has been achievedin [15], where several group-blind linear multiuser detectorswere presented. These detectors first exploit the known users’spreading sequences to estimate their signature vectors basedon the methods in (18) and (19). By using the estimated sig-nature vectors, several group-blind detectors can be developedwith the following group constraint:

(20)


TABLE IALGORITHMS OF SUBSPACE BLIND LINEAR DECORRELATING AND

SUBSPACE BLIND LINEAR MMSE DETECTORS

where is the number of columns in is the detector foruser contains the estimated signature vectors of a group ofknown users, which is a subset of in (10), and is denotedas a dimension column vector with all zero entries exceptfor the th entry. The constraint in (20) is the key idea of thegroup-blind zero-forcing detector and the group-blind hybriddetector in [15]. Equation (20) is equivalent to the followingconstraints:

(21)

where contains the remaining column vectors of exceptfor . In (21), the first constraint is to assure the existence ofthe desired user’s signal in the detector, and the second con-straint is used to zero force the interference of the known user’ssignals. Hence, the essence of the group-blind multiuser detec-tion is to construct a group constraint, which exactly removesthe influence of some interfering users, thereby achieving per-formance gains. Since the group constraint in [15] requires theknown user’s spreading sequences and the estimations of theirmultipath channels and delays, such techniques are only fit forthe CDMA uplink, where the base station knows the spreadingsequences of the intracell users.

Inspired from the previously mentioned group methods, wedevelop two improved blind linear multiuser detectors in thissection. The basic idea for the proposed detectors is to con-struct a new group constraint, which is similar to the group con-straint in (21), by exploiting the intersymbol information of thereceived signals. In this method, we only need the channel esti-mation of the desired user. Therefore, our detectors can provideperformance gain over the blind linear decorrelating detectorand the blind linear MMSE detector for both the downlink anduplink cases.

Let us assume without loss of generality that user 1 is thedesired user, and the receiver is synchronized to user 1, i.e.,

. Then, the signal in (10) can be rewritten as

(22)

where andare, re-

spectively, the signature matrix and received signals of theinterfering users. We next consider the channel estimation of

, which is the signature vector of the desired signal .Meanwhile, we will also estimate since it is required by thefollowing derived detectors. With the assumption that user 1 issynchronized, we denote

.... . .

.... . .

...(23)

.... . .

......

(24)

where both and are matrices.is constructed by the first rows of defined in (8),

and is formed by the remaining rows of plus anzero matrix. According to the definitions

of and in Section II and the denotation in (23) and (24),it is easy to obtain

(25)

(26)

where is the multipath channel of user 1, which is definedin (8). Since and are easy to construct with theprior knowledge of the spreading sequences of the desired user,we only need to consider the estimation of . The multipathchannel vector can be estimated by exploiting the orthogo-nality between the signal subspace and the noise subspace. Since

is in the subspace of in (15), and the subspace of isorthogonal to , we have

(27)

Hence, the estimation of can be uniquely determined by com-puting the principle eigenvector of matrix[12], [19], i.e.,

Max-eigenvector (28)

Based on (28), the signature vectors and can be deter-mined according to (25) and (26).

After the channel estimation, we construct a useful cross-cor-relation matrix , which is given by

(29)


where is generated by the cross correlation of the receivedsignal. It is seen that is an matrix and is irrelevantto the background noise. This matrix plays an important rolein this paper since it will be exploited to develop the proposeddetectors.

Recalling that the basic idea of the group-blind multiuserdetection is to suppress the interference of some known usersbased on the group constraint, we naturally expect that the moresubspace of the interfering users the constraint removes, thebetter performance improvement the detector achieves. In thissection, we utilize to develop a constraint similar to the groupconstraint in (20) for the desired user, which is given by

(30)

Let us now consider the properties of the above constraint withthe following propositions to analytically demonstrate the per-formance improvement brought by the proposed constraint.

Proposition 1: Based on the signal model in (10) and thedefinition of in (29), the constraint impliesthe constraint , i.e.,

(31)

Proof: The Proof of Proposition 1 is straightforward bydirectly substituting the constraint into ,i.e.,

(32)

where the last equality follows from the definition of in (10),i.e., . The constraint is theoptimal group constraint for group-blind multiuser detectionsince it ensures that the detector is orthogonal to the sub-space spanned by all interfering users except the desired user.The result of Proposition 1 shows that the proposed constraintis the necessary condition of the optimal group constraint. Itmeans that the proposed constraint will not re-move the subspace spanned by the desired user and it is pos-sible to achieve the optimal group constraint based on this con-straint. Therefore, we are interested in the feature of the pro-posed constraint and want to know if this constraint implies theoptimal group constraint or to what extent it removes the sub-space spanned by the interfering users. Propositions 2–4 sum-marize the proposed constraint’s ability to resist the interferencebrought by the interfering users in different cases.

Proposition 2: The constraint implies the con-straint if has full column rank .

Proof: We proceed with a proof by contradiction. As-suming that and , then canbe expressed as

(33)

where is a nonzero dimensional columnvector, and and are, respectively, the first entries and

the last entries of . By denoting and substi-tuting (33) into , we have

(34)

where the third right arrow follows from the fact that

and the fifth right arrow is based on the fact that is of fullcolumn rank. Hence, (34) contradicts (33).

Proposition 2 shows that the proposed constraint is equiva-lent to the optimal group constraint which removes all MAI andISI on condition that is of full column rank. Recalling that thesignature matrix in the asynchronous (uplink) CDMA model,which is given in Section II, is assumed to be full rank, we nat-urally expect that the proposed constraint provides the greatestperformance gain in the CDMA uplink.

As mentioned before, the full-rank condition on is hardlysatisfied in the CDMA downlink, where the intracell users aretransmitted synchronously and the intercell users are trans-mitted asynchronously. In this case, we divide into and

according to (10), where contains the subspace spannedby the desired user and MAI, and contains the subspacespanned by ISI. In the CDMA downlink, most treatmentsassume that is full rank and independent of , whileis not full rank when the number of active users exceeds themaximum length of the multipath channel .

Proposition 3: For nonfull-rank , the constraintimplies if is independent of with full

column rank.Proof: The proof is similar to the Proof of Proposition 2,

where

(35)

where the fourth arrow follows from the fact thatrange and is independent of , and the last arrow isbased on the fact that is of full column rank.

Proposition 3 shows that the proposed constraint can totallyremove the ISI with nonfull-rank in the CDMA downlink.In particular, if we assumed that is independent of the other


vectors in , a more satisfying result can be achieved by theproposed constraint.

Proposition 4: For nonfull-rank , the constraintimplies if is independent of and

is independent of the rest vectors in .Proof: Similar to the Proof of Proposition 3, we consider

(36)

where the third arrow is derived from the independence betweenand , and the last arrow is based on the fact that cannot

be constructed by the linear combination of the other vectors in. Proposition 4 shows that the proposed constraint can remove

all MAI if the signature vector of the desired user in ISI isindependent of the other signature vectors in ISI. In this case, wedo not require the full rank of . However, if we also assumedthat isoffull rank, theconstraint inProposition4caneliminateboth MAI and ISI according to Proposition 3, which means ourproposed constraint may achieve the optimal group constraint oncertain conditions even without a full-rank .

The results of Propositions 2–4 show that the proposed con-straint is able to remove the MAI and ISI to dif-ferent extents according to the properties of the signature matrix

. We then develop two improved blind linear detectors by con-sidering this constraint.

A. Improved Blind Linear Decorrelating Detector

The improved blind linear decorrelating detector considersthe minimization of the following constraint cost function:

subject to: (37)

where is the proposed detector. The first constraint in (37)is used to keep the information of the desired user, and thesecond constraint is used to suppress the MAI and ISI. Thetwo constraints compose an equivalent group-blind constraint to(21). It was shown in [15] that the improved blind linear decorre-lating detector in (37) is equivalent to the decorrelating detectorin (12), i.e., . Proposition 5 presents the resolution ofthe proposed linear decorrelating detector.

Proposition 5: Based on the eigendecomposition in (15), thesolution of the improved blind linear decorrelating detector in(37) is given by

(38)where

Proof: See Appendix A.

B. Improved Blind Linear Hybrid Detector

The improved blind linear hybrid detector considers the fol-lowing constrained optimization problem:

range

subject to (39)

where is the proposed detector. Proposition 6 gives theimplementation for the proposed blind linear hybrid detector.

Proposition 6: Based on the eigendecomposition in (15), thesolution of the improved blind linear hybrid detector in (39) isgiven by

(40)

where

Proof: See Appendix B.In the above results, two improved blind linear detectors are

expressed in subspace forms. In these detectors, the influence ofthe interfering users are mitigated by the two constraints, whichare constructed from the correlation matrix and the channelestimation of the desired user. After suppressing the interferingsignals, the signals of the desired user are identified by the trans-formation of the subspace based on the zero-forcing or the hy-brid criteria. Unlike the group constraint in [15], our proposedconstraints utilize the correlation matrix to improve the per-formance of the detectors and then avoid the channel estima-tion of other users except the desired user. Therefore, our pro-posed detectors can be implemented blindly for both downlinkand uplink CDMA channels, while the method in [15] is onlysuitable for uplink channels. In addition, the performance gainprovided by the proposed constraint depends on the property ofthe signature matrix . When is of full column rank (up-link-CDMA), the proposed constraint can totally remove MAIand ISI to achieve the best performance gain. When is notof full rank (downlink-CDMA), the proposed constraint can atleast remove the influence of ISI. Table II summarizes the algo-rithms of the proposed detectors.

V. SIMULATION RESULTS

In this section, simulation results are provided to demonstratethe performance of the improved blind linear detectors. We testthe proposed methods in a CDMA system with spreading gain

. The spreading sequences for all users are generatedby Gold codes. Both the asynchronous and synchronous casesare considered. In each case, user 1 is assumed to be the desireduser which is synchronized, and each user has multipathdelays. The multipath gains in each user’s channel are randomlychosen and kept fixed, which have been normalized with equalpower. In addition, we simulate a severe near–far case in whichthe power of each interfering user is 10 dB more than that ofthe desired user. In each simulation, an eigendecomposition isperformed on the autocorrelation matrix of the received signals.


TABLE IIALGORITHMS OF IMPROVED BLIND LINEAR DECORRELATING DETECTOR

AND BLIND LINEAR HYBRID DETECTOR

The length of the signal frame is . The algorithmic de-tails of the matrix operations and parameter estimation involvedin computing the various detectors can be seen from the Tables Iand II.

A. Performance of the Proposed Detectors for AsynchronousCDMA Systems

In the CDMA uplink, the received signal at the base stationis created by the asynchronous intercell and intracell users. Foran asynchronous CDMA system, we assume that the delay ofeach interfering user is randomly distributed in where

is the chip period. Hence, the maximum delay spread is lim-ited within one symbol interval. Meanwhile, it is also assumedthat the signature matrix in (10) is of full column rank asthe existence of the delays of interfering users. We compare theperformance of six detectors, i.e., the conventional blind decor-relating and MMSE detectors implemented by the algorithmsin Table I, the group-blind zero-forcing and hybrid detectors in[15] and the proposed improved blind linear decorrelating andhybrid detectors implemented by the algorithm in Table II.

Fig. 1 compares the bit-error rates (BERs) of various detec-tors, which only require us to estimate the channel of the de-sired user, in an asynchronous system with seven users. It showsthat the proposed blind linear decorrelating and hybrid detectorsprovide substantial BER improvement over the conventionaldecorrelating and MMSE detectors. It is also observed that thereexist trivial differences in performance between the proposeddecorrelating detector and hybrid detector. This is because thetwo detectors use the same constraint to suppress the ISI and

Fig. 1. BER comparison of different detectors for asynchronous CDMA (P =500; K = 7).

Fig. 2. BER comparison of different detectors for asynchronous CDMA (P =500;K = 10).

MAI. When is of full rank, the proposed constraint totallyremoves the ISI and MAI. Hence, both proposed detectors canzero force all signals from interfering users to achieve similarperformance.

We next compare the performance of the group-blind de-tectors in [15] with our proposed detectors. The results of thesimulation are shown in Figs. 2 and 3. In Fig. 2, a ten-userasynchronous system is considered. It is assumed that there areeight known intracell users and two unknown intercell usersfor the group-blind detectors. It is seen that the group-blindzero-forcing detector is a little superior to the proposed decorre-lating detector. The group-blind hybrid detector also achieves abetter performance than the proposed hybrid detector. It shouldalso be noted that our proposed detectors only need the channelestimation of the desired user, while the group-blind detectorsrequire the channel estimation of all known users. Fig. 3 con-siders a 15-user system with eight intracell known users andseven intercell unknown users. Such situation may occur at theedge of the cell, where the number of the intracell users is close


Fig. 3. BER comparison of different detectors for asynchronous CDMA (P =500; K = 15).

to the number of the intercell users. It is observed that both thegroup-blind detectors and the proposed detectors have a deteri-oration compared with Fig. 2. In this case, the performance ofproposed decorrelating detector and hybrid detector is similar tothat of the group-blind zero-forcing detector and the group-blindhybrid detector, respectively. It means that the group-blind de-tectors are more sensitive to the number of the unknown users.This is because the group-blind detectors use the channel es-timation of all known users to construct the group constraintto improve the performance. Hence, the more users the detec-tors know, the better performance gain the detectors achieve.However, our proposed detectors utilize the cross correlation ofthe received symbols and the channel estimation of the desireduser to construct the corresponding constraint. Such constraint,which can remove both MAI and ISI in asynchronous CDMAcommunication, is irrelevant to the number of unknown users.

B. Performance of the Proposed Detectors for SynchronousCDMA Systems

Next, we consider the performance of the proposed detectorsin the CDMA downlink, i.e., synchronous CDMA. In down-link-CDMA communication, the signal received by the desireduser is made up of the synchronous intracell users and someasynchronous intercell users. In this case, the signature matrix

, and hence the signature matrix , may not be of full rankwhen the number of synchronous intracell users surpasses themultipath length .

It is noted from the above simulation results that the perfor-mance of the conventional blind MMSE detector and the pro-posed hybrid detector are close to the performance of the con-ventional blind decorrelating detector and the proposed decor-relating detector, respectively. However, the two decorrelatingdetectors require the estimate of the noise level. In addition, thegroup-blind detectors in [15] cannot be used for downlink chan-nels because they need to know the spreading sequences of theother intracell users. Therefore, we only compare three detec-tors for synchronous CDMA communication, i.e., the conven-tional MMSE detector, the proposed hybrid detector, and the

Fig. 4. BER comparison of different detectors for synchronous CDMA (P =500;K = 10).

true linear MMSE detector. The true linear MMSE detector isgiven by

(41)

where is created by the average of 10 000 received samplesat a high signal-to-noise ratio (SNR).

Fig. 4 compares the performance of the above three detectors.The system is assumed to have eight synchronous intracell usersand two asynchronous intercell users. It is seen from this figurethat the proposed hybrid detector still achieves a better perfor-mance than that of the conventional MMSE detector, althoughboth are inferior to the true linear MMSE detector. However,the performance gain of the proposed hybrid detector over theconventional MMSE detector is not as obvious as that in Fig. 1.This is because the constraint in the proposed hybrid detectorcan only remove the ISI for synchronous CDMA, while it canremove both MAI and ISI for asynchronous CDMA. In Fig. 5,the system has seven asynchronous intercell users and eight syn-chronous intracell users. In this case, the signature matrixhas more column rank since the number of the asynchronoususers is increased. It implies that the proposed constraint canremove more MAI. It is observed as expected that the proposedhybrid detector provides more performance gain over the con-ventional MMSE detector compared with the result of Fig. 4 andis closer to the true linear MMSE detector.

Theoretically, both the conventional detectors and the pro-posed detectors converge to the true linear MMSE detector athigh SNR when the signal frame length is infinite. However,for a finite frame length , the proposed detectors significantlyoutperform the conventional detectors no matter in the CDMAuplink or in the CDMA downlink. The reason for such a perfor-mance improvement is that more multiuser environment infor-mation is incorporated in the derivation of the detection process.

VI. CONCLUSION

New improved blind linear multiuser detection techniques areproposed for both downlink of the synchronous CDMA system


Fig. 5. BER comparison of different detectors for synchronous CDMA (P =500; K = 15).

and uplink of the asynchronous CDMA system. The new tech-niques make use of an important cross correlation matrix be-tween the adjacent symbols as a constraint to suppress the influ-ence from the interfering users. Two improved blind linear mul-tiuser detectors are developed based on different optimal criteriawith the constraint constructed by the cross correlation matrix.The proposed detectors can be implemented blindly only withthe channel estimation of the desired user. Simulation resultsshow that the proposed improved blind linear detectors providea substantial performance gain over the conventional blind de-tectors.

APPENDIX APROOF OF PROPOSITION 5

Referring to the method in [15], we use Lagrange multiplierto resolve the problem in Proposition 5. For the cost function in(37), we have

(42)

where denotes the pseudo-inverse. Combining the constraintwith (42), we obtain

(43)

Hence, it is easy to obtain

(44)

Substituting (44) into (42), we have

(45)

Combining the constraint with (45) and denoting

, we obtain

(46)

According to the eigendecomposition in (15), the matrixcan be replaced by . We then

rewrite and as

(47)

where is denoted as . Sub-stituting and into (44), it is easy to obtain

(48)

Finally, the proposed detector is given by

(49)

APPENDIX BPROOF OF PROPOSITION 6

Similar to the Proof of Proposition 5, we use Lagrange multi-plier to resolve the problem in Proposition 6. For the cost func-tion in (39), we have

(50)

Denoting and substituting into (50), we obtain

(51)

Referring to the methods in (44)–(46) and denoting, we have

(52)

It is seen from (22) and (29) that and range ,

i.e., and . Thus, by denoting


, we can rewrite and in the followingsubspace forms:

(53)

Finally, the proposed detector can be given by

(54)

ACKNOWLEDGMENT

The authors are grateful to the referees and the associate ed-itor for their careful review, insightful comments, and construc-tive suggestions.

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Gaonan Zhang received the B.S. degree in elec-trical engineering from Xi’an Highway University,Xi’an, China, in 1995 and the M.S. degree in systemengineering from Xi’an Jiaotong University, in 2000.He is working toward the Ph.D. degree at the Schoolof Electrical and Electronic Engineering, NanyangTechnological University, Singapore.

His research interests include multiuser detectionfor CDMA systems and advanced signal processingfor wireless communication. Since May 2003, he hasserved as a Research Staff member in Flextronics

Asia Design Pte Ltd., Singapore. His current research includes the design andthe development of GSM and third-generation CDMA products.

Guoan Bi (M’85–SM’89) received the B.Sc. degreein radio communications from the Dalian Universityof Technology, China, and the M.Sc. degree intelecommunication systems and Ph.D. degree inelectronics systems from Essex University, U.K., in1982, 1985, and 1988, respectively.

Since 1991, he has been with the School ofElectrical and Electronic Engineering, NanyangTechnological University, Singapore. His currentresearch interests include DSP algorithms andhardware structures and digital signal processing for

communications.

Liren Zhang received the B.Eng. degree from Shan-dong University, the M.Eng. degree from the Univer-sity of South Australia, and the Ph.D. degree from theUniversity of Adelaide, Australia, in 1982, 1988, and1990, respectively, all in electrical engineering.

He is currently an Associate Professor in theSchool of Electrical and Electronic Engineering,Nanyang Technological University, Singapore. From1990 to 1995, he was a Senior Lecturer in theDepartment of Electrical and Computer SystemsEngineering, Monash University, Australia. He has

vast experience as an Engineer, Academic, and Researcher in the field ofmultimedia communications, switching and signaling, teletraffic engineering,network modeling and performance analysis for ATM networks, high-speeddata networks, mobile networks, satellite networks, and optical networks. Hehas published more than 100 research papers in international journals andconferences.

Dr. Zhang has been the Associate Editor for the Journal of Computer Com-munications since 2000.

group-blind intersymbol multiuser detection for downlink cdma with multipath

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