blind equalization and multiuser detection in dispersive cdma channels

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 1, JANUARY 1998 91

Blind Equalization and MultiuserDetection in Dispersive CDMA Channels

Xiaodong Wang and H. Vincent Poor,Fellow, IEEE

Abstract—The problem of blind demodulation of multiuserinformation symbols in a high-rate code-division multiple-access(CDMA) network in the presence of both multiple-access inter-ference (MAI) and intersymbol interference (ISI) is considered.The dispersive CDMA channel is first cast into a multiple-input multiple-output (MIMO) signal model framework. By ap-plying the theory of blind MIMO channel identification andequalization, it is then shown that under certain conditions themultiuser information symbols can be recovered without anyprior knowledge of the channel or the users’ signature waveforms(including the desired user’s signature waveform), although thealgorithmic complexity of such an approach is prohibitively high.However, in practice, the signature waveform of the user ofinterest is always available at the receiver. It is shown that byincorporating this knowledge, the impulse response of each user’sdispersive channel can be identified using a subspace method.It is further shown that based on the identified signal subspaceparameters and the channel response, two linear detectors thatare capable of suppressing both MAI and ISI, i.e., a zero-forcing detector and a minimum-mean-square-errror (MMSE)detector, can be constructed in closed form, at almost no extracomputational cost. Data detection can then be furnished byapplying these linear detectors (obtained blindly) to the receivedsignal. The major contribution of this paper is the developmentof these subspace-basedblind techniques for joint suppression ofMAI and ISI in the dispersive CDMA channels.

Index Terms— High-rate CDMA, intersymbol interference,multiple-access interference, multiuser detection.

I. INTRODUCTION

T HERE is currently a significant interest in the designof wireless code-division multiple-access (CDMA) net-

works that would give users access to data rates on the order of1–10’s of megabits per second [3], [12], or even higher asyn-chronous transfer mode (ATM)-compatible rates for wirelessmultimedia applications [10]. Due to bandwidth constraints, itis anticipated that for high-rate wireless data applications, theprocessing gain may be an order of magnitude lower than thatin the voice communications [3]. In the design and analysisof the low-rate CDMA systems, the presence of intersymbolinterference (ISI) due to the multipath nature of the wirelesschannels is often neglected [14]. However, for the high-rate

Paper approved by S. L. Miller, the Editor for Spread Spectrum of theIEEE Communications Society. Manuscript received April 30, 1997; revisedSeptember 8, 1997. This work was supported by the the U.S. Office of NavalResearch under Grant N00014-94-1-0115. This paper was presented in part atthe Sixth Communication Theory Mini-Conference (CTMC’97) in conjunctionwith GLOBECOM’97, Phoenix, AZ, November 5–7, 1997.

The authors are with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(98)01067-8.

CDMA systems, the ISI is no longer negligible and, in fact,together with the multiple-access interference (MAI) whichis inherent to any nonorthogonal CDMA system, constitutesthe major impediment to the overall system performance. Inthis paper, we developblind techniques for joint channelequalization and multiuser detection in such dispersive CDMAchannels.

It has been demonstrated that multiuser detection providesvery substantial performance gains over detection techniquesconventionally used in multiple-access channels. Recently,blind techniques for adapting the linear multiuser detectorshave been proposed [5], [19], [22]. These blind techniquesessentially allow one to use a linear multiuser detector fora given user with no knowledge beyond that required forimplementation of the conventional detector for that user. Thatis, the informational complexity of these detectors is identicalto that of the conventional detector, while the performanceof the blind multiuser detector is substantially better. Theblind method in [5] for adapting the linear minimum-mean-square-error (MMSE) detector is based on minimizing theoutput energy of a linear filter subject to a constraint. Thatwork was extented in [19] to incorporate the multipath effect.The blind approach proposed in [22] is based on the signalsubspace tracking technique and can blindly adapt both thedecorrelating (i.e., zero-forcing) detector and the linear MMSEdetector. It has been shown that a major limitation of theminimum output energy (MOE) approach to blind multiuserdetection is that there is a saturation effect in the steadystate, which causes a significant performance gap between theconverged blind MOE detector and the true MMSE detector[5], [13]. In the subspace-based blind approach, however, thelinear detectors are constructed in closed form once the signalsubspace components are computed, and it is seen in [22]that the subspace-based detector outperforms the blind MOEdetector in the steady state.

The method in [22] is targeted primarily at the low-rateCDMA systems, e.g., single-path channels or frequency-selective multipath channels with negligible ISI. In thispaper, we extend that work and develop subspace-basedblind techniques to combat both MAI and ISI in the high-rate dispersive CDMA systems. Several recent works haveaddressed the use of the subspace-based MUSIC-type ofmethods for parameter estimation in CDMA systems, such asdelay and channel estimation [1], [17]. The major contributionof this work is the development of blind techniques forblind joint suppression of MAI and ISI in dispersive CDMAchannels.

0090–6778/98$10.00 1998 IEEE

92 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 1, JANUARY 1998

In this paper, the communication channels are assumed tobe slowly varying, such that they remain constant for a certainnumber of symbol intervals. The signal processing techniquesdeveloped here process the received signal in a block-by-block fashion and therefore incur a larger delay comparedwith the symbol-by-symbol processing methods. Hence, thescenarios considered here are delay-insensitive CDMA com-munication applications in slowly varying channels, such asindoor wireless data communications.

The rest of the paper is organized as follows. In Section II,we introduce the dispersive CDMA channel model, and castit into a multiple-input multiple-output (MIMO) signal modelframework. In Section III, by applying the theory of blindMIMO channel identification and equalization, we show thatunder certain conditions, the multiuser information symbolscan be recovered without any prior knowledge of the channelresponse and the users’ signature waveforms, although thealgorithmic complexity of such an approach is prohibitivelyhigh. In Section IV, we develop subspace-based blind channelestimation techniques for both the forward link and the reverselink, which exploit the knowledge of the signature waveformof the desired user available at the receiver. In Section V,we further show that based on the identified signal subspaceparameters and the channel response, a linear zero-forcingdetector and a linear MMSE detector for joint suppression ofMAI and ISI can be constructed in closed-form, with almostno extra computational overhead. In Section VI, several relatedissues, including initial delay estimation, signal subspace rankestimation, and robustness against channel estimation errors,are discussed. In Section VII, we provide some simulationexamples to demonstrate the performance of the proposedblind channel equalization and multiuser detection techniques.Section VIII contains the conclusion.

II. SIGNAL MODELS

A. Dispersive CDMA Channel Model

Consider a -user binary communication system, employ-ing normalized modulation waveforms andsignaling through their respective dispersive channels withadditive white Gaussian noise. The transmitted signal due tothe th user is given by

(1)

where denotes the information symbol interval and ,, and denote, respectively, the amplitude, symbol

stream, and delay of theth user In the direct-sequence spread-spectrum (DS/SS) multiple-access format, theuser signaling waveforms are of the form

(2)

where is the processing gain, is a signaturesequence of 1’s assigned to the th user, and is anormalized chip waveform of duration .

The th user’s signal propagates through a dispersivechannel with complex impulse response . At the receiver,the received signal due to theth user is then given by

(3)

where denotes convolution, and

(4)

The total received signal at the receiver is the superpositionof the data signals of the users plus the additive whiteGaussian noise, given by

(5)

where is a zero mean complex white Gaussian noise withpower spectral density .

The signal model given by (3)–(5) represents a dispersiveasynchronous CDMA channel, which is typical for thereversechannel (i.e., mobile to base station) of a CDMA network.The forward channel(i.e., base station to mobile) of a CDMAnetwork is a special case of this model, where the data signalsof the users are synchronous, i.e.,and they propagate through a single dispersive channel, i.e.,

.

B. Discrete-Time MIMO Model

We proceed to develop a discrete-time MIMO signal modelfor the dispersive CDMA channel, which is key to developingthe blind equalization and multiuser detection algorithms fordispersive CDMA channels in the subsequent sections.

Suppose that in (4) has finite support of length, i.e., it is zero outside the interval . At

the receiver, the received signal is first filtered by a chip-matched filter and then sampled at the chip rate. The resultingdiscrete-time signal component due to theth user at the thchip period of the th symbol interval is then

(6)

(7)

WANG AND POOR: BLIND EQUALIZATION AND MULTIUSER DETECTION 93

where in (6), we denote

(8)

and in (7) we denote for

(9)

(10)

In (10), is the composite signal waveform of thethuser, resulting from the convolution of the original spreadingsequence with the total channel response .The received discrete-time signal during theth chip periodof the th symbol interval is then given by

(11)

where

Define the following quantities:

......

...

......

...

...

Then (7) and (11) can be written respectively as

(12)

(13)

Define By stacking successive sam-ples of the received data (where is called the smoothingfactor), we further define the quantities shown at the bottomof the page.

The block Toeplitz matrix is called the generalizedSylvester matrix. We can then write (12) and (13) in matrixform as

(14)

(15)

Suppose that a block of received data samplesare processed together. Then define the

following block Hankel matrices:

By (14) and (15) we have

(16)

(17)

......

...

......

......

......


Fig. 1. MIMO modeling of the dispersive CDMA channel.

The MIMO model (14) and (15) for the dispersive CDMAchannel is illustrated in Fig. 1. We still need to specify the val-ues of and First, in order to achieve blind identifiability,the Sylvester matrix should have full column rank, whichnecessitates that it be a “tall” matrix, i.e., .Therefore, should satisfy

(18)

As will be shown later [cf. (26)], another necessary conditionthat should satisfy is

(19)

The number of samples should chosen such that the datasymbol matrix has full row rank [cf. (C3) in Proposition1], which necessitates that it be a “wide” matrix, i.e.,

i.e.,

(20)

III. B LIND MULTIUSER DETECTION

WITHOUT USING SPREADING SIGNATURES

The basic problem here is to estimate the transmittedmultiuser information symbols from the received data .From the data formulation in the previous section, it is seenthat this problem is equivalent to separating multiple signals ina finite-impulse-response (FIR)-MIMO channel. The problemof blind MIMO channel identification and equalization arisesin array signal processing, and has received considerable recentinterest [6], [7], [18], [21]. In this section, by applying thetheory of blind MIMO channel identification and equaliza-tion, we show that under certain conditions it is possible torecover the multiuser information symbols without knowingthe channel response and the users’ signature waveforms.The results in this section are mainly of theoretical interest,due to the high complexity of the algorithm for separatingan instantaneous linear mixture of finite alphabet signals.Moreover, in practice, the signature waveform of the desireduser is always available at the receiver, and this informationcan be exploited to develop more efficient and powerful blindinterference suppression algorithms, as will be discussed inthe subsequent sections.

A. Blind Identifiability

Let be the -transform of For nowwe assume that there is no noise and consider the issue ofidentifying the channel response directly from thedata matrix in (16). The following result, found in [7],states that under certain conditions the total channel responsecan be identified up to a nonsingular ambiguity matrix.

Proposition 1: [7] Suppose that the following conditions aresatisfied:

(C1) is of full column rank;(C2) is of full column rank for all ;(C3)(C4) has full row rank.

Then if there are a Sylvester matrix and a matrixwith the same dimensions as and , respectively, suchthat then

(21)

where is a nonsingular matrix and arethe filter coefficient matrices associated with

Now suppose that the channel has been identified up to anonsingular ambiguity matrix, i.e., we have obtained

where for somenonsingular matrix where denotes the Kronecker matrixproduct and denotes an identity matrix. Using (14),we have

(22)

where denotes the pseudomatrix inverse and is avector of the form

where is a vector. Denote

andThen (22) indicates that, at this point, we have identified

the multiuser data symbols up to an ambiguity factor ,i.e., Given the fact that the data symbols belongto a finite alphabet, e.g., , they are identifiable withsufficient data samples [25]. We can obtainby solving thefollowing optimization problem:

(23)


where the elements of are constrained to be 1. This is anonlinear separable optimization problem with mixed integerand continuous variables. It is proven in [4] that this optimiza-tion can be carried out in two steps. First, (23) is minimizedwith respect to , which is unconstrained minimization,then, substituting the solution backinto (23) and minimizing with respect to by enumeration.It is clear that the complexity of this procedure is exponentialin ( ), which is certainly prohibitive. Notice that severaliterative algorithms have been proposed in [18] to solve suchproblems of separating an instantaneous linear mixture of finitealphabet signals. However, our simulations indicate that thesealgorithms fail to provide satisfactory performance, at least forthis particular application (e.g., the probability of error is wellabove 10% even in the noise-free case.).

We next outline briefly a subspace method for computingthe MIMO channel estimate up to a nonsingular ambiguitymatrix factor as in (21).

B. Subspace Method for Channel Identification

The following subspace algorithm for MIMO FIR channelidentification is an extension of the algorithm in [11] tomultiple signals [20]. Let be a basis of the left null spaceof Since is of full row rank, we have

(24)

Since it is assumed that has full column rank andhas full row rank, both equal to , it thenfollows that the also has rank and therefore its left nullspace has dimension Now partition thematrix as

(25)

where has dimension forDenote

...

......

......

......

Then, it can be readily verified that

(26)

If the matrix is tall, i.e., [Notethat this condition gives rise to the second condition that thesmoothing factor should satisfy in (19).], then its right nullspace specifies up to a right nonsingular matrixfactor , i.e.,

IV. BLIND CHANNEL ESTIMATION BASED

ON USER SPREADING SIGNATURES

It is seen from the previous section that without the knowl-edge of the users’ spreading waveforms, the composite sig-nature waveforms of all users can be identified only up to anonsingular ambiguity matrix factor. To further resolve thisambiguity, the finite alphabet property of the input data hasto be exploited, and the corresponding algorithm is prohib-itively expensive. Furthermore, the practical utility of suchan approach is probably restricted to noncooperative appli-cations such as eavesdropping. For any conventional CDMAcommunication applications, the signature waveform of thedesired user is always available at the receiver. Specifically,in the forward channel, the receiver at the mobile unit hasthe knowledge of its own spreading signature, whereas, inthe reverse channel, the receiver at the base station hasthe knowledge of all users’ spreading signatures. In thissection, we show that by incorporating the knowledge of users’signature waveforms, it is possible to determine uniquely theimpulse response of each user’s dispersive channel through asubspace method. Note that the channel estimate (and thus thecomposite signature waveform estimate) obtained through ablind method always has an arbitrary phase ambiguity, whichcan be easily resolved by differentially encoding and decodingthe information data.

A. Spectral Decomposition

Consider the autocorrelation matrix of thereceived signal Since the additive noise is assumedto be white and independent of the signals, from (15) we get

(27)

where we have used the assumption that each user’s informa-tion symbols are independently identically distributed (i.i.d.),and the symbol streams of different users are independent,therefore (recall that

It is also assumed that the Sylvester matrix has fullcolumn rank In [7] it is shown that has full column ifthe conditions (C1)–(C3) in Proposition 1 hold.

Let be the eigenvalues of Sincehas full rank , the signal component of the covariance

matrix in (27), i.e., , has rank , therefore we have

for

for

By performing an eigendecomposition of the matrix weobtain

(28)

wherecontains the largest eigenvalues of in

descending order, contains the correspondingorthonormal eigenvectors, and and

contain the orthonormal eigenvectorsthat correspond to the eigenvalue It is easy to see that


The range space of is calledthe signal subspaceand its orthogonal complement, thenoisesubspace, is spanned by Now it is clear that the left nullspace of in (24) is given by

(29)

Define the diagonal matrixFrom (27) and (28) we

obtain

(30)

B. Forward Link Channel Estimation

In the forward link the signals of all users are synchronousand propagate through the same dispersive channel. We as-sume that the receiver at the mobile unit has synchronizedwith the base station transmitter, i.e.,We also assume that the total channel response definedin (8) has support of length where

Notice that the matrix in (26) has the formof where isthe composite signature of theth user, with definedas [cf. (10)]

(31)

Therefore, can be expressed in matrix form as

(32)

where

......

......

......

......

...

...

Now substituting (32) into (26), we have

(33)

Therefore is the right null vector of the matrixFurthermore, under certain conditions specified by the follow-ing result, the right null vector of the matrix uniquelydetermines the channel response vectorup to a multiplicativeconstant.

Proposition 2: Suppose that in addition to the four condi-tions (C1)–(C4), as in Proposition 1, the following conditionalso holds.

(C5) The matrix has fullcolumn rank.

Then if there is a vector such that , thenwhere is a scalar constant.

Proof: Since conditions (C1)–(C4) hold, by Proposition1, we have

Now by assumption (C5), we must have andThus,

Next we consider computing the channel response vectorat the th mobile unit. Assume a general case in which the

th user has the knowledge of the signature sequences of someusers whose index form a set withSince in practice only the sample estimateis available, wechoose to solve (26) in the least squares sense. This leads tothe following minimization problem:

(34)

where Therefore, is given by theeigenvector corresponding to the smallest eigenvalue of

C. Reverse Link Channel Estimation

We now consider the reverse link scenario in which thebase station estimates the channel response of each reverselink. Assume that the impulse response of the physicalchannel for the th user is zero outside the intervalSince , and has support only on the interval

, using (4) we obtain the total channel response as

(35)

That is, is nonzero only on the interval

Denote and It thenfollows that the total channel response vector in (8) isnonzero only for Assume that the channeldelay spread is known and the receiver at the base stationhas estimated the delay of each user within a chip period,i.e., and are available. Let be the submatrix of


consisting of the columns indexed from to and letbe the nonzero subvector of consisting of the elements

indexed from to Then we have

(36)

Similarly as before, given an estimateof , an estimate ofis given by the eigenvector corresponding to the smallest

eigenvalue of the matrix

V. BLIND JOINT SUPPRESSION OFMAI AND ISI

It is demonstrated in the previous section that the dispersivechannel of the desired user can be estimated based on the spec-tral decomposition of the autocorrelation matrix of the receivedsignal. In this section, we consider further constructing twoforms of linear detectors for joint suppression of both MAIand ISI, based on the identified signal subspace parametersand the channel response.

Consider the problem of detecting theth user’s informationsymbols from the received signal A linear detectorfor this purpose is in terms of a correlator followed by a hardlimiter, such that the is demodulated according to thefollowing rule:

(37)

whereSuppose that the channel response is known. The

composite signature waveform is normalized to have aunit norm, i.e., define where

Then the total channel response has

the form Denote

and Let and be the filtercoefficient matrix and the Sylvester matrix, respectively,corresponding to . Then we have

(38)

(39)

Substituting (39) into (27), we get

(40)

Denote a -vector with all entries zeros except for theth

entry, which is one Define anvector

ifif

(41)

Recall from Section IV-A that the diagonal

matrix Define

Let thesingular value decomposition (SVD) of the matrix be

(42)

where the matrix has for all ,and The numbers are

the positive square roots of the eigenvalues of and,hence, are uniquely determined. The columns of thematrix are the orthonormal eigenvectors of and thecolumns of the matrix are the orthonormal eigenvectorsof The following result can be found in [22].

Lemma 1: The diagonal matrix is given by

(43)

where the is the transpose of in which the positivesingular values of are replaced by their reciprocals, and

is a unitary matrix consisting of the eigenvectors of asin (28).

In what follows we derive two forms of linear detectorsfor joint suppression of MAI and ISI in dispersive CDMAchannels, namely, the linear zero-forcing detector and thelinear MMSE detector.

A. Linear Zero-Forcing Detector

The linear zero-forcing detector for detecting theth bitof the th user has the form of (37) with the weight vector

such that, in the absence of noise, both the MAI andthe ISI are completely eliminated at the correlator output.

Proposition 3: A linear zero-forcing detector for detectingthe th user’s data bit from the received signal isgiven by

(44)

where and are defined in (28).Proof: By definition (41) In the

absence of noise, using (44) and (14), we have

(45)

where the second equality follows from (28), the third equalityfollows from (42) and (43), and the fourth equality followsfrom the following facts:

Therefore, completelyeliminates the MAI and the ISI, and it is indeed a zero-forcingfilter.

B. Linear MMSE Detector

The linear MMSE detector for detecting theth bit of the thuser has the form of (37) with the weight vectorwhere minimizes the output mean-square error(MSE), defined as

Proposition 4: A scaled version of the linear MMSE de-tector for detecting the th user’s data bit from thereceived signal is given by

(46)


Proof: Let be a -vector of all zeros except for theth entry, which is unity. Using (12) and (15), we have

...

... (47)

Therefore, the linear MMSE detector for user is given by

(48)

where the second to last equality follows from (28) and thelast equality follows from the fact that

and thereforeRemark: The two linear detectors given by (44) and (46)

can be interpreted as follows. First the received signalis projected onto the signal subspace to get a-vector

The desired user’s composite signaturewaveform is also projected onto the signal subspace to

obtain The projection of a linear detector in thesignal subspace is then a signal such that the data bitis demodulated as According to (44)and (46), the projection of the linear zero-forcing detector andthat of the linear MMSE detector in the signal subspace aregiven, respectively, by

...

... (49)

Note that as the two linear detectors become identical.

C. Near–Far Resistance

A commonly used performance measure for a multiuserdetector is the asymptotic multiuser efficiency (AME) [8],defined as1

1Pk(�) is the probability of error of the detectorQ(x)�=

(1=p2�) s1x e�(x =2) dx:

, which measures the exponential decay rateof the error probability as the background noise approacheszero. A related performance measure, the near–far resistance,is the infimum of AME as the interferers’ energies are allowedto vary The near–far resistance of the

linear detectors derived in the previous section is given bythe following result.

Proposition 5: The near–far resistance of the zero-forcingdetector in (44) and that of the linear MMSE detectorin (46) is given by

Proof: Since as the two linear detectors becomeidentical, they have the same AME and near–far resistance.Hence, it suffices to find the near–far resistance of the zero-forcing detector in (44). By Proposition 3 the output ofcontains only the useful signal and the ambient Gaussian noise.The amplitude of the useful signal at the output is

; the variance of the noise is where

(50)

where the second equality follows from the facts thatand (42); the third equality follows

from (43); the fourth equality follows from the facts that

and ;and the sixth equality follows from (42). The probability oferror is then given by

It then follows that

VI. DISCUSSION

A. Initial Delay Estimation

So far we have assumed that the receiver has estimatedthe delay of the user of interest within a chip inter-val. Next we consider how to estimate this timing informa-

tion Since takes values from the set, we propose the following minimization

procedure for initial delay estimation:

(51)


where as in Section IV-C, for a given , is the submatrixof starting from the th column. The rationale for using(51) to estimate the delay as follows. For the right hypothesisof , the minimized cost function will be close to zero,whereas, for the other hypothesis, the cost is higher.

Next we consider a special case of (51) where the signals areasynchronous and there is no multipath, i.e., we consider delayestimation in an asynchronous CDMA system. In [9], a blindadaptive technique for joint acquisition and demodulationis proposed, which is based on the MOE blind multiuserdetection approach. In the subspace-based blind approach, theuser delay estimate can also be obtained once the subspacecomponents of the received signal covariance matrix arecomputed. Define and

In our framework, for this case the physical channelimpulse response is given by Assuming the chipwaveform is rectangular, then from (4) and (8) we obtain

otherwise.(52)

Denote as the th column of , andfor Then for the th hypothesis (i.e.,

), we compute the fractional delay by

ifif and

(53)

The corresponding minimum cost is given by

ifif

(54)

Finally, the delay estimate is given by

(55)

B. Signal Subspace Rank Estimation

In the previous discussion, we have assumed that thedimension of the signal subspace isknown to the receiver. Alternatively, it can also be estimatedbased on the eigenvalues of the sample covariance matrix. Let

be the eigenvalues of the sample covariancematrix . Informationtheoretic criteria such as the Akaike information criterion(AIC) or minimum description length (MDL) criterion can be

used to estimate the rank of the signal subspace [23]. Theyare defined as follows:

(56)

(57)

where

(58)

The estimate of the rank is given by the value thatminimizes the corresponding quantity.

C. Robustness Against Channel Estimation Errors

The expressions for the linear zero-forcing detector (44) andthe linear MMSE detector (46) derived in the previous sectionare based on the true signal subspace parametersand the true composite signature waveform of the desired user

In practice, these exact quantities are replaced by theirrespective sample estimates. In the high signal-to-noise region,the performance of the two exact linear detectors tends to bethe same. However, as will be illustrated by the simulationexamples in the next section, when the two linear detectors areconstructed from the sample estimates of the signal subspaceparameters and the estimated composite signature waveform,the linear MMSE detector considerably outperforms the linearzero-forcing detector. The reason is that the form of the linearMMSE detector in (46) is more robust against the channelestimation error. To see this, we write the estimated compositewaveform of the desired user as

(59)

where is the estimation error vector. Replacing the truecomposite waveform in (44) and (46) by the estimatedone , we obtain the two estimated linear detectors, given,respectively, by

(60)

(61)

Comparing (60) and (61) it is clear that the composite signaturemismatch causes more error on the zero-forcing detectorthan on the MMSE detector

An alternative way to see that in (46) is more robustto the mismatched composite signature thanin (44) is asfollows. Consider constructing the linear MMSE detector inthe presence of mismatch. To alleviate the detrimental effectscaused by the mismatch, the norm of the linear detector


(a)

(b)

Fig. 2. Performance of the subspace-based blind multiuser detectors insynchronous nondispersive CDMA channels(L = 1;m = 1):

should be constrained [5], i.e., the optimization problem canbe formulated as

Using the method of Lagrange multipliers, this can be solvedas follows:

(62)

where is a positive number. Therefore, we see that in order

(a)

(b)

Fig. 3. Performance of the subspace-based blind multiuser detectors inasynchronous nondispersive CDMA channels(L = 2;m = 3):

to make the detector robust against the mismatch, the diagonalmatrix should be perturbed by a positive number. However,it is seen from (44) that in the linear zero-forcing detector, thediagonal matrix is perturbed by a negative number, whichis just the opposite of the way toward robustness. Therefore,it can be inferred that the linear zero-forcing detector is notrobust under channel estimation error or composite signaturemismatch.

VII. SIMULATION EXAMPLES

In this section, we provide some simulation examples todemonstrate the performance of the blind channel equaliza-tion and multiuser detection algorithms developed in thispaper. The simulated CDMA system has a processing gain

with randomly generated spreading sequences. Ineach example, samples are used for estimatingthe channel response and the linear multiuser detectors. The


Fig. 4. Performance of the exact linear zero-forcing detector in syn-chronous/asynchronous nondispersive CDMA channels, under the samerespective conditions as in Figs. 2 and 3.

desired user is user 1. The performance measure plotted here isthe probability of bit error versus the desired user’s signal-to-noise ratio (SNR), (SNR is defined as ) averaged over500 independent runs.

A. Example 1: Synchronous/AsynchronousNondispersive CDMA System

We first illustrate the performance of the subspace-basedblind multiuser detectors in nondispersive CDMA channels.Two cases of system load are simulated: and

For each case, both the perfect power control situation(i.e., all users have equal powers) and a near–far situation(i.e., each interfering user is 10 dB stronger than the userof interest) are simulated. The simulated performance in thesynchronous channel is plotted in Fig. 2. The performance inthe asynchronous channel is plotted in Fig. 3, where a randomdelay is assigned to each user The performanceof the conventional matched-filter is also plotted in the figures.As expected, the matched-filter is near–far limited and itperforms poorly even in the case of perfect power control sincethe signature sequences are not optimized, whereas the blindlinear detectors are near–far resistant, though they have thesame informational complexity as the matched-filter (i.e., theyonly assume the knowledge of the desired user’s signaturewaveform). Moreover, shown in Fig. 2 is the performanceof the detectors using either the true delay or the estimateddelay. It is seen that there is almost no performance loss whenthe user delay is not known to the receiver. Therefore, thedelay estimate provided by the algorithm is accurate even ina severe near–far situation. As a comparison, in Fig. 4, theperformance of the exact zero-forcing detector which assumesperfect knowledge of all user’s signature waveforms, delays,and channel gains, under the same respective conditions as inFigs. 2 and 3, is plotted. (Note that the performance of theexact linear MMSE detector is similar, especially in the highSNR region).

(a)

(b)

Fig. 5. Performance of the subspace-based blind multiuser detectors insynchronous dispersive CDMA channels(L = 2;m = 3):

B. Example 2: Synchronous/AsynchronousDispersive CDMA System

We next consider a dispersive channel with multipath delayspread of one symbol interval. The complex channel gains

are generated according to the complex Gaussian distri-bution with zero means and unit variances, and normalizedwith respect to the zero-delay component, i.e.,Again a five-user system and a ten-user system are considered,and, for each case, both the perfect power control situation(i.e., all users have equal powers) and a near–far situation (i.e.,each interfering user is 10 dB stronger than the user of interest)are simulated. The performance in the synchronous channel isplotted in Fig. 5, and the performance in the asynchronouschannel is plotted in Fig. 6. The performance of the conven-tional matched-filter which is matched to the true compositesignal waveform is also plotted in the figures. It is seen thatthe blind linear detectors are near–far resistant in the presence


(a)

(b)

Fig. 6. Performance of the subspace-based blind multiuser detectors inasynchronous dispersive CDMA channels(L = 3;m = 4).

of both MAI and ISI. Again as a comparison, in Fig. 7 theperformance of the exact zero-forcing detector which assumesperfect knowledge of all user’s signature waveforms, delays,and channel gains, under the same respective conditions as inFigs. 5 and 6, is plotted.

VIII. C ONCLUSION

In this paper, we have considered the problem of blinddetection of the multiuser information symbols in a dispersiveCDMA network where the ISI together with the MAI consti-tute a major impediment to the system performance. We havefirst cast the dispersive CDMA channel model into a MIMOsignal model framework. By applying the theory of blindMIMO channel identification and equalization, we have shownthat it is possible to recover the multiuser information symbolswithout any prior knowledge of the channel or the users’signature waveforms, although the computational complexity

Fig. 7. Performance of the exact linear zero-forcing detector in syn-chronous/asynchronous dispersive CDMA channels, under the same respectiveconditions as in Figs. 5 and 6.

of such an approach is prohibitive. However, in practice, thesignature waveform of the desired user is always known to thereceiver. We have developed subspace-based techniques forblindly estimating the channel response for both the forwardlink and the reverse link. We have also considered initialdelay estimation within the framework of subspace approach.Furthermore, we have shown that based on the estimated signalsubspace components and the identified channel response, alinear zero-forcing detector and a linear MMSE detector canbe constructed in closed form, at almost no extra computationalcost. Data detection can thus be furnished by applying theselinear detectors (obtained blindly) to the received signal. Itis observed that the linear MMSE detector is more robustagainst the composite signature waveform mismatch causedby the channel estimation error. Finally it is worth noting thatit is possible to replace the batch eigenvalue decomposition bythe computationally more efficient adaptive subspace trackingalgorithms [2], [15], [16], [24], which update the subspacecomponents as increasingly more columns of are takeninto account.

REFERENCES

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[3] R. O. LaMaireet al, “Wireless lans and mobile networking: Standardand future directions,”IEEE Commun. Mag., pp. 86–94, Aug. 1996.

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[5] M. Honig, U. Madhow, and S. Verd́u, “Blind multiuser detection,”IEEETrans. Inform. Theory, vol. 41, pp. 944–960, July 1995.

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[7] , “On blind MIMO channel identification using second-order sta-tistics,” in Proc. 1996 Conf. Information Sciences and Systems, Prince-ton, NJ, Mar. 1996, pp. 1166–1171.


[8] R. Lupas and S. Verd´u, “Linear multi-user detectors for synchronouscode-division multiple-access channels,”IEEE Trans. Inform. Theory,vol. 35, pp. 123–136, Jan. 1989.

[9] U. Madhow, “Blind adaptive interference suppression for the near-far resistant acquisition and demodulation of direct-sequence CDMAsignals,” IEEE Trans. Signal Processing, vol. 45, pp. 124–136, Jan.1997.

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[12] K. Pahlavan, T. J. Probert, and M. E. Chase, “Trends in local wirelessnetworks,” IEEE Commun. Mag., pp. 88–95, Mar. 1995.

[13] H. V. Poor and X. Wang, “Code-aided interference suppression inDS/CDMA communications: Parallel blind adaptive implementations,”IEEE Trans. Commun., vol. 45, pp. 1112–1122, Sept. 1997.

[14] J. G. Proakis,Digital Communications, 3rd ed. New York: McGraw-Hill, 1995.

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Xiaodong Wangreceived the B.S. degree in electri-cal engineering and applied mathematics (with high-est honors) from Shanghai Jiao Tong University,Shanghai, China, in 1992, and the M.S.E.E. degreefrom Purdue University, West Lafayette, Indiana, in1995. He is now working toward the Ph.D. degree atthe Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ.

He has worked in the areas of digital commu-nications, parallel and distributed computing, andquantum computing. His current research interests

include multiuser communications theory and advanced signal processingtechniques for wireless communications.

H. Vincent Poor (S’72–M’77–SM’82–F’87)received the Ph.D. degree in electrical engineeringfrom Princeton University, Princeton, NJ, in 1977.

From 1977 until 1990, he was a Faculty Memberat the University of Illinois, Urbana, IL. In 1990 hejoined the faculty at Princeton University, Princeton,NJ, where he is a Professor of electrical engineering.He has also held visiting and summer appointmentsat several universities and research organizations inthe U.S., Great Britain, and Australia. His researchintersts are in the area of statistical signal processing

and its applications, primarily in wireless communications. His publicationsin this field include the graduate textbookAn Introduction to Signal Detectionand Estimation(New York: Springer-Verlag, 1988 and 1994).

Dr. Poor is a Fellow of the Acoustical Society of America, and of theAmerican Association for the Advancement of Science. He served as Presidentof the IEEE Information Theory Society in 1990, and as a member of theIEEE Board of Directors in 1991 and 1992. In 1992 he recieved the TermanAward from the American Society for Engineering Education, and in 1994 hereceived the Distinguished Member Award from the IEEE Control SystemsSociety.

blind equalization and multiuser detection in dispersive cdma channels

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