blind turbo multiuser detection for long-code multipath cdma

112 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002

Blind Turbo Multiuser Detection for Long-CodeMultipath CDMA

Zigang Yang and Xiaodong Wang

Abstract—We consider the problem of demodulating and de-coding multiuser information symbols in an uplink asynchronouscoded code-division multiple-access (CDMA) system employinglong (aperiodic) spreading sequences, in the presence of unknownmultipath channels, out-cell multiple-access interference (OMAI),and narrow-band interference (NBI). A blind turbo multiuser re-ceiver, consisting of a novel blind Bayesian multiuser detector anda bank of MAP decoders, is developed for such a system. The effectof OMAI and NBI is modeled as colored Gaussian noise with someunknown covariance matrix. The main contribution of this paperis to develop blind Bayesian multiuser detectors for long-codemultipath CDMA systems under both white and colored Gaussiannoise. Such detectors are based on the Bayesian inference of allunknown quantities. The Gibbs sampler, a Markov chain MonteCarlo procedure, is then used to calculate the Bayesian estimatesof the unknowns. The blind Bayesian multiuser detector computesthe a posterioriprobabilities of the channel coded symbols, whichare differentially encoded before being sent to the channel. Beingsoft-input soft-output in nature, the proposed blind Bayesian mul-tiuser detectors and the MAP decoders can iteratively exchangethe extrinsic information to successively refine the performance,leading to the so-calledblind turbo multiuser receiver.

Index Terms—Coded system, Gibbs sampler, long-code CDMA,multipath, narrow-band interference, out-cell multiple-access in-terference, turbo multiuser detection.

I. INTRODUCTION

EXISTING code-division multiple-access (CDMA) stan-dards (such as IS-95 [1]) employ long spreading codes on

the reverse link, i.e., PN sequences with very long periods. Thetheme of this paper is on the design of a blind multiuser receiverfor an uplink asynchronous coded CDMA system employinglong spreading sequences. It is assumed that the blind receiverhas only the knowledge of the spreading sequences and theinitial delays of the desired users within the cell. The multipathchannels are unknown to the receiver. No pilot symbols areused by any users.

Paper approved by A. Goldsmith, the Editor for Wireless Communication ofthe IEEE Communications Society. Manuscript received September 15, 2000;revised April 15, 2001. This work was supported by the National Science Foun-dation under Grant CAREER CCR-9875314, Grant CCR-9980599, and GrantDMS-0073651. This paper was presented in part at the 34th Asilomar Confer-ence on Signals, Systems, and Computers (Asilomar’00), Pacific Grove, CA,November 2000, and at the 2001 International Conference on Acoustics, Speechand Signal Processing (ICASSP’01), Salt Lake City, UT, May 2001.

Z. Yang is with the Department of Electrical Engineering, Texas A&M Uni-versity, College Station, TX 77843 USA.

X. Wang was with the Department of Electrical Engineering, Texas A&MUniversity, College Station, TX 77843 USA. He is now with the Department ofElectrical Engineering, Columbia University, New York, NY 10027 USA.

Publisher Item Identifier S 0090-6778(02)00526-3.

Some recent works haveaddressed blind channel estimation inlong-code CDMA systems [2], [3]. In these approaches, channelparameters are first estimated and receivers are then constructedbased on the estimated channels. This is suboptimal due to theseparation of channel estimation and data detection (as opposedto joint estimation of both channels and data). Moreover, thesemethods are primarily targeted at uncoded systems and they donot attempt to exploit the signal structures induced by channelcoding existing in most communication systems.

On the other hand,iterative processinghas recently attractedvast attention. In [4]–[6], turbo multiuser detection schemes forcoded CDMA systems are developed, which iterate betweenmultiuser detection and channel decoding to successively im-prove the receiver performance. In these works, the user chan-nels are assumed perfectly known at the receiver.

In this paper, we address the problem of blind turbo multiuserdetection in unknown multipath channels for asynchronouscoded CDMA systems employing long spreading sequences.A novel blind Bayesian multiuser detector is proposed, whichcomputes the MAP estimates of the channel coded multiusersymbols, that are differentially encoded before being sent tothe channel. This technique is based on the Gibbs sampler [7],a Markov chain Monte Carlo (MCMC) technique for Bayesiancomputation. Although originated in the field of statistics, theGibbs sampler has recently been investigated for the optimalreceiver design in various communication systems [8]–[10].

Another issue addressed in this paper is blind Bayesian mul-tiuser detection in the presence of unknown out-cell multiple-ac-cess interference (OMAI) and narrow-band interference (NBI),a scenario that occurs in CDMA overlay systems. Various tech-niques for interference suppression in CDMA overlay systemsare reviewed in [11]. Existing methods include frequency-do-main techniques [12], predictive techniques [13], [14], the linearMMSE estimation technique [15], [16] and the maximum-like-lihood technique [17]. In this paper, we propose a blind ap-proach to interference suppression, where the total effect ofwhite Gaussian noise, OMAI and NBI is modeled as coloredGaussian noise with some unknown covariance matrix and theGibbs sampler is then used to calculate the Bayesian estimatesof all unknowns.

The rest of this paper is organized as follows. In Section II,the system model for a long-code multipath CDMA system isdescribed; In Section III, the turbo multiuser receiver structureand the Gibbs sampler are introduced; blind Bayesian multiuserdetection algorithms under white and colored Gaussian noiseare derived in Section IV. Simulation results are provided inSection V. Finally, Section VI contains the conclusions.

0090–6778/02$17.00 © 2002 IEEE

YANG AND WANG: BLIND TURBO MULTIUSER DETECTION FOR MULTIPATH CDMA 113

II. SYSTEM DESCRIPTION

A. Channel Model

Consider a -user uplink CDMA system, employing nor-malized long pseudorandom spreading sequences and signalingthrough multipath channels with additive white Gaussian noise(AWGN) and other unknown interference. The transmittedsignal due to the th user is given by

(1)

where denotes the length of the data frame,is the pro-cessing gain, denotes the symbol interval, is asignature sequence assigned to theth user for the th symbol,

, and denote, respectively, the am-plitude, the symbol stream, and the delay of theth user’s signal,

is a normalized chip waveform of duration . Theth user’s signal propagates through a multipath channel

whose impulse response is given by

(2)

where is the total number of resolvable paths in the channel,and and are, respectively, the complex path gain andthe delay of the th user’s th path, .The received continuous-time signal at the receiver is given by

(3)

where denotes convolution, , , andis the ambient noise plus the unknown interfering signals,

as will be explained in Section II-B.At the receiver, the received signal is filtered by a chip-

matched filter and sampled at the chip-rate. Let

(4)

be the maximum delay spread among theusers in terms ofsymbol intervals. The signal sample at the matched filter outputat time is given by (5), shown at the bottom of the

page. where .Since the chip waveform has a duration, is nonzero

only for . For con-

venience, define as the initial delayin terms of number of chips for theth user’s signal; define

as the maximum channel delay

among all users; and defineas the channel response for theth user. Throughout the paper,assume that both the maximum initial delay andthe maximum channel delay are less than . Hence, themaximum symbol delay satisfies .

It is convenient to express the signal model (5) in a vectorform as

(6)

where , ,

and . It is easy to verify thatcan be expressed as

......

...

. . ....

(7)

(5)


Fig. 1. A coded uplink CDMA communication system.

B. Noise Model

1) White Gaussian Noise:In the simplest case, it is assumedthat in (3) contains the channel ambient noise only, which isa white complex Gaussian process. It is further assumed that thechip waveform is a rectangle pulse with duration. Hence,all the noise samples are i.i.d. zero mean complexGaussian random variables with variance. Moreover,is a sequence of zero-mean i.i.d. complex Gaussian vectors, i.e.,

(8)

2) Unknown Interference and Colored Gaussian Assump-tion: In cellular DS-CDMA, the same uplink/downlink pair offrequency bands are reused for each cell. Therefore, a signaltransmitted in one cell may cause interference in neighboringcells, resulting in out-cell multiple-access interference (OMAI).In addition, narrow-band communication systems sometimescan overlay with CDMA systems and thus cause narrow-bandinterference (NBI) to the latter. Hence, in general, the noisecomponent in (6) consists of white Gaussian noise (WGN),OMAI, and NBI, i.e., . The WGNhas zero mean and covariance . The OMAI hasthe same structure as the in-cell CDMA signals, i.e.,

(9)

where denote the total number of out-cell users. Whenis large, by the central limit theorem approaches aGaussian vector with zero mean and a covariance matrix, de-noted by . Note that both the encoded bitsand the elements of the spreading sequences areindependent random variables. After some manipulations, theelement of can be written as

(10)

The NBI signal is typically modeled as a correlated Gaussianprocess. For example, it can be represented as anth-orderautoregressive (AR) signal [15], where , i.e.,

(11)

where denotes the noise sample component due to theNBI signal and is a white Gaussian process with variance

. Henceis Gaussian with zero mean and a covariance matrix,

denoted by .Combining these three components, the noise vectors

can be modeled as colored Gaussian vectors with zero mean anda covariance matrix , i.e.,

(12)

Note that, when the OMAI and the NBI are present, the noisevectors are correlated. However, when developing theblind Bayesian multiuser detectors in Section IV-B, we ignoresuch temporal correlations to simplify the algorithms.

C. System Model

The block diagram of the transmitter end of a typical CDMAsystem is shown in Fig. 1, where is the total number of in-cellusers. The binary information bits for user are en-coded using some channel code (e.g., block code, convolutionalcode, or turbo code), resulting in a code bit stream . Acode-bit interleaver is used to reduce the influence of the errorbursts at the input of the channel decoder. The interleaved codebits are then mapped to BPSK symbols . TheseBPSK symbols are differentially encoded to yield the symbolstream . Differential encoding is used to resolve thephase ambiguity inherent in any blind receiver and is given by

.(13)

Each symbol is then modulated by a spreading waveform, which is a segment of the long spreading waveform of

the th user. Each modulated signal is transmitted through amultipath channel . The received signal is given by (6).

III. RECEIVER STRUCTURE

A. Turbo Multiuser Receiver

The receiver under consideration is an iterative receiverstructure as shown in Fig. 2. It consists of two stages: ablind Bayesian multiuser detector, followed by parallelsingle-user channel decoders. The two stages are separated bydeinterleavers and interleavers.


Fig. 2. Turbo multiuser receiver.

Define . The blind Bayesianmultiuser detector computes the log-likelihood ratios (LLRs) ofthe interleaved code bits

(14)

The second term in (14), denoted by , represents thea priori LLR of the code bit , which is computed by thechannel decoder in the previous iteration, interleaved, and thenfed back to the Bayesian multiuser detector. (The superscript

indicates the quantity obtained from the previous iteration).For the first iteration, it is assumed that all code bits are equallylikely. The first term in (14), denoted by , representsthe extrinsic information delivered by the Bayesian multiuserdetector, based on the received signals, the structure of signalmodel (6) and the prior information about all other code bits

. The extrinsic information , is thendeinterleaved and sent to the channel decoder. Assume that

is mapped to after deinterleaving.Each user’s channel decoder is based on the MAP decoding

algorithm [18]. It computes thea posterioriLLR of each codebit [6] as shown in (15), at the bottom of the page. It is seenfrom (15) that the output of the MAP decoder is the sum of theprior information and theextrinsic information

delivered by the channel decoder. This extrinsicinformation is the information about the code bitgleaned from the prior information about the other code bits,

, based on the constraint structure ofthe code. After interleaving, the extrinsic information deliveredby the channel decoder is then fed back tothe blind Bayesian multiuser detector as the refined priorinformation about the code bits for the next iteration. Note that,at the first iteration, the extrinsic information and

are statistically independent. But subsequently,

since they use the same information indirectly, they willbecome more and more correlated and finally the improvementthrough iteration will diminish.

B. Problem Statement of Blind Bayesian Multiuser Detector

The MAP decoder employed in the turbo multiuser receiveris the standard BCJR algorithm [18]. The main purpose of thispaper is to develop the blind Bayesian multiuser detector. Thisdetector is assumed to have the knowledge of the spreading se-quence and the initial delay information for each in-cell user,i.e., in (6) are known to the receiver.Note that the initial delay is the first nonzero channel coefficient,which may be obtained through CDMA timing acquisition tech-niques [19]–[21]. If the estimated initial delay is larger than theactual value, then some signal energy is lost. On the other hand,if the estimated initial delay is smaller than the actual value, theneffectively more channel parameters need to be estimated. Nev-ertheless, in this paper, we will assume the perfect knowledgeof the initial delays.

For convenience, define thea priori LLRs of the interleavedcode bits as

(16)

which is essentially the extrinsic information, , fedback by the channel decoder in the turbo multiuser receiver [cf.Fig. 2].

In Section IV, we consider the problem of estimating theaposterioriprobabilities of the code bits

(17)based on the received signals, the signal structure (6) andthe prior information , without knowing thechannel response and the noise parameters (i.e.,for white Gaussian noise, for colored Gaussian noise). Notethat, although is directly determined by the differentiallyencoded symbols as seen in the signal model (6), thechannel decoders require the posterior distributions of the codebits [cf. Fig. 2].

code constraints

code constraints(15)


C. The Gibbs Sampler

The blind Bayesian multiuser detectors developed in thispaper are based on the Gibbs sampler [7], a Markov chainMonte Carlo (MCMC) procedure for numerical Bayesiancomputation. Let be a vector of unknownparameters. Let be the observed data. Suppose that we areinterested in finding thea posteriori marginal distribution ofsome parameter, say, conditioned on the observation, i.e.,

. Direct evaluation involves integrating out the rest ofthe parameters from the joint posterior density , whichin most cases is computationally infeasible. The basic ideabehind the Gibbs sampler is to generate random samples fromthe joint posterior distribution and then to estimate anymarginal distribution using these samples. Given the samplesat time , , at the thiteration, this algorithm performs the following operation toobtain samples at time, :

• For , draw from the conditional distribu-tion

It is known that under regularity conditions [22]–[25]:• The distribution of converges geometrically to ,

as .• , as

, for any integrable function .

IV. BLIND BAYESIAN MULTIUSER DETECTION

A. White Gaussian Noise

In this section, we consider the problem of computing theaposterioribit probabilities in (17) under the assumption that theambient noise distribution is white and Gaussian. That is, thepdf of in (6) is given by

(18)

Denote

Then (6) can be written as

(19)

The problem is solved under a Bayesian framework, by treatingthe unknown quantities , , and as realizations of randomvariables with some prior distributions. The Gibbs sampler isthen employed to calculate the marginal distribution of thoseunknown parameters. Note that, although the code bitsareof interest, it is more convenient to sample the differentiallyencoded bits in the Gibbs sampler.

1) Prior Distributions: In principle, prior distributions areused to incorporate the prior knowledge about the unknown pa-rameters and less restrictive (i.e., noninformative) priors shouldbe employed when such knowledge is limited. The priors should

also be chosen such that the conditional posterior distributionsare easy to compute and simulate. To that end,conjugate priorsare usually used to obtain simple analytical forms for the re-sulting posterior distributions. The property that the posteriordistribution belongs to the same distribution family as the priordistribution is called conjugacy. Following the general guide-lines in Bayesian analysis [26]–[28], we choose the conjugateprior distributions for the unknown parameters , and

, as follows.

1) For the unknown channel , a complex Gaussian priordistribution is assumed,

(20)

Note that large value of corresponds to a less infor-mative prior.

2) For the noise variance , an inverse chi-square prior dis-tribution is assumed

(21)

Small values of correspond to the less informativepriors.

3) The data bit sequence is a Markov chain, encoded from. Its prior distribution can be expressed as

(22)

where (22) follows from the definition (16). Notice thatwe set to count for the phase ambiguityin .

2) Conditional Posterior Distributions:The followingconditional posterior distributions are required by the blindBayesian multiuser detector. The derivations can be found inAppendix A.

1) The conditional distribution of theth user’s channel re-sponse given , , , and is [where .Note that denotes the exclusive operator.] shown in(23)–(23)–(25) at the bottom of the next page.

2) The conditional distribution of the noise variancegiven , , and is given by

(26)

with

(27)

3) The conditional distribution of the data bit given, , , and can be obtained from (28)–(30),

where , shown at the bottom of the next

page,where.


TABLE IBLIND BAYESIAN MULTIUSER DETECTION ALGORITHM IN WHITE GAUSSIAN NOISE

3) Gibbs Multiuser Detector in White GaussianNoise: Using the above conditional posterior distribu-tions, the Gibbs sampling implementation of the blind Bayesianmultiuser detector in white Gaussian noise proceeds iterativelyas shown in Table I. Note that the samples of code bitsarecomputed based on the samples of differentially encoded bits

in ( ).To ensure convergence, the above procedure is usually car-

ried out for ( ) iterations and samples from the lastiterations are used to calculate the Bayesian estimates of theunknown quantities. By the second convergence property men-tioned in Section III-C, the posterior symbol probabilities (17)can be approximated as

(31)

where is an indicator such that , ifand , if .

The complexity of the above blind Bayesian multiuserdetectorper iteration is , where the dominant

complexity comes from the computation of the inverse co-variance matrix in (24). The total complexity is then

. On the other hand, the exactaposteriori probabilities (17) of the code bits can be calculatedaccording to

(32)Clearly, the computation in (32) involves computingmultidimensional integrals. Therefore, the blind Bayesian mul-tiuser detector achieves a substantial complexity reduction com-pared with the direct evaluation of Bayesian estimates in (32).

B. Colored Gaussian Noise

In this section, we develop the blind Bayesian multiuser de-tector for colored Gaussian noise, due to the existence of OMAIand NBI. It is assumed that in (6) have a complex jointGaussian distribution, i.e.,

(33)

(23)

with

(24)

(25)

(28)

with

(29)

(30)


As mentioned before, the noise vectors are temporallycorrelated. However, in what follows, we ignore this correlationto reduce the receiver complexity.

1) Prior Distributions: The unknown quantities in this caseare ( , , ), which are assumed to be independent witheach other. As in the case of Gaussian noise, the prior distribu-tion of and are given, respectively, by (20) and (22). Forthe noise covariance matrix, conjugate prior with the inversecomplex Wishart distribution [29] is assumed, i.e.,

(34)

where denotes the trace of a matrix;. Small values of and correspond to less informative

prior. The inverse of the covariance matrix has a complexWishart distribution, i.e.,

(35)

According to [29], a random matrix with a Wishart distri-bution with degrees of freedom (35) can be generated by

, where are i.i.d. Gaussian random vectorswith zero mean and covariance .

2) Conditional Posterior Distributions:The followingconditional posterior distributions are required by the blindBayesian multiuser detector. The derivations can be found inAppendix B.

1) The conditional distribution of theth user’s channelresponse given , , and is shown in

(36)–(38), at the bottom of the page, where .2) The conditional distribution of the noise covariance ma-

trix given , , and is

(39)

with

(40)

Therefore, the conditional distribution of the inverse co-variance matrix given , , and is

(41)

3) The conditional distribution of the data bitgiven , , , and can be obtained from(42) to (44), shown at the bottom of the page, where

.where.

3) Gibbs Multiuser Detector in Colored GaussianNoise: Using the above conditional posterior distribu-tion, the Gibbs sampling implementation of the blind Bayesianmultiuser detector in colored Gaussian noise proceeds itera-tively as shown in Table II. As in the case of white Gaussiannoise, thea posteriorisymbol probabilitycan also be computed by (31). The complexity of this blindBayesian multiuser detectorper iteration is ,where the dominant complexity comes from the computationof inverse covariance matrix in (37). The total complexityis then . This is again a substantialcomplexity reduction compared with the direct implementa-tion of Bayesian symbol estimates, which involvesvery-high-dimensional integrals according to

(45)

(36)

with

(37)

(38)

(42)with

(43)

(44)


TABLE IIBLIND BAYESIAN MULTIUSER DETECTION ALGORITHM IN THE PRESENCE OFWGN, OMAI, AND NBI

Fig. 3. Samples drawn by the Gibbs sampler for the case of white Gaussian noise withK = 3,E =N = 8 dB for all the users.

V. SIMULATION RESULTS

Simulation Setup:In this section, we provide a number ofsimulation examples to illustrate the performance of the blindBayesian multiuser detectors and the blind turbo receiver de-veloped in this paper. We consider a CDMA system with pro-cessing gain . The long spreading sequences of all usersare generated randomly.

In all the simulations described in this section, the followingnoninformative conjugateprior distributions are used in theGibbs sampler. For the case of white Gaussian noise

and for the case of colored Gaussian noise

In all the simulations related to colored Gaussian noise, SNRis used to denote the in-cell user signal to WGN ratio, and SIRis used to denote the in-cell user signal to NBI ratio. The NBIis modeled as a second-order AR model with coefficients

, in (11). The OMAI is generated according to

(9) with energy 12 dB below the in-cell user and the numberof out-cell users is set as . Note that the Bayesianmultiuser receivers in this paper are developed under the gen-eral channel conditions. Although we choose in the simulationsto have equal-power scenario, the proposed techniques performequally well in near–far situations.

The channel code is a rate of 1/2 constraint length-5 convolu-tional code (with generators 23, 35 in octal notation). The inter-leaver is generated randomly and fixed for all simulations. Theblock size of the information bits is set to be 50. After channelcoding, an extra bit is needed to begin the differential en-coding. Therefore, the data block size for each user is ,the number of path for each user is , and the transmitterdelay is generated randomly with the restriction . Foreach data block, the Gibbs sampling is performed for 100 itera-tions, with the first 50 iterations as the “burning-in” period, i.e.,

in (31).Convergence Behavior of the Blind Bayesian Multiuser De-

tectors: We first illustrate the convergence behavior of the pro-posed blind Bayesian multiuser detector in white Gaussian noise( ). The channel responses are generated ran-domly with normalized energy. Due to the limited space, inFig. 3, we just plot the first 100 samples drawn by the Gibbssampler of and . The corresponding true values of thesequantities are also shown in the same figure with dashed lines.


Fig. 4. Samples drawn by the Gibbs sampler for the case of colored Gaussian noise withK = 2,K = 18; for each in-cell user,SNR = 20 dB andSIR =

�15 dB.

(a) (b) (c)

Fig. 5. Performance of blind Bayesian multiuser detector in white Gaussian noise, assuming all the in-cell users have same energy. (a)K = 3. (b)K = 5.(c) K = 7.

It is seen that the Gibbs sampler reaches convergence within thefirst several iterations.

Next, we illustrate the convergence behavior of the proposedblind Bayesian multiuser detector in colored Gaussian noise( ). The channel responses of in-cell users aregenerated randomly with normalized energy and the channelresponse of the out-cell users are generated randomly with en-ergy 12 dB below. In Fig. 4, we plot the first 100 samples drawnby the Gibbs sampler of and . The correspondingtrue values of and are also shown in the samefigure with dashed lines. Again, it is seen that the Gibbs sam-pler reaches convergence within the first several iterations. Thechannel response samples converges toor randomly dueto the phase ambiguity. It is seen that is far from 0,which indicates that the noise covariance matrix is not diagonalany more with the existence of OMAI and NBI.

Performance of the Blind Bayesian Multiuser Detectorin White Noise : Fig. 5 illustrates the performance of

with different numbers of in-cell users. The biterror rate (BER) of the code bits is averaged amongall the users and then plotted. The RAKE receiver and thenonlinear parallel interference concellation (PIC) receiver [30]are also implemented assumingperfect channel knowledge.The performance of the RAKE receiver and the performanceof PIC after five iterations are also shown in Fig. 5 for thepurpose of comparison. It is seen that, at reasonable SNR, theperformance of is better than that of the othertwo methods where perfect channel knowledge is assumed.The performance gain of over the other twomethods increases as the number of users increases.

Performance of the Blind Bayesian Multiuser Detector inColored Noise: In order to demonstrate the performance of


Fig. 6. Performance of blind Bayesian multiuser detector in colored Gaussiannoise withK = 3,K = 18 and fixedSNR = 15 dB for all in-cell users.

, in Fig. 6, we compare the performance (interms of averaged code BER versus SIR) ofwith that of the following receiver schemes:

• Linear MMSE multiuser detector: In this case, we assumethat the multipath channels for all in-cell users are known

to the receiver. Define ,which gathers all the received signal which is related todata bits , . Furthermore, we define

Then, the signal model (6) becomes

(46)

Note that is the summation of white ambient noise,OMAI, and NBI, which is again modeled as coloredGaussian noise, however, with known covariance matrix

. It is easy to show that the linear MMSE detector forthe th user is given by [15], [16]

(47)where denotes a (5K) vector with all-zeros elementsexcept for the th entry which is one.

• Genie-Aided PIC detector: In this case, we assume thata genie provides the receiver with an observation ofsignal- free NBI corrupted by additive ambient noise and

Fig. 7. Performance of blind Turbo multiuser receiver in white Gaussian noisewithK = 3, assuming all users have the same energy.

OMAI with the same statistics, i.e.,, where , is gener-

ated according to (9) with independent sets of randomsymbols. The NBI signal samples follow the AR model(11). Based on , we can then use a Kalman filterto obtain an estimate of the NBI signal [14]. Aftersubtracting the estimated NBI signal from the observation

, a parallel interference concellation (PIC) receiveris implemented assuming perfect channel knowledge.

• Single-User Bound Without NBI: In this case, we assumethat there is no NBI. Rake receiver is implemented forsingle user CDMA system with the same component ofwhite ambient noise and OMAI. It is clear that this de-tector provides a lower bound to the system we discussedhere.

Note that the three approaches given above assume perfectchannel knowledge as well as other side information about thechannel. For example, in the linear MMSE detector, the covari-ance matrix of the combined NBI, OMAI and noise is assumedknown; in genie-aided PIC detector, a genie observation is as-sumed to be available for estimating NBI signal; in single userbound, both NBI and other in-cell users are assumed perfectlyknown. Hence such performance comparisons are unfavorableto our proposed blind methods. Nevertheless, it is seen in Fig. 6that, at reasonable SIR, outperforms the othertwo receivers (linear MMSE detector and genie-aided PIC de-tector) and approaches a near single-user bound performance,which demonstrates that the proposed blind Bayesian multiuserdetector under the colored Gaussian noise assumption is effec-tive for combating unknown NBI and OMAI.

Performance of the Turbo Multiuser Receivers :Figs. 7 and8 illustrate the performance of turbo multiuser receiver underwhite and colored Gaussian noise, respectively. The BER of thecode bits is averaged among all the users and is plottedfor the first three iterations. It is seen that by incorporating theextrinsic information provided by the channel decoders, the pro-posed blind Bayesian multiuser detectors ( and

) both achieve significant performance improve-ment by the turbo procedure.


Fig. 8. Performance of blind turbo multiuser receiver in colored Gaussiannoise withK = 3, K = 18: (a) with fixedSNR = 15 dB for all in-cellusers and (b) with fixedSIR = �20 dB for all in-cell users.

VI. CONCLUSION

We have developed a blind turbo multiuser receiver for anasynchronous CDMA system employing long spreading se-quences in the presence of unknown multipath channel, out-cellmultiple-access interference (OMAI), and narrow-band inter-ference (NBI). A novel blind Bayesian multiuser detector isderived for joint multiuser detection and differential decoding,which is “soft-input soft-output” in nature and fits well intothe proposed turbo multiuser receiver framework. This blindmultiuser detector is based on the Bayesian inference of allunknown quantities and can be efficiently implemented usingthe Gibbs sampler, a Markov chain Monte Carlo procedurefor computing Bayesian estimates. Such a blind Bayesianmultiuser detector is derived for long-code multipath CDMAsystems under both white and colored Gaussian noise, wherecolored Gaussian noise is used to model the effects of OMAIand NBI. Finally, we have provided simulation examples todemonstrate the effectiveness of the proposed techniques.

APPENDIX I

The derivation of (23) is shown at the bottom of the page. Thederivation of (26) is

The derivation of (28) is shown on the next page.

Derivation of (23)


Derivation of (28)

Derivation of (36)

APPENDIX II

The derivation of (36) is shown at the top of the page. The derivation of (39) is shown at the top of the next page. The derivationof (42) is shown at the top of the next page.


Derivation of (39)

Using the fact that

Derivation of (42)

where

with

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Zigang Yang received the B.S. degree in electricalengineering and applied mathematics in 1995and the M.S. degree in electrical engineering in1998 from Shanghai Jiao tong University (SJTU),Shanghai, China, 1995. She is currently workingtoward the Ph.D. degree in the Department ofElectrical Engineering, Texas A&M University,College Station.

Her research are in the area of statistical signalprocessing and its applications, primarily in wirelesscommunications

Xiaodong Wang received the B.S. degree in elec-trical engineering and applied mathematics (with thehighest honor) from Shanghai Jiao Tong University,Shanghai, China, in 1992 and the M.S. degree in elec-trical and computer engineering from Princeton Uni-versity, Princeton, NJ, in 1998.

From July 1998 to December 2000, he was withthe Department of Electrical Engineering, TexasA&M Univesity, College Station, as an AssistantProfessor. His research interests fall in the generalareas of computing, signal processing and commu-

nications. He has worked in the areas of digital communications, digital signalprocessing, parallel and distributed computing, nanoelectronics and quantumcomputing. His current research interests include multiuser communicationstheory and advanced signal processing for wireless communications. Heworked at the AT&T Labs— Research, in Red Bank, NJ, during the summerof 1997. He joined the Department of Electrical Engineering, ColumbiaUniversity, New York, NY, in January 2001 as an Assistant Professor.

Dr. Wang is a member of the American Association for the Advanced ofScience. He has also received the 2001 IEEE Information Theory Society andCommunications Society Joint Paper Award. He is now an Associate Editor forIEEE TRANSACTION ONCOMMUNICATION and IEEE TRANSACTION ONSIGNAL

PROCESSING. He has received the 1999 NSF CAREER Award.

blind turbo multiuser detection for long-code multipath cdma

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