Turbo multiuser detection for turbo-coded CDMA in multipath fading channels

1096 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 51, NO. 5, SEPTEMBER 2002

Turbo Multiuser Detection for Turbo-Coded CDMAin Multipath Fading Channels

Qinghua Li, Xiaodong Wang, and Costas N. Georghiades, Fellow, IEEE

Abstract—Multiuser detection and turbo coding are two of themost powerful techniques for enhancing the performance of next-generation wide-band code-division multiple-access (CDMA) sys-tems. In this paper, we develop an iterative multiuser receiver forturbo-coded CDMA systems with aperiodic spreading sequencesoperating in multipath fading, which exploits the power of bothtechniques. The key innovation in the proposed receiver is a low-complexity soft multiuser detector which uses the same decisionstatistic as the conventional RAKE receiver (i.e., the outputs of themaximum ratio combiners for all the users), and performs soft-in-terference cancellation and instantaneous minimum mean-squareerror (MMSE) filtering. The soft multiuser detector has a com-plexity of ( 2) per bit per iteration, where is the number ofusers. A single-user receiver which employs a soft RAKE detectorfollowed by a turbo decoder is also considered. Simulation resultsdemonstrate that the proposed multiuser receiver offers more than0.5-dB gain over the single-user RAKE receiver at a bit-error rate(BER) of 10 3 and even outperforms the single-user RAKE re-ceiver operating in asingle-userenvironment.

Index Terms—Code-division multiple access (CDMA), minimummean-square error (MMSE) filtering, multipath fading, multiuserdetection, soft interference cancellation, turbo coding.

I. INTRODUCTION

OVER the past decade, it has been demonstrated thatvarious multiuser detection schemes can offer significant

gain in multiple-access interference (MAI) suppression overconventional techniques [1]. Since error-control coding isapplied in most practical CDMA systems, recent researchhas addressed multiuser detection for coded code-divisionmultiple-access (CDMA) systems [2]–[7]. In [2], Giallorenziand Wilson derived the optimal decoding scheme for anasynchronous convolutionally coded CDMA system. Thisscheme has a prohibitive computational complexityper bit, where is the number of users and is the codeconstraint length. In [4], Moher derived an iterative multiuserdetector for synchronous convolutionally coded multiaccesschannels with high user correlation. The time complexity of

Manuscript received July 31, 2000; revised October 22, 2001. The work ofQ. Li was supported in part by a grant from the Texas Telecommunications En-gineering Consortium (TxTEC).

Q. Li is with Intel Labs, Intel Corporation, RNB-6-49, Santa Clara, CA 95052USA.

X. Wang was with the Department of Electrical Engineering, Texas A&MUniversity, College Station, TX 77843-3128 USA. He is now with the Depart-ment of Electrical Engineering, Columbia University, New York, NY 10027USA.

C. N. Georghiades is with the Department of Electrical Engineering,Texas A&M University, College Station, TX 77843-3128 USA ([email protected]).

Digital Object Identifier 10.1109/TVT.2002.801749

this detector is per bit per iteration, which isexponential in the number of users, and, thus, it is prohibitivefor channels with medium to large number of users. In [5],Alexander, Grant, and Reed derived an iterative multiuserdetector based on Expectation–Maximization algorithm. In[6], Reed and the other authors derived a low-complexitymultiuser detector for synchronous turbo-coded CDMA chan-nels using M-algorithm [8]. More recently, in [7], Wang andPoor proposed a low-complexity iterative multiuser receiverfor asynchronous convolutionally coded CDMA in multipathchannels. The time complexity of the multiuser detector inthis receiver is per bit per iteration, where is theprocessing gain of the CDMA system and the complexity ofthe receiver is per bit per iteration. One ofthe main contributions of the work reported in this paper is tofurther reduce the complexity of the soft multiuser detectorin the iterative receiver to per bit per iteration. Sincein practical systems, [9], [10], this is a considerablereduction in computational complexity.

Parallel- and serial-concatenated codes, the so-called turbocodes, are considered the most important breakthrough in thecoding community in the 1990s. Since these powerful codes canachieve near-Shannon limit error-correction performance withrelatively low complexity, they have been adopted as an optionalcoding technique for next-generation (3G) CDMA systems [11].

In this paper, a low-complexity iterative multiuser receiver isdeveloped for turbo-coded1 asynchronous CDMA systems overmultipath fading channels. The iterative receiver consists of twostages: a soft multiuser detector, followed byparallel codeextrinsic information (CEI) decoders [19], [20]. There are twotypes of iterations in the receiver, where the outer iteration ex-changes information between the multiuser detector and the CEIdecoders and the inner iterations compute thea posterioriprob-ability (APP) of each code bit within the CEI decoders. Themultiuser receiver developed is applicable to CDMA systemsemployingaperiodicspreading sequences. Moreover, the softmultiuser detector operates on the same decision statistics asthe conventional RAKE receiver (i.e., the outputs of the max-imum ratio combiners for all the users), and it achieves signif-icant performance gains by performing interference cancella-tion/suppression at a moderate computational complexity.

1Although the iterative receiver structure proposed can be applied to convo-lutionally coded or block-coded systems that employ soft-output channel de-coding, we believe that the extrinsic information delivered by turbo decodersare more reliable than the one of convolutional maximuma posteriori(MAP)decoders, because of the concatenated structure of turbo codes. And the reliableextrinsic information benefits not only demodulation in fading channels but alsothe resistance of the near–far problem that is one of the major detrimental ef-fects in CDMA systems.

0018-9545/02$17.00 © 2002 IEEE

LI et al.: TURBO MULTIUSER DETECTION FOR TURBO-CODED CDMA 1097

Fig. 1. The turbo-coded CDMA system.

The rest of the paper is organized as follows. In Section II,the signal model is described, and the transmitter and receiverstructures are outlined. In Section III, the low-complexity softmultiuser detector is developed. Section IV contains the presen-tation of the code-extrinsic-information decoder, and Section Vthe description of a soft RAKE receiver. Simulation results arepresented in Section VI, and Section VII concludes the paper.

II. SYSTEM DESCRIPTION

We consider a -user turbo-coded CDMA system signalingthrough multipath fading channels, where each user employsbinary phase-shift keying (BPSK) spread-spectrum modulation.The block diagram of the system is illustrated in Fig. 1.

A. The Transmitter Structure

The transmitter end is shown in the upper left part of Fig. 1.The binary information bit stream foruser , , is encoded by a rate turbo en-coder. The output of the turbo encoder is the code bit stream

. For each user, a different code bit in-terleaver is used to reduce the influence of the error bursts atthe input of each channel decoder. The interleaved bit of useris denoted by , where the superscript indicates an inter-leaved quantity. Using BPSK, each interleaved code bit is modu-lated by a signature waveform of duration, denoted by .That is, is mapped to and ismapped to . The spread-spectrum signal is then trans-mitted through a multipath fading channel.


The transmitted signal generated by theth user is given by

(1)

where is the number of code bits per user per frame, andand denote, respectively, the amplitude and normal-

ized signature waveform of theth user for the th code bit.It is assumed that is unit energy and is supported onlyon the interval . Note that we assume anaperi-odic spreading sequence, which means the spreading waveform

varies with symbol index.The Turbo Encoder:The turbo encoder works as follows.

For user , the frame of input binary information bits, denotedby , where , the size of the informa-tion bit frame, is encoded by the constituent encoders for parallelconcatenated codes (PCCs) or by the outer and inner encodersfor serial concatenated codes (SCCs). To generate a desired coderate , the output codeword is punctured. The remainingbits after puncturing are then shifted to new levels, i.e.,and , and they are then transmitted in a serial way. Theoutput bit frame is denoted by , where

is the size in bits of the encoded frame.

B. The Multipath Fading Channel Model

We assume atime-variant, asynchronous multipath channel.The th user’s signal is transmitted through the multipathchannel with impulse response

(2)

where is the number of paths in theth user’s channel,are complex fading processes, andis the delay of theth pathof the th user’s signal. It is assumed that the fading processesare known to the receiver and do not vary during one codedsymbol interval, but may vary from symbol to symbol, i.e.,

for (3)

Using (2) and (3), the received faded signal due to theth useris

(4)

where denotes convolution. The total received signal is then

(5)

where is zero mean, complex, white Gaussian noise ofpower spectral density .

C. The Turbo Receiver Structure

The iterative (turbo) receiver structure is shown in the lowerhalf of Fig. 1. It consists of two stages: a soft multiuser detector,followed by parallel single-user CEI decoders, defined later

in (7). The two stages are linked by deinterleavers and inter-leavers. The soft multiuser detector deliversmultiplex extrinsicinformation(MEI), which can be expressed as

(6)

The first term in (6) represents thea posteriori log-likelihoodratio (LLR) of the code bit, and the second term, denoted by

, represents thea priori LLR of the code bit ,which is computed by the CEI decoder of theth user in theprevious iteration, interleaved, and then fed back to the softmultiuser detector. The superscriptindicates the quantity ob-tained from the previous iteration. For the first iteration, as-suming equally likely code bits, i.e., no prior information avail-able, we have: , for and .

The soft multiuser detector computes the MEIbased on the received signal , the structure of the multiusersignal given by (4) and (5), and thea priori information ofall other code bits in the frame. The extrinsic information

, which is not influenced by thea priori informa-tion of provided by the CEI decoder, is then deinterleavedand fed into the th user’s CEI decoder as thea priori infor-mation in the current iteration.

Based on the prior information and the structureof the turbo code (as indicated by the conditioning onbelow),the th user’s CEI decoder computes the CEI of each code bitusing

(7)

As will be seen in Section IV, this extrinsic information is theinformation about code bit gleaned from the prior infor-mation about the other code bits , based onthe constraint of the turbo code. The CEI decoder also com-putes thea posterioriLLR of every information bit, which isused to make a decision on the decoded bit at the last itera-tion. After interleaving, the extrinsic information delivered bythe CEI decoders is then fed back to thesoft multiuser detector as the prior information about the codebits of all users in the next iteration. Note that at the first itera-tion and are statistically independent.Subsequently, however, since they use the same information in-directly, they become more and more correlated and finally theimprovement through the iterations diminishes.

III. T HE SOFT MULTIUSER DETECTOR

The diagram of the soft multiuser detector is shown in Fig. 2.In this section, we first show that is the suffi-cient statistic for detecting the code bits , and de-rive an expression for it in terms of channel parameters; we then


Fig. 2. The soft multiuser detector.

describe the three blocks, soft cancellation, minimum mean-square error (MMSE) filtering, and computing MEI, in Fig. 2.Finally, we discuss the complexity of the detector.

A. The Sufficient Statistic

Let

and

and the received signal without noise denoted as

(8)

Using the Cameron–Martin formula [12], the likelihood func-tion of the received waveform in (5), conditioned on allthe transmitted symbols of all users and the fading, can bewritten as

(9)

where is some positive scalar constant, and

(10)The first integral in (10) can be expressed as

(11)

where is the output of maximum ratio combiner; andis the output of user ’s th RAKE finger.

Since the second integral in (10) does not depend on the re-ceived signal , by (11), the sufficient statistic for detectingthe multiuser symbols is . It is seen from (11)that the sufficient statistic is obtained by passing the receivedsignal through a bank of maximum-ratio multipath com-biners (i.e., RAKE receivers), as shown in Fig. 2. We next de-rive an explicit expression for this sufficient statistic in terms ofthe multiuser channel parameters and the transmitted symbols,which is instrumental to developing the turbo multiuser receiverin subsequent sections.

Assume that and the multipathspread of any user signal is limited to at mostsymbol inter-vals, for some positive integer. That is,

(12)

Define the following correlation of the delayed user signalingwaveforms

(13)

Since and is nonzero only for , it thenfollows that , for . Now substituting(5) into (11), we have

(14)


where is a sequence of zero-mean, complex, Gaussianrandom variables with covariance given (after some manipula-tion) by

(15)

Now see the equations shown at the bottom of the page, whereis the RAKE filter output for all paths of all users;

consists of the channel gains of all paths for theth symbol; isthe diagonal amplitude matrix; is the output of maximumratio combiners (MRC) for users. By definition, the sufficientstatistic can be written as . We can then express (14)as

(16)

and from (15), the covariance matrix of the complex Gaussianvector sequence is

(17)

Substituting (16) into (11) we obtain the expression for as

(18)

where is a sequence of zero-mean complex Gaussian vec-tors with covariance matrix

(19)

Note that by definition (13) we have

It then follows that .

B. Soft Multiuser Detection

We next specify the detailed operations of the three blocks onthe right side of Fig. 2.

1) Soft Interference Cancellation:Even though the generalcase of arbitrary can be handled, for simplicity here we as-sume that , i.e., each user’s multipath delay spread iswithin one symbol interval. We define the following quantities:

matrix

matrix

vector

We can then write (18) in a matrix form as

(20)

where by (19) .Based on thea priori LLR of the code bits of all users,

provided by the CEI decoder from theprevious iteration, we first form soft estimates of the user codebits

(21)

Let

(22)

(23)

......

......

......

...

matrix

-vector

vector

vector

matrix

matrix

vector


(24)

where denotes a vector of all zeros, except for theth element, which is. The elements of are the estimates

of all users’ th, th, and th code bits, except that azero taking the place of user’s th code bit.

At symbol time , for each user , soft interference cancel-lation is performed on the received discrete-time signal in(20) to obtain

(25)

where is the estimates of the MAI from all theother users.

2) Instantaneous MMSE Filtering:To further mitigate theresidual multiaccess interference and speed up convergence ofthe detector, an instantaneous linear MMSE filter is then appliedto to obtain

(26)

where the filter is chosen to minimize the mean-square error (MSE) between the code bit and the filteroutput , i.e.,

(27)

The two expectations are given by

(28)

(29)

where

Substituting (28) and (29) into (27) and taking the derivativeof the MSE in (27) with respect to the filter gives thesolution of (27) as

(30)3) Computing the MEI:In [13], it is shown that the distri-

bution of the MAI-plus-noise at the output of a linear MMSEmultiuser detector is approximately Gaussian. In what follows,we assume that the output of the instantaneous MMSE filter

in (26) represents the output of an equivalent additive

white Gaussian noise (AWGN) channel having as its inputsymbol. This equivalent channel can be represented as

(31)

where is the equivalent amplitude of theth user’s signalat the output, and is a Gaussian noisesample. Using (26) and (30), the parameters andcan be computed as follows, where the expectation is taken withrespect to the code bits of interfering users [7]:

(32)

(33)

Thus, the extrinsic information delivered by the soft instanta-neous MMSE filter is given by

(34)

4) A Fast Algorithm for MEI Computation:From (34), toobtain the MEI , we must first compute and

, which involves inverting a matrix [cf. (26), (30),(32)], i.e.,

(35)

Note that can be written as:

(36)

where

Substituting (36) into (35), we have

(37)

where denotes the th column of .Using the matrix inversion lemma, (37) can be written as

(38)

where

(39)


Fig. 3. Bit-error rate (BER) performance comparison between the turbo multiuser receiver and the RAKE receiver in a multipath fading channel withK = 6,processing gainN = 16, vehicle speed= 120 km/h, data rate= 9.6 kb/s, carrier frequency= 2.0 GHz.

Equations (39) and (38) constitute the procedure for computingin (35) efficiently.

Complexity: Next, we examine the computational com-plexity of the soft multiuser detector in Fig. 2. By (39), ittakes multiplications to obtain using direct matrixinversion. After is obtained, by (38), it takes moremultiplications to get for each . Since at each time,

is computed only once for all users, it takesmultiplications per user per code bitto obtain . After

is computed, by (26), (32), and (34), it takesmultiplications to obtain . Therefore, the total timecomplexity of the soft multiuser detector is per userper code bit. In contrast, the algorithm in [7] operates on thechip samples (instead of the sufficient statistic) with antime complexity. Since, in practice, we have , thesoft multiuser detector developed here achieves a significantcomplexity reduction over the method in [7].

IV. THE CEI DECODER

From Fig. 1, after deinterleaving, the output LLRs of the softmultiuser detector which contain all users’ MEI are sent to thecorresponding users’ CEI decoders. Each CEI decoder is essen-tially an APP calculator using standard soft-input–soft-outputdecoders [14]. It computes the CEI [defined in (7)] of eachcode bitbased on the input LLRs and the structure of the turbocode. For each bit, the CEI decoder has the same complexityas the standard turbo decoder. The output of each CEI decoderis then fed back to the soft multiuser detector as thea prioriinformation for the next iteration. After some iterations, the

LLRs of the information bits are computed and hard decisionsare finally made. The number of iteration within each CEI de-coder is estimated by simulations in single-user AWGN chan-nels. Simulation results in Section VI demonstrate that threeouter iterations (between the soft multiuser detector and CEI de-coders) are sufficient. Therefore, the complexity of the receiveris , where is the constraint length of the con-stituent convolutional codes.

V. THE SINGLE-USERRAKE RECEIVER

In order to compare the performance of the proposed turbomultiuser receiver with the conventional technique used in prac-tical systems, a single-user RAKE receiver employing max-imum-ratio combining for turbo-coded CDMA systems is de-scribed next. In such a receiver, the decision statistic for thethuser’s th code bit is as shown in Fig. 2.

To obtain the LLR of the code bit, a Gaussian assumption ismade on the distribution of . Further, assume that the userspreading waveforms contain independent and identically dis-tributed (i.i.d.) random chips and that the time delayis uni-formly distributed over a symbol interval. Assuming the multi-path fading gains are independent between different users andare normalized such that , it is shownin the Appendix that, the LLR of for the RAKE receiveris given by

(40)


Fig. 4. BER performance comparison between the turbo multiuser receiver with five inner iterations and one inner iteration,K = 6, processing gainN = 16,vehicle speed= 120 km/h, data rate= 9.6 kb/s, carrier frequency= 2.0 GHz.

The LLRs of each user’s code bits are then sent to a corre-sponding turbo decoder to obtain the estimated information bits.

Remark: Note that the soft multiuser detector discussed inSections III-B1–III-B4 operates on the same decision statisticas the conventional RAKE receiver [i.e., the outputs of the max-imum ratio combiners ]. The RAKE receiver de-modulates user ’s symbols based on , whereas the softmultiuser detector demodulates all users’ symbols jointly usingall the decision statistics .

VI. SIMULATION RESULTS

In this section, we demonstrate the performance of the pro-posed turbo multiuser receiver in multipath fading CDMA chan-nels by simulation. The multipath channel model is given by (2).The delays of users’ paths were randomly generated and thenfixed for the simulation. The number of paths for each user isthree . The time-variant fading coefficients were ran-domly generated to simulate channels with different data ratesand vehicle speeds. The parameters were chosen based on theprospective services of W-CDMA systems [15].

We consider a reverse link of an asynchronous CDMA systemwith six users . The spreading sequences for each userand each encoded bit are independently and randomly gener-ated. The processing gain is . Each user uses a differentrandom interleaver , shown in Fig. 1, to permute their codebits. In all simulations, the same set of interleavers was used andall users have equal signal amplitudes (i.e.,

). The number of iterations within each CEI decoder is five.

The code employed is a rate binary parallel concate-nated turbo code. The two recursive convolutional constituentencoders have a generator polynomial

with effective free distance [16]. An -random interleaverwith is used. The interleaver size is and, thus,

.In the following three examples, the performances of the

turbo multiuser receiver and the conventional single-userRAKE receiver are compared. The single-user RAKE receivercomputes the code bit LLRs of userusing (40), which arefed to a turbo decoder to decode the information bits. Theaverage bit-error rates (BERs) of six users versus areplotted, where is the energy for each information bit and

is the power spectral density of the complex AWGN. Thesimulation results are based on collecting more than 200 errorsper simulation point.

The number of inner iterations other than five with differentnumbers of outer iterations are tested for the turbo multiuserreceiver for all three examples. There is no significant perfor-mance gain using more than five inner iterations, but significantdegradation is observed with less than five inner iterations. Forcomparison between inner iteration numbers, the average BERswith one inner iteration and up to five outer iterations are plottedin Figs. 4, 6, and 8, and the average BERs with five inner itera-tions and up to three outer iterations are plotted in Figs. 3, 5, and7. It is seen that the performances with five inner iterations aresignificantly better than the corresponding ones with one inner


Fig. 5. BER performance comparison between the turbo multiuser receiver and the RAKE receiver in a multipath fading channel withK = 6, processing gainN = 16, vehicle speed= 60 km/h, data rate= 38.4 kb/s, carrier frequency= 2.0 GHz.

Fig. 6. BER performance comparison between the turbo multiuser receiver with five inner iterations and one inner iteration,K = 6, processing gainN = 16,vehicle speed= 60 km/h, data rate= 38.4 kb/s, carrier frequency= 2.0 GHz.


Fig. 7. BER performance comparison between the turbo multiuser receiver and the RAKE receiver in a time-invariant multipath channel withK = 6 andprocessing gainN = 16.

Fig. 8. BER performance comparison between the turbo multiuser receiver with five inner iterations and one inner iteration in a time-invariant multipath channelwith K = 6 and processing gainN = 16.


iteration when the total number of inner iterations is five. FromFigs. 4, 6, and 8 it is seen that there is almost no performanceimprovement after three outer iterations with one inner iterationper outer iteration. Therefore, if the total number of inner itera-tions is fixed to 15, the performances with 3 outer iterations and5 inner iterations per outer iteration (shown in Figs. 3, 5, and 7)are still better than the corresponding ones with 15 outer itera-tions and one inner iteration per outer iteration.

Example 1 (Fast Vehicle Speed and Low Data Rate):In thisexample, we consider a Rayleigh-fading channel with vehiclespeed 120 km/h, data rate 9.6 kb/s, carrier frequency 2.0GHz, (the normalized Doppler is ). The resultswith five inner iterations and one inner iteration are plotted inFigs. 3 and 4, respectively.

Example 2 (Medium Vehicle Speed and Medium DataRate: Next, we consider a multipath Rayleigh-fading channelwith vehicle speed 60 km/h, data rate 38.4 kb/s, carrierfrequency 2.0 GHz, ( ). The results with fiveinner iterations and one inner iteration are plotted in Figs. 5and 6, respectively.

Example 3 (Very Slow Fading):Finally, we consider a veryslow fading channel (a time-invariant channel). The channelgains are randomly generated and fixed in simulation. Everyuser has equal received signal energy (i.e., ,for and ). The results with fiveinner iterations and one inner iteration are plotted in Figs. 7 and8, respectively.

From Examples 1, 2, and 3, it is seen that a significantperformance gain is achieved by the proposed iterative receivercompared with the conventional noniterative receiver (i.e.,the RAKE receiver followed by a channel decoder). Thereare 0.5–0.8-dB gains at BER . The performance of theproposed receiver with two outer iterations is very close to thatof the RAKE receiver in a single-user channel. Moreover, athigh signal-to-noise ratios, the detrimental effects of the MAIand intersymbol interference (ISI) in the channel can almostbe completely eliminated. Furthermore, it is seen from thesimulation results that the proposed multiuser receiver in amultiuser channel even outperforms the RAKE receiver in asingle-user channel. This is because the RAKE receiver makesthe assumption that the delayed signals from different pathsfor each user are orthogonal and no ISI cancellation is per-formed, which effectively neglects the ISI. In the contrast, byperforming (25), the ISI contained in (20) is “softly” canceledin the iterative multiuser receiver.

VII. CONCLUSION

We have proposed a low-complexity receiver for turbo-coded,asynchronous direct-sequence CDMA (DS-CDMA) systemsusing aperiodic spreading sequences. The receiver consists of asoft multiuser detector and a bank of modified turbo decoders.The soft multiuser detector uses the same decision statisticsas the conventional RAKE receiver and has a complexity

per bit per user, which is a considerable reductionover the method previously reported in [7]. Simulation resultsdemonstrate that, in asynchronous multipath fading channels,the proposed turbo multiuser receiver outperforms significantly

the conventional noniterative RAKE receiver within threeiterations. Moreover, at high signal-to-noise ratios, after threeiterations, the performance of the multiuser receiver in amultiuserenvironment is even better than that of a single-userRAKE receiver in asingle-userenvironment.

APPENDIX

PROOF OF(40)

To obtain the LLR of the code bit, a Gaussian assumption ismade on the distribution of in (11), i.e., we assume that

(41)

where is the equivalent signal amplitude; ,which is a complex random variable; and is the conditionalvariance of .

As in typical RAKE receivers [18], we assume that the sig-nals from different paths are orthogonal for a particular user.Conditioned on , using (4) and (5), the mean in (41)is given by

where the expectation is taken with respect to channel noise andthe code bits of the interfering users. The variancein (41) canbe computed as in (42) shown at the top of the following page.Using the previous orthogonality assumption, it is easy to checkthat in (42). Since terms and in (42) are zero-meanindependent random variables, we have

(43)

Due to the orthogonality assumption, the second term in (43) isgiven by

(44)

For simplicity, we assume that the time delay equals somemultiple of the chip duration. Expanding the integral above overthe -second long chip intervals, the first term in (43) canbe written as

(45)

where


(42)

Assuming that signature waveforms contain i.i.d. antipodalchips, then is an i.i.d. binary random variabletaking values of with equal probability. Since ,

, and are independent, (45) can bewritten as

(46)

Substituting (44) and (46) into (43), we have

Then, the LLR of can be written as

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers forhelpful suggestions and discussions which have enhanced thequality of the final manuscript.

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Qinghua Li received the B.S. degree in electrical engineering from South ChinaUniversity of Technology, Guangzhou, China, in 1992, the M.S. degree in signalprocessing from Tsinghua University, Beijing, China, in 1995, and the Ph.D.degree in electrical engineering from Texas A&M University, College Station,TX, in 2001.

From 1995 to 1996, he was with a government institution and a company,the Telecommunications Research Bureau of Guangdong Province and EricssonNetwork System, Dallas, TX. From 1996 to 2001, he was a Research Assistantwith the Department of Electrical Engineering, Texas A&M University, CollegeStation. Since 2001, he has been a researcher with Intel Labs. His research in-terests are in the general areas of communications and signal processing.

Xiaodong Wang received the B.S. degree in electrical engineering and ap-plied mathematics (with the highest honor) from Shanghai Jiao Tong Univer-sity, Shanghai, China, in 1992; the M.S. degree in electrical and computer en-gineering from Purdue University in 1995; and the Ph.D. degree in electricalengineering from Princeton University in 1998. From July 1998 to December2001, he was an Assistant Professor in the Department of Electrical Engineering,Texas A&M University. In January 2002, he joined the Department of ElectricalEngineering, Columbia University, as an Assistant Professor.

Dr. Wang’s research interests fall in the general areas of computing, signalprocessing and communications. He has worked in the areas of digital com-munications, digital signal processing, parallel and distributed computing, na-noelectronics and quantum computing, and has published extensively in theseareas. His current research interests include multiuser communications theoryand advanced signal processing for wireless communications. He worked at theAT&T Labs-Research, in Red Bank, NJ, during the summer of 1997. He re-ceived the 1999 NSF CAREER Award, and the 2001 IEEE CommunicationsSociety and Information Theory Society Joint Paper Award. He currently servesas an Associate Editor for the IEEE TRANSACTIONS ONCOMMUNICATIONS, theIEEE TRANSACTIONS ONSIGNAL PROCESSING, and the IEEE TRANSACTIONS

ON WIRELESSCOMMUNICATIONS.

Costas N. Georghiades(S’82–M’85–SM’90–F’98) received the B.E. degreewith distinction from the American University of Beirut in June 1980, and theM.S. and D.Sc. degrees from Washington University in May 1983 and May1985, respectively, all in electrical engineering. Since September 1985 he hasbeen with the Electrical Engineering department at Texas A&M Universitywhere he is a Professor and holder of the Delbert A Whitaker Chair. Since1997 he serves as director of the telecommunications and signal processinggroup in the Electrical Engineering Department. His general interests are in theapplication of information, communication and estimation theories to the studyof communication systems, with particular interest in receiver design, iterativedetection, space-time coding and optical systems.

Dr. Georghiades is a Fellow of the IEEE and a registered ProfessionalEngineer in Texas. Over the years he held editorial positions with the IEEETRANSACTIONS ON INFORMATION THEORY, the IEEE TRANSACTIONS ON

COMMUNICATIONS, the IEEE Journal on Selected Areas in Communications,and the IEEE Communication Letters. He has been involved in organizing anumber of conferences, including as Technical Program Chair for the 1997IEEE Communication Theory Mini Conference, the 1999 IEEE VehicularTechnology Conference, the 2001 Communication Theory Workshop and asChair of the Communication Theory Symposium within Globecom 2001. Heis currently a member of the Communication Society’s Awards Committee andthe Information Theory Society’s Fellows Committee.

Dr. Georghiades was the recipient of the 1995 Texas A&M University Collegeof Engineering Halliburton Professorship and in 1997 the J. W. Runyon Jr. En-dowed Professorship. In 2000 he received the Clear Lake Council of TechnicalSocieties Educator of the Year Award. He is the Keynote Speaker at the Interna-tional Conference onWireless and Optical Communications(WOC-2002), July2002, Banff, Canada.

Documents

Turbo multiuser detection for turbo-coded CDMA in multipath fading channels