Decision-feedback soft-input/soft-output multiuser detector for iterative decoding of coded CDMA

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 50, NO. 1, JANUARY 2001 25

Decision-Feedback Soft-Input/Soft-Output MultiuserDetector for Iterative Decoding of Coded CDMA

Mao-Ching Chiu, Associate Member, IEEE

Abstract—The optimal decoding scheme for a code-divisionmultiple-access (CDMA) system that employs convolutional codesresults in a prohibitive computational complexity. To reduce thecomputational complexity, an iterative receiver structure was pro-posed for decoding multiuser data in a convolutional coded CDMAsystem. At each iteration, extrinsic information is exchanged be-tween a soft-input/soft-output (SISO) multiuser detector and abank of single-user SISO channel decoders. However, a directimplementation of the full-complexity SISO multiuser detectoralso has the exponential computational complexity in terms ofthe number of users. This paper proposes a low-complexity SISOmultiuser detector based on tentative hard decisions that are madeand fed back from the channel decoders in the previous iteration.The computational complexity of the proposed detector is linearin terms of the number of users and can be adjusted accordingto the complexity/performance tradeoff. Simulation results showthat even with this simple feedback scheme, the performance ofthe coded multiuser system approaches that of the single-usersystem for moderate to high signal-to-noise ratios (SNRs).

Index Terms—Code-division multiple access, multiuser detector.

I. INTRODUCTION

M OST of the previous work on multiuser detection fo-cused on the area of the uncoded receiver design [1],

i.e., on the demodulation of multiuser signals. In practice, mostcode-division multiple-access (CDMA) systems employ error-control coding and interleaving. When the uncoded receiver de-sign is applied to the coded CDMA systems, the multiuser inter-ference is first resolved using the conventional uncoded receiver,and then the channel decoders are applied independently. In thiscase, hard decisions on the coded bits of each user are madeprior to decoding. This separated design philosophy is subop-timal, since we lose information by making hard decisions.

The computational complexity of the optimal decodingscheme for a convolutional coded CDMA system was shownto be [2], where is the number of users andis the code constraint length. Due to the high computationalcomplexity, it is not practical to decode the CDMA channelcode using the full-complexity decoding technique [3]. Thismotivates the study of a number of low-complexity suboptimaldecoders [3]–[10].

A synchronous (respectively, asynchronous) CDMA channelcould be viewed as a block code [8] (respectively, convolu-tional code [6]). Therefore, the iterative decoding scheme that

Manuscript received April 11, 2000; revised September 6, 2000. This paperwas presented in part at the IEEE International Symposium on InformationTheory and Its Applications, Honolulu, HI, November 2000.

The author is with the Department of Electrical Engineering, National ChiNan University, Nantou, Taiwan 545 R.O.C. (e-mail: [email protected]).

Publisher Item Identifier S 0018-9545(01)01921-1.

exchanges soft information between the multiuser detector andthe channel decoders can be employed. This technique has beensuccessfully applied to many detection/decoding problemssuch as the parallel concatenated code (turbo code), serialconcatenated code, equalization, and coded modulation. Forcoded CDMA systems, the maximuma posterioriprobability(MAP) technique is used as the multiuser detector that passesthe reliability information of the code bits to the single-userdecoders, and the single-user codes are decoded indepen-dently using the full-complexity MAP algorithm [4]–[6]. Thefull-complexity MAP algorithm can be carried out by the BCJRalgorithm proposed by Bahlet al. [12]. These results showthat, with the iterative decoding scheme, the performance ofthe coded multiuser system approaches that of the single-usersystem for moderate to high signal-to-noise ratio (SNR).The computational complexity of these methods, however, is

per bit per iteration, which is still prohibitivefor channels with a medium to large number of users.

In this paper, we focus attention on reducing the computa-tional complexity of the SISO multiuser detector. A low-com-plexity suboptimal multiuser detector based on tentative harddecisions that are made and fed back from previously decodingoutput is proposed for iterative decoding of convolutional codedmultiuser data. This new scheme gives a total computationalcomplexity of , where is anarbitrary integer with . If is fixed, the computa-tional complexity of the multiuser detector is linear with. If

, the computational complexity is equal to that proposedin [4] and [6]. Simulations show that, even with , the per-formance of the coded multiuser system approaches that of thesingle-user system.

The concept of iterative hard-decision feedback has also beeninvestigated in [11] for a so-called code-spread CDMA system,which is operated as follows. The hard decisions are obtainedfrom the previous decoding outputs of all hard-output Viterbidecoders and are combined to produce an estimate of the inter-ference for each user. The estimated interference for each useris subtracted from the received signal. The result is then fedto each single-user decoder for the next iteration of decoding.However, in our scheme, the hard-decision feedback is not di-rectly used. Instead, we leave an uncertainty on some bits ofthe hard decisions by calculating thea posteriorilog-likelihoodratio (LLR) over some neighbors of the hard decisions. The re-sulting soft-output LLR is then fed to each single-user SISO de-coder for the next iteration of decoding.

Similar iterative decoding schemes for convolutional or turbo-coded CDMA systems were proposed in [4]–[10]. The main dif-ference on the structure of those schemes as compared to our

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26 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 50, NO. 1, JANUARY 2001

Fig. 1. A coded CDMA system with iterative multiuser receiver.

scheme is the type of the SISO multiuser detector used. In [4] and[5], a full-complexity SISO multiuser detector was proposed forconvolutional coded synchronous CDMA systems, resulting in acomputational complexity of for the multiuser detector.The work was then extended to a coded asynchronous CDMAsystem[6],where the asynchronous randomCDMA channelwasviewed as a time-varying convolutional code. Based on the con-cept of the time-varying convolutional code, forward and back-ward recursions, similar to those of the BCJR algorithm [12],were proposed for the SISO multiuser detector, and analgo-rithm was presented to reduce the computational complexity. In[7],astep-wisedifferencecalculationwaspresented todetermineall the 2 possible likelihood values for synchronous CDMAsystems. In addition, a reduced-complexity multiuser detectorwas proposed to keep a variable number of likelihood values ateach step as determined by comparing likelihood values with athreshold. The max-log-map algorithm [13] is then used to ob-tain the LLR of each code bit. Another type of iterative decodingis based on the soft interference cancellation [8]–[10], where thesoft estimates of code bits are represented by real numbers ratherthan hard-decision values and are combined to produce estimatesof LLRs of the code bits. The LLRs of code bits are then fed tothe SISO channel decoders for the next iteration of decoding.

This paper is organized as follows. Section II describes themodel of the system and the iterative decoding scheme. A de-cision-feedback SISO multiuser detector is proposed in Sec-tion III. Section IV presents the simulation results about the newscheme. The conclusion of this paper is given in Section V.

II. SYSTEM DESCRIPTION

We consider a coded synchronous CDMA system whereusers access the channel, each employing a normalized modula-

tion waveform of duration chips. The baseband model of thissystem is shown in Fig. 1. The binary information data(for ) are encoded by the channel encoders withcode rate , respectively. A code-bit interleaver is then usedto reduce the influence of the error bursts at the input of eachchannel decoder. The interleaved code bits for all users arebinary phase-shift keying modulated, yielding code symbols ofduration . Each code symbol is then modulated by a spreadingwaveform and then transmitted through the channel. It isassumed that is supported only in the interval [0,] withunit energy given by

for

where is a signature sequence of as-signed to the th user and is a chip waveform of duration

. We assume symbol-synchronous transmission,in which all users transmit their signals with reference toa common clock. The received signal at the receiver is thesuperposition of the users’ signals plus the additive whiteGaussian noise, given by

wherenumber of code symbols per user per frame;

th user’s amplitude;white Gaussian noise with single-side power spectraldensity .

CHIU: DECISION-FEEDBACK SOFT-INPUT/SOFT-OUTPUT MULTIUSER DETECTOR 27

To convert the continuous-time signal to a discrete timesignal at the receiver, is first filtered by a bank of matchfilters that matches to the signals for andsampled at symbol rate. Let , where

for . The sampled signal atthe th symbol interval can be expressed by the-tuple vector

(1)

where;

diag ;;

zero-mean Gaussian noise vector with.

The superscript denotes the transpose operation. It is wellknown that represents a sufficient statistic for demodulatingthe th code bit of all the users.

The structure of the iterative multiuser receiver is shown inthe lower half of Fig. 1. Similar descriptions of this structurecan be found in [4], [6], and [10]. It consists of two stages:a SISO multiuser detector followed by parallel single-userSISO channel decoders. The SISO multiuser detector calcu-lates the marginal probabilities for the th de-coder. Then all the single-user SISO channel decoders, basedon the marginal probabilities, generate thea posterioricodedbit probabilities, which are then used as thea priori informa-tion for the SISO multiuser detector on the next iteration. Inlog-likelihood notations, the SISO multiuser detector evaluatesthea posterioriLLR of every code bit of every user, i.e.,

(2)

By Bayes’ rule, (2) can be written as

(3)

The first term in (3), denoted by , represents the ex-trinsic information [14] that will be delivered by the SISO mul-tiuser detector. The second term in (3), denoted by ,represents thea priori LLR of . For iterative decoding,

is computed by the SISO channel decoder of thethuser in the previous iteration, interleaved, and then fed back tothe SISO multiuser detector. (The superscriptindicates thequantity obtained from the previous iteration.) For the first iter-ation, no prior information is available; we havefor all and . The extrinsic information is reverse in-terleaved and then fed into theth user’s channel decoder as thea priori information of the code bit.

We assume that the error control codes employed are binaryconvolutional codes for simplicity. The extension of this resultto turbo codes is straightforward. Based on thea priori LLR

obtained from the SISO multiuser detector and the

trellis structure of the convolutional code, theth SISO channeldecoder computes thea posterioriLLR of each code bit

and outputs the extrinsic information

The extrinsic information is then interleaved and fedback to the SISO multiuser detector as thea priori informationabout the code bits on the next iteration. Thea posterioriLLR

is used to make a decision on the decoded bit at thelast iteration, as well as to make a tentative hard decision at eachiteration in our decision-feedback scheme. Regarding the com-putation of , the full-complexity BCJR algorithm [12]is employed. Some possible reduced-complexity algorithms forcomputing will be discussed in Section III-C.

III. I TERATIVE MULTIUSER DECODER

A. Full-Complexity SISO Multiuser Detector

Without ambiguity, we omit the symbol indexfor simplicity.From (1) and (2), thea posterioriLLR for the th symbolcan be expressed as

where

Note that we use the notation . Theapriori probabilities of the code bits can be expressed interms of their LLR . After some manipulation, we have

(4)

The first term in (4) is independent of , and hence canbe eliminated in the LLR calculation. Define a vector

. After simplification, wehave

(5)


where

The summations in the numerator and denominator of (5)are over all the 2 possible vectors with and

, respectively. The computational complexity of thefull-complexity SISO multiuser detector is exponential in termsof the number of users, which is certainly prohibitive forchannels with a medium to large number of users.

B. Decision-Feedback SISO Multiuser Detector

We propose a low-complexity suboptimal approach to calcu-late (5). In particular, we intend to approximate (5) by summinga relatively small number of terms that dominate the summa-tions. This is approached by employing tentative decisions thatare fed back from the channel decoders in the previous iteration.

At each iteration, a tentative hard decisionis made based onthe LLR outputs of the channel decoders, i.e.,

if ,if ,

for

The tentative decision is then fed back to the SISO multiuserdetector as a starting vector for likelihood calculation. Sinceisobtained from the channel decoders and is likely to be estimatedwell, is likely to have large likelihood value . The sum-mations are then carried out by summing the likelihoods oversome neighbors of.

Assume that and are -tuple vectors over . De-note as the vector with for .Let be a set of -tuple vectors over . Thea poste-riori LLR (5) is approximated by

(6)

The computational complexity of (6) is proportional to, thecardinality of . For example, if , then (6) isequivalent to the full-complexity SISO multiuser detector, andthe result is totally independent of the feedback. Now, theproblem is how to select the subset. First, a simple restrictionon is that for all there must exist two elements

and in such that ; otherwise, either the nu-merator or the denominator of (6) is zero, resulting a positive ornegative infinity LLR. Secondly, for all , rangesover the most likelihoods so that the selected terms dominatethe summations. However, the complexity for determining thissubset is still . Therefore, a heuristic scheme is proposedto select the subset.

Define the reliability of as the absolute value of .If a bit, say, , has a large reliability, then is likelyto have a relatively large value if theth element of isfixed to be , i.e., with . Therefore, most of the vec-tors of should have “1’s” at the relatively reliable positions.Consequently, the summations of (6) are carried out over the

terms that have the most reliable positions fixed and less reli-able positions varied. Without loss of generality, the reliabili-ties of are assumed to have nonincreasing order, i.e.,

for . Let be an integer with. Define . Let be a -tuple vector

with all elements being 1 except for theth element, which is1. Define ,

where is the all-one -tuple vector. Furthermore,define as the concatenation of two vectors. The setischosen to be

and

With this selection, the computational complexity of (6) is pro-portional to . For example, assume and thereliabilities of are in nonincreasing order. Let ;then can be determined to be

If is fixed, the computational complexity of (6) is linear with. Now, consider the special case when . The subset

contains 1 vectors. Each vector in except theall-one vector has one element that equals1; otherwise, 1.Consequently, the result from the calculation of (6) much relieson the feedback, since only has at most one bit differencefrom the feedback.

The development of the decision-feedback scheme does notspecify the first stage, which delivers the initial estimate of.Yet, conventional multiuser detectors can be employed for theinitial estimate of . We will focus on the class of linear detec-tors. The linear detectors form the initial estimatebased on theoutput of various linear filters

In particular, we are interested in the decorrelator and the linearminimum mean-squared error (MMSE) filter [1]. Both lineardetectors were given to be

for the decorrelator and

for the MMSE filter. The calculation of either the MMSE filteror the decorrelator requires a matrix inversion, which is a largecomputational burden. Iterative implementations for the decor-relator and MMSE filter can significantly reduce the computa-tional complexity [15]–[17]. However, these implementationsare outside the scope of this paper, and hence direct implemen-tations of the decorrelator and MMSE filter are adopted in thisstudy.

In uncoded multiuser detection [18]–[20], conventional mul-tiuser detectors are also employed to obtain an initial estimateof multiuser data. In [18], a set of nearest neighbors determined


by the Voronoi region of a signal point is considered for sub-optimal detection, and a decorrelator is used to find a startingsignal point in the multiuser signal constellation. In the firststage, the detected signal point is initially set to the startingsignal point, and is then iteratively updated by the signal point inits nearest neighbors that is closer to the received signal point. In[19] and [20], multistage detectors with hard-decision parallelcancellation for synchronous and asynchronous CDMA systemswere proposed. The initial estimate of the multiuser data is alsoobtained by conventional detectors. The estimates of multiuserdata at current stage are combined to produce an estimate ofmultiuser interference for each user. The multiuser interferencefor each user is then subtracted from the received signal. Theresults are then used to make a new estimate of the multiuserdata for the next stage. Although those multistage detectors areproposed for uncoded CDMA systems, they are quite similar toour scheme, as the hard-decision feedback is repeatedly updatedby some kinds of iterative processes. However, our goal is notto make a hard decision on the multiuser data in the stage of theSISO multiuser detector. We use the neighbors of the hard-de-cision feedback that are likely to have larger likelihood value togive an accurate estimate of thea posterioriLLR of the codebits.

C. SISO Channel Decoders

In order to identify the effect upon using the decision-feed-back SISO multiuser detector, we use the full-complexity BCJRalgorithm [12] in this study. Nevertheless, to further reduce thecomputational complexity, some simplified suboptimal BCJRalgorithms may be employed in this system. The main simpli-fication of the BCJR algorithm is the soft-output Viterbi algo-rithm (SOVA) [13], in which a simplified estimate of the datasymbol probability by using the difference between path metricsin a Viterbi algorithm is associated. In [21], to reduce the exces-sive memory requirement of the BCJR algorithm, the forwardand backward recursions are carried out within small windowsize rather than the entire code sequence, saving the memory re-quired to store the entire state metric history. In addition, it wasshown that the modified BCJR algorithm can be implementedwith no more than four times of the complexity of a Viterbi de-coder for the same code [21]. In [22], it is observed that mostprobabilities obtained in the BCJR algorithm are very small.These small probabilities can be truncated to zero, saving therelated computation that stems from them without losing errorperformance.

A more closely related work that may incorporate with ourdecision-feedback SISO multiuser detector is possibly the onereported in [23], originally proposed for turbo decoding. Themethod is based on the observation that the values of certaininformation symbols, state variables, and codeword symbolsoften can be ascertained reliably early during iterative decodingprocess. If such candidates can be identified and deleted early,the computational complexity of subsequent decoding iterationscan be reduced. A possible method that combines our deci-sion-feedback SISO multiuser detector with the early detectionscheme is described as follows.

In the SISO channel decoder, some code bits whose LLRsare larger than a threshold can be early detected in the decoding

process. In the SISO multiuser detector, the early detected bitsin parts of the feedback are assigned to have infinite relia-bility (without uncertainty). The interference from the early de-tected bit is cancelled from the received signal, saving the com-putation of LLRs of the early detected bits. The early detectedbits are then identified in the SISO channel decoders, signifi-cantly reducing the branch complexity of the BCJR algorithm.This scheme is attractive for reducing the computational com-plexity. However, this paper focuses on the evaluation of the de-cision-feedback SISO multiuser detector, and only simulationresults with the full-complexity BCJR algorithm are presented.

IV. SIMULATION RESULTS

In this section, simulation results of the proposed iterative de-coding for coded CDMA systems are presented for five, ten, and15 users. Of particular interest are the receivers that employ thedecision-feedback SISO multiuser detectors. All the users in thesystem employ the same rate 1/2 constraint length convo-lutional code with generators 23, 25 in octal notation. Each useruses a different pseudorandom interleaver, and the same set ofinterleavers is used for all simulation runs. All simulations use ablock size of 500 information bits for each user. Both the decor-relator and MMSE filter are tested in the simulation as the firststage that delivers the initial estimate of the code bits. However,the results show no significant difference between the perfor-mance curves of both cases. Therefore, in the following, onlyresults for decorrelators as the first stage are presented.

First consider a five-user system with equal cross-correla-tion , . All the users have thesame power. The simulation is tested for the first five itera-tions. However, for clarity, only the performance curves withone iteration and five iterations are shown in Fig. 2. Three de-cision-feedback SISO multiuser detectors withare considered in the simulation. The performance of the iter-ative decoding scheme that employs the full-complexity SISOmultiuser detector is also shown in the figure. In addition, thesingle-user performance ( ) that serves as a lower boundfor multiuser systems is given. At the first iteration, the systemthat employs the decision-feedback SISO multiuser detector isobviously worse than the system using the full-complexity onefor all SNRs considered. After five iterations, the performancecurves for the decision-feedback SISO multiuser detector havenoticeable loss at low SNR. However, the loss becomes invis-ible for moderate SNR, where all performance curves convergetoward the single-user performance. Moreover, even with thishigh correlation, the performance loss due to the decision-feed-back SISO multiuser detector is small, and the curves convergerapidly to the single-user performance. Fig. 3 shows the perfor-mance with five users and . In this example, the loss be-comes apparent at low SNR. However, the performance curvealso converges to the single-user performance for sufficientlyhigh SNR.

Consider a near–far situation for the five-user system with. The signal strength of the desired user, which in our

case is the first user, is fixed at 3.5 dB. In Fig. 4, the perfor-mance curves of the desired user are depicted as functions ofSNR /SNR ( ) ranging from 8 to 8 dB. After five


Fig. 2. Performance of multiuser iterative decoders for five-user system with� = 0:6 and an equal power for all users.

Fig. 3. Performance of multiuser iterative decoders for five-user system with� = 0:7 and an equal power for all users.

iterations, the performance of user 1 that adopts the full-com-plexity SISO multiuser detector actually improves as the inter-fering users become stronger, a phenomenon that is consistentwith the finding of [1]. Roughly speaking, when the interfering

users are sufficiently strong, the code bits of the interfering userscan be correctly estimated with high probability. Therefore, thechannel noise, rather than the randomness of the code bits of theinterfering users, becomes the primary source of errors. When


Fig. 4. Near–far performance of multiuser iterative decoders for five-user system with� = 0:6. The desired user is the first user with SNR= 3:5 dB.

Fig. 5. Performance of multiuser iterative decoders for ten-user system with� = 0:6 and an equal power for all users.

the power of the interfering users is less than the power of thedesired user, the performance loss after five iterations is signif-icant but not dramatic. The loss becomes invisible when the in-terfering users are stronger than the desired user and all the per-

formance curves converge to the single-user performance. Fur-thermore, when the power of the interfering users is sufficientlyless than that of the desired user, increasing the number of iter-ations cannot result in a significant performance improvement.


Fig. 6. Performance of multiuser iterative decoders for 15-user system with� = 0:6 and an equal power for all users.

Now, consider a ten-user system with . For large-usersystems, the computational complexity of the full-complexitySISO multiuser detector is extremely high, and hence the exactSISO multiuser detector is not feasible for practical implemen-tation; whereas the proposed decision-feedback scheme has areasonable and adjustable complexity that can be easily im-plemented even for very large. Fig. 5 shows the simulationresults for the ten-user system. It is obvious that after five it-erations, the performance loss for the decision-feedback SISOmultiuser detector is significant at low SNR and becomes invis-ible for sufficiently high SNR. In addition, all the performancecurves converge to the single-user performance for sufficientlyhigh SNR.

Fig. 6 shows the performance of a 15-user system with. Since the computational complexity of the full-complexity

SISO multiuser detector is too high even for computer simula-tion, only the results for the decision-feedback scheme are givenin the figure. In this example, a threshold effect is becoming ap-parent at the low SNR after five iterations. Below the threshold,the performance is unusable. Above the threshold, the perfor-mance rapidly improves. The performance curves forand converge to the single-user performance for suffi-ciently high SNR. In addition, from the trend of the curve, weconjecture that the performance for also converges to thesingle-user performance.

V. CONCLUSION

A decision-feedback soft-input/soft-output multiuser de-tector has been proposed for iterative decoding of a codedsynchronous CDMA system. This scheme is developed based

on tentative hard decisions that are made and fed back fromthe channel decoders in the previous iteration. The complexityof the proposed detector is linear in terms of the number ofusers and can be adjusted according to the complexity/per-formance tradeoff. At moderate SNR, it performs closely to,but is less complex than, the full-complexity SISO multiuserdetector. The simulation results showed that even with a largecross-correlation value and/or a large number of users, theperformance curves for the proposed detector converge to thesingle-user performance. We notice that there is a visible loss inthe low SNR, compared with that of the full-complexity one. Inaddition, a near–far situation showed that the performance lossdue to using the decision-feedback SISO multiuser detectoris significant but not dramatic and becomes invisible whenthe power of the interfering users is stronger than that of thedesired user.

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Mao-Ching Chiu (S’92–A’96) was born in Chiayi,Taiwan, R.O.C., on April 12, 1968. He received theB.S. degree in electronic engineering from NationalTaiwan University of Science and Technology,Taipei, Taiwan, in 1990 and the M.S. and Ph.D.degrees in electrical engineering from NationalTsing Hua University, Hsinchu, Taiwan, in 1992 and1996, respectively.

Since August 1998, he has been with the De-partment of Electrical Engineering, National ChiNan University, Nantou, Taiwan, as an Assistant

Professor. His current research interests include digital transmission, error-cor-recting codes, and spread-spectrum communications.

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Decision-feedback soft-input/soft-output multiuser detector for iterative decoding of coded CDMA