IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 5, MAY 1997 605

A Linear Receiver for Coded Multiuser CDMAPaul D. Alexander,Member, IEEE,Lars K. Rasmussen,Member, IEEE,and Christian B. Schlegel,Member, IEEE

Abstract—In this paper we consider a CDMA system witherror-control coding. Optimal joint decoding is prohibitivelycomplex. Instead, we propose a sequential approach for han-dling multiple-access interference and error-control decoding.Error-control decoding is implemented via single-user soft-inputdecoders utilizing metrics generated by linear algebraic multiusermetric generators. The decorrelator, and a new scheme termedthe projection receiver, are utilized as metric generators. For asynchronous system, the coded performance of the projectionreceiver metric is shown to be superior to the decorrelatoreven though they are equally complex. Also, the theoreticaldegradation relative to the single user bound is derived.

Index Terms—CDMA, convolutional codes, random codes.

I. INTRODUCTION

T HE majority of research on receivers for CDMA commu-nications systems has been in the area of uncoded receiver

design [1]–[5]. Once the multiuser interference (MUI) has beenresolved, error-control coding can be applied independently.In this case, hard decisions on the coded bits of each user aremade prior to decoding. Clearly, this segmented design philos-ophy is suboptimal since we are throwing away information bymaking hard decisions. Recent results in multiuser informationtheory reveal that, in order to approach the channel capacity,efforts should be invested in the design of a joint error-controlcodebook for the users rather than a joint spreading codebook[6]. The system that jointly considers MUI and error-controlcoding, however, is prohibitively complex. Furthermore, theuse of spreading codes has other system benefits, e.g., syn-chronization, design complexity, etc. We therefore persistconsidering spreading-code-based CDMA systems. To limitthe complexity of the joint decoder, we propose a sequentialapproach. A multiuser metric generator eliminates the MUIfrom all but a set of users, and generates soft output reliabilitymetrics for all possible transmissions of the set. These metricsare then fed to an optimal joint error-control decoder for thelimited sized set. The paper will focus on the case where theset size is one, leading to a single-user error-control decoder.The effects of the multiuser device is included in the decoder

Paper approved by G. L. Stuber, the Editor for Spread Spectrum of theIEEE Communications Society. Manuscript received May 4, 1996; revisedOctober 9, 1996. This work was supported in part by Telstra Australia underContract 7368 and the Commonwealth of Australia under International S &T Grant 56. This paper was presented in part at the 33rd Annual AllertonConference on Communications, Control, and Computing, IL, October 1995.

P. D. Alexander is with the Mobile Communications Research Centre, Sig-nal Institute for Telecommunications Research, University of South Australia,SA 5095, Australia.

L. K. Rasmussen is with the Strategic Research Group, Center for WirelessCommunications, Singapore Science Park II, 117674 Singapore.

C. B. Schlegel is with the Department of Electrical Engineering, Universityof Utah, Salt Lake City, UT 84112 USA.

Publisher Item Identifier S 0090-6778(97)03717-3.

through the decoder metrics. The resulting structure is thus alow-complexity, suboptimal approach to joint decoding.

As mentioned by Duel-Hallenet al. [7], system design forcoded CDMA is scarce in the literature. In this paper, weaddress this area by presenting a new type of metric generator.The operation of this device is geometrical in nature, and isthus termed the projection receiver. The initial derivation of theprojection receiver for the synchronous channel by Schlegeland Xiang can be found in [8, 9]. A similar receiver for theuncoded case has been suggested independently by Varanasiin [10]. We compare the coded performance of the projection-receiver- and decorrelator- [2, 11] fed decoders, and show thatthe projection-receiver-fed system performs at least as well asthe decorrelator-fed system.

For the synchronous channel, where there is no unintentionalcorrelation between subsequent symbols transmitted by anygiven user, the appropriate decoding metric is the individualcodeword bit distances. When unintentional correlation isencountered over adjacent symbol intervals, an intersymbolinterference (ISI)-type receiver is employed where the metricdepends not only on the current hypothesized symbol, but alsoon some number of previous symbols. This type of metricgeneration is not considered in this paper.

The paper is organized as follows. In Section II, we intro-duce the uplink CDMA communication system where error-control coding is employed. A new linear metric generatoris introduced in Section III, along with the decorrelator. Theuncoded and coded performance of the system under thedifferent metric sources are theoretically derived in SectionIV. The averaged performance degradation on the randomcode CDMA channel compared to the single-user bound iscomputed in Section V for the synchronous case. In SectionVI, the theoretical results are verified via simulation.

Throughout this paper, scalars are lower case, vectors areunderlined lower case, and matrices are underlined uppercase. Subscripting is dropped where no ambiguities arise.The symbols are the transposition and inversionoperators, respectively. The delimiter defines a space ofdimension Vector subscripting can be of the formdenoting the th component of vector When is someconstant, then the subscript implies vector size, i.e.,is acolumn vector of zeros of size

II. UPLINK MODEL FOR SYMBOL SYNCHRONOUS CDMA

In this section, the uplink model for the CDMA commu-nication system considered throughout this paper is brieflydescribed. The uplink model is based on a discrete-time sym-bol synchronous, chip synchronous CDMA system assuming

0090-6778/97$10.00 1997 IEEE

606 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 5, MAY 1997

single-path channels and the presence of stationary additivewhite Gaussian noise with zero mean and variance

The users in this communication system first encodetheir length information sequences viaan encoder with output frame length into a lengthsequence where and and arethe user index and the coded bit interval index, respectively.Each user then spreads this encoded sequence by choosing aspreading code which modulates the codedbit and is transmitted using BPSK over theAWGN channel.

When each user picks a different spreading code for eachtransmission, we call this channel the random-code (RC)CDMA channel, and when the user picks one code and usesthat for each bit in the entire frame, we call the resultingchannel the fixed-code (FC) CDMA channel.

Since the waveforms are chip aligned at the receiver, theoutput of a chip waveform matched filter can be expressed asa linear combination of spreading codes. Note that we haveassumed that the received energy contributed by each user inone symbol interval is normalized to i.e.,the received energy in one symbol interval from each user isequal to The output of the chip matched filter of this chipsynchronous system in symbol intervalcan be expressed as

(1)

The single-symbol interval channel matricesand the datavector are defined as follows:

The sampled noise corrupting the output of thechip matched filter is independent in each sample since thechannel noise is assumed to be white and the chip waveformsare assumed to fulfill the Nyquist criterion (e.g., rectangularchip pulses). The noise vector in symbol intervalis

wheredefines the Gaussian distribution with meanand variance

and is the one-sided noise power spectral density. Atthe input to the CDMA channel are encoders, and at theoutput is the multiuser receiver that generates metrics for the

soft input single-user decoders. The single-user decodersform their branch metric for a particular codeword by taking

metrics from the codeword bit metric streams feeding thedecoder. For example, if101 were a codeword in user’scodebook, then the corresponding branch metric would be

This concept is easilyextended to multiuser decoding.

III. L INEAR METRIC GENERATION

For each user, we aim to produce a streamconsisting of codeword bit Euclidean distances. The distancesare computed as the distance from the output of the multiuserreceiver to the two hypotheses, namely,and In general,any multiuser receiver could be used as the metric generator.We will study the decorrelator and the projection receiver.

A. Decorrelating Receiver Metric

The well-known decorrelating filter or decorrelator [2, 11]eliminates all of the multiuser interference inat the expenseof noise enhancement. The output of the decorrelator is

(2)

A Euclidean metric for the hypothesis based on thedecorrelator statistic is

(3)

A Euclidean metric for some sequence is simply the sum ofthe squared individual symbol distances

B. Projection Receiver Metric

The constrained maximum likelihood sequence estimate ofis

where The complexity of such a constrainedsearch is prohibitive. Let us partition into two sets as

where, without loss of generality, we have chosen convenientsets.

An important property of the projection receiver is thatcomplexity is saved by estimating over the real (orunconstrained) domain, i.e., and only overthe constrained domain, i.e., This is a complexitysaving proportional to over the optimal MLSE case.Note that the number of transmissions in isThe channel now can be rewritten as

(4)

where has as its columns some subset of sizeof thecolumns of and has the complementary set of size

The suboptimal hybrid constrained/unconstrainedMLSE of is now

(5)

The solution to the inner minimization1 of (5) is

The constrained minimization problem for is now

(6)

where

(7)

1Via quadratic minimization over^dU;i [8, 9].

ALEXANDER et al.: LINEAR RECEIVER FOR CODED MULTIUSER CDMA 607

is the projection matrix onto the null space of Themetric in (6) is computed by first canceling the hypothesisand then projecting out all other symbols, hence, we have theterminology projection receiver. Since the noise inis white,we can interpret (6) as the Euclidean metric for the hypothesis

Although we have restricted the unconstrained symbols tobe at the end of any set of transmissions can be detectedby the projection receiver. It is straightforward to isolate anydesired transmissions through column and row permutationsof (4).

The constrained minimization in (6) can be modified toinclude: 1) a single symbol of one given user, 2) a singlesymbol of several users, or 3) several symbols of a given userin asynchronous CDMA.2 Case 1) is considered in this paperwhere we are interested in only one symbol at a time, saysymbol of user The metric is then generated as

(8)

where projects out all transmissions in symbol intervalbut user ’s.

C. Uncoded Performance Comparison

For the purpose of comparison between the projectionreceiver- and the decorrelator-fed system, let us considerthe component of as required in (3). Asmentioned, through permutations, componentcan be isolatedat the end of Using a well-known matrix inversion lemma[12, eq. (7), p. 931], we have

where

(9)

We note that is nothing more than theth diagonalelement of the decorrelation matrix [2] for symbol intervali.e., where We note that isnot the correlation between two users’ spreading codes as issometimes defined in the literature. By (3), we now have themetric generated by the decorrelator for the hypothesisin the form

(10)

Lemma 1:The performance of the decorrelator and theprojection receiver are identical in uncoded communication.

Proof: In uncoded operation, hard decisions are madeon as follows:

2The synchronous CDMA channel decouples by symbol interval, andsequence metrics are redundant.

When using metrics for hypothesis testing, all terms in themetric that are independent of the particular hypothesis underconsideration can be considered as a constant. The decisionsmade by the projection receiver using (8) become

and the decorrelator decisions from (10) are

where Since is a positive constant (w.r.t.the detection of the decisions made by the projectionreceiver will be the same as the decisions made by thedecorrelator.

IV. DECODING BER ON THE

SYNCHRONOUS CDMA CHANNEL

The branch metrics for the decoders may be developed asa sum of squared codeword bit distances. We consider a path

through the single-user trellis corre-sponding to user In order to use the developed performanceanalysis for convolutional codes (e.g., [13]), we must derivean expression for the probability that a Hamming weight

path has a better metric than the all zero path (or all-1path) which we assume to be transmitted. To this end, wedefine the metric for the path with length corresponding to

information bits, each encoded intochannel symbols, as

where the branch metric is expressed as a sum of codeword bitdistances, and is the distance metric for codewordbit in codeword transmitted by user The Hamming

weight error vector has elementsequal to 2. We place the symbol numbers of these errors intothe set So the first error occurs at bit the second atbit etc.

A. Decorrelator-Fed Decoder BER

Using (2), we can redefine the decorrelator metric as

(11)

For the decoder of user fed by the decorrelator metric

608 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 5, MAY 1997

stream,

(12)

where

and thus

(13)

where we have utilized the fact that the statistical propertiesof are independent of

B. Projection-Receiver-Fed Decoder BER

From Lemma 1, we see that

(14)

where the path selected by the projection-receiver-fed decoderis given in terms of the decorrelator metric. We can then getthe probability for the projection receiver from (12)

where

(15)

We therefore get

(16)

Lemma 2:For RC synchronous CDMA using Euclideanmetrics for decoding,

Proof: The arguments of the functions in (13) and(15) satisfy

(17)

due to Jensen’s inequality since is convex in

Corollary 1: For an FC synchronous CDMA channel,

Proof: In the FC channel, each user is assigned a par-ticular spreading code. It follows that It thenfollows that thus, equality is achieved in (17),and the proof is complete.

The result stated in Lemma 2 may seem surprising since,for uncoded communication, the error rates are the same forthe projection receiver and decorrelator systems (Lemma 1).The difference in the coded case arises because the metrics arenow sums over symbol intervals rather than one-shot single-symbol metrics. The decoders build their metrics as the sumof squared codeword bit distances. For the sake of discussion,consider a metric in the decoder for a path of length 2. Theprojection receiver is based on orthogonal projections, andhence has the property that equiprobable metric contours arecircles. The signal power of the projection receiver is de-creased (length of projection), but the noise power is invariantunder these orthogonal projections [14]. The decorrelator cangeometrically be described by a nonorthogonal projection. Ineach symbol interval, the signal power remains constant, butthe noise power is enhanced according to the nonorthogonalprojection, leading to elliptic equiprobable metric contours.3

A Euclidean distance metric is therefore not appropriate forsequence decoding for the decorrelator. This accounts forthe degradation of the decorrelator relative to the projectionreceiver system in the coded case, despite the fact that, bit bybit, the SNR’s are identical for the two receivers.

It is clear that the difference between the projection receivermetrics and the decorrelator metrics is that the projectionreceiver has normalized the variance of the distance metricacross the symbol intervals.

V. AVERAGE PROJECTION RECEIVER

DEGRADATIONS FROM SINGLE-USE BER

In this section, we determine the losses relative to single-user performance for communication over the random codeCDMA channel using the PR to drive the single-user decoders.

A. Gold Code Synchronous CDMA Channel

A special case of the CDMA channel is of interest since itis often used in simulation studies in the literature. Considera system where Gold codes [15] are utilized as spreadingcodes. It can be shown that the ’s for such a channel areindependent of and and are given by

(18)

3The axis lengths are different due to the variation of� with symbol interval.

ALEXANDER et al.: LINEAR RECEIVER FOR CODED MULTIUSER CDMA 609

Fig. 1. Decoder performance for theK = 10; N = 15 synchronous CDMA channel.

Since, for the single-user system, then using (16),the loss in decibels from the single-user bound for a systemusing single-symbol metric generation and Gold codes is

(19)

For example, when and the loss is 0.39dB. This is verified by simulations presented in Section VI. Itis apparent from (19) that as the loss is

If then there is no degradation from thesingle-user bound. This is not surprising since these conditionsyield a very lightly loaded system. Conversely, as the systemapproaches full capacity (i.e., the loss will be 3 dB.

B. RC Synchronous CDMA Channel

For an FC channel, the’s for a particular user are constant,and the bounding follows easily as in Section V-A. However,for an RC channel, statistics of the distribution of the’s mustbe found in order to evaluate Concentrating onthe argument of the function of (16), we get

where is the orthonormal basis for the null space ofof dimension are

the columns of and Consider the expected

value of the scalar over

since and will be independent in the case of interest

It follows that isalso computed as a lower bound in [16]. Letting part of theargument of the function of (16) be defined as

the expected value of is easily found to be

For the average performance of the projection-receiver-fedsystem, we take the expectation of over as follows:

where the inequality arises from the application of Jensen’s in-equality for the convex function.4 The loss in decibelsrelative to the single-user bound is

(20)

As an example, consider the systemsimulated in Section VI. The loss in decibels from single-userperformance is lower bounded by 4.0 dB according to (20).According to the simulations, the loss is approximately 4.5 dB.

4Althoughpx is concave, the convexity ofQ(x) is stronger, and therefore

Q(px) is convex rather than concave.

610 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 5, MAY 1997

VI. SIMULATION

In this section, we simulate the projection receiver anddecorrelator metric stream versions of the coded system onthe synchronous CDMA channel with both random and fixedspreading codes. The error-control code used is a rate 1/2 codewith constraint length generators and

A system consisting of ten users with spreading codes oflength 15 is considered. Two additional systems are includedfor comparison, namely, the matched filter receiver and anonlinear approximate MAP (AMAP) probability approach[17]. The matched filter system is a very low complexitymetric generation scheme which contrasts with the AMAPsystem which has the highest complexity of all four systems.We should note that the maximum likelihood solution cannotbe simulated due to its enormous complexity, and only theasymptotic performance is known.

As was shown in Section IV, the performance of theprojection receiver and the decorrelator is identical for anFC CDMA channel. This is verified in Fig. 1 where theperformance of the single-user system and the synchronouschannel, where each user transmits using a Gold code oflength 15, is shown. It is also verified that for the RC CDMAchannel, the projection receiver provides better performanceover the synchronous channel. The losses in performance fromthe single-user bound are also shown, and correspond with thetheoretical results of Section V.

We see that a substantial price is paid for allowing randomspreading codes. The advantage of the synchronous channel isthat spreading code design is possible, and the Gold codes area good example. It is true, however, that the management ofthe synchronous channel with such special codes may negatethe large benefits it has over the asynchronous channel.

VII. CONCLUSION

We have derived a new multiuser linear complexity receiver,and have used it to supply metrics to soft-input single-userdecoders in a coded synchronous CDMA communications sys-tem. It was shown analytically that the projection-receiver-feddecoders will perform better than the decorrelator-fed decodersfor randomly selected spreading codes, and identically for afixed assignment of spreading codes.

REFERENCES

[1] A. Duel-Hallen, “A family of multiuser decision-feedback detectorsfor asynchronous code-division multiple-access channels,”IEEE Trans.Commun., vol. 43, pp. 421–434, Feb. 1995.

[2] R. Lupas and S. Verd́u, “Linear multiuser detectors for synchronouscode-division multiple-access channels,”IEEE Trans. Inform. Theory,vol. 35, pp. 123–136, Jan. 1989.

[3] Z. Xie, C. K. Rushforth, and R. T. Short, “Multiuser signal detectionusing sequential decoding,”IEEE Trans. Commun., vol. 38, pp. 578–583,May 1990.

[4] M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronouscode-division multiple-access communications,”IEEE Trans. Commun.,vol. 38, pp. 509–519, Apr. 1990.

[5] L. Wei and C. Schlegel, “Synchronous DS-SSMA with improveddecorrelating decision-feedback multiuser detection,”IEEE Trans. Veh.Technol. (Special Issue on Future PCS Technologies), vol. 43, pp.767–772, Aug. 1994.

[6] A. J. Grant and P. D. Alexander, “Randomly selected spreading se-quences for coded CDMA,” inIEEE Int. Symp. Spread SpectrumTechniques Appl., Mainz, Germany, vol. 1, Sept. 1996, pp. 54–57.

[7] A. Duel-Hallen, J. Holtzman, and Z. Zvonar, “Multiuser detection forCDMA systems,”IEEE Personal Commun. Mag., pp. 46–58, Apr. 1995.

[8] C. Schlegel and Z. J. Xiang, “Multiuser projection receivers,” inProc.IEEE Int. Symp. Inform. Theory, Whistler, Canada, Sept. 1995, p. 318.

[9] C. Schlegel, S. Roy, P. Alexander, and Z. Xiang, “Multi-user projectionreceivers,”IEEE J. Select. Areas Commun., vol. 14, Oct. 1996.

[10] M. K. Varanasi, “Group detection for synchronous Gaussiancode–division multiple-access channels,”IEEE Trans. Inform. Theory,vol. 41, pp. 1083–1096, July 1995.

[11] P. Jung and J. Blanz, “Joint detection with coherent receiver antennadiversity in CDMA mobile radio systems,”IEEE Trans. Veh. Technol.,vol. 44, pp. 76–88, Feb. 1995.

[12] K. Ogata, Discrete-Time Control Systems. Englewood Cliffs, NJ:Prentice-Hall, 1987.

[13] S. B. Wicker, Error Control Systems for Digital Communication andStorage. Englewood Cliffs, NJ: Prentice-Hall, 1995.

[14] P. D. Alexander and P. Jung, “A unified approach to multiuser receiversand their geometrical interpretations,” inInt. Symp. Personal, Indoor,and Mobile Radio Commun., vol. 3, Sept. 1995, pp. 970–974.

[15] R. Gold, “Optimum binary sequences for spread spectrum multiplexing,”IEEE Trans. Inform. Theory, vol. IT-13, pp. 619–621, Oct. 1967.

[16] U. Madhow and M. L. Honig, “MMSE detection of direct-sequenceCDMA signals: Analysis for random signature sequences,” inProc.IEEE Int. Symp. Inform. Theory, San Antonio, TX, Jan. 1993, p. 49.

[17] P. D. Alexander and L. K. Rasmussen, “Sub-optimal MAP metrics forsingle user decoding in multiuser CDMA,” inInt. Symp. Inform. TheoryAppl., vol. 2, Sept. 1996, pp. 657–660.

Paul D. Alexander (S’89–M’96) was born in Syd-ney, Australia, on April 11, 1969. He received theB.E. and M.Eng.Sc. degrees from the Universityof Adelaide, South Australia, in 1991 and 1995,respectively. In early 1996, he completed his Ph.D.dissertation entitled “Coded multiuser CDMA,” atthe University of South Australia and it is now underexamination.

Since his dissertation submission, he has been aPostdoctoral Fellow in the Mobile CommunicationsResearch Centre, University of South Australia. His

research interests include multiuser communication theory in general andmultiuser CDMA systems specifically.

Lars K. Rasmussen(S’92–M’93), for a photograph and biography, pleasesee p. 220 of the February 1997 issue of this TRANSACTIONS.

Christian B. Schlegel (S’86–M’88), was born in St. Gallen, Switzerland,on August 22, 1959. He received the Dipl.El.Ing. ETH from the FederalInstitute of Technology, Z̈urich, in 1984, and the M.S. and Ph.D. degrees inelectrical engineering from the Uiversity of Notre Dame, IN, in 1986 and1988, respectively.

In 1988 he joined the Communications Group at the research center of AseaBrown Boxeri, Ltd., Baden, Switzerland, where he was involved in mobilecommunications research. He spent the 1991/1992 academic year as VisitingAssistant Professor at the University of Hawaii at Manoa, HI, before joiningthe Digital Communications Group at the University of South Australia,Adelaide, Australia, where he supervised research in mobile communications.In 1994 he joined the University of Texas at San Antonio. He is currently withthe Department of Electrical Engineering, University of Utah, Salt Lake City.His interests are in the area of digital communications, coded modulation,mobile radio, and multiple access communications. he recently completedthe research monographTrellis Coding (Piscataway, NJ: IEEE Press, 1997)and is currently working on a book entitledCoordinated Multiple UserCommunications,jointly with Dr. Alex Grant.

Dr. Schlegel is a member of the IEEE Information Theory and Commu-nications Societies. His work is widely published and he consults for andcooperates with diverse national and international industrial and academicpartners.