ImprovedMultistageDetectorby Mean-FieldAnnealingin Multi-UserCDMA

ThomasFabricius,andOle WintherInformaticsandMathematicalModelling

TechnicalUniversityof DenmarkCopenhagen,Denmark

Abstract— We proposea multistage multi-user detector for Code DivisionMultiple Accessthat implements Mean-Field Annealing. Annealing is a wellknown technique to avoid local minima in detection and optimisation prob-lems. Local minima certainly occurs in CDMA multi-user detection leadingto anomalousBit Err or Rate curves. Annealing is implementedasa standardmulti-user detector implementing interfer encecancellation,but with a controlparameter determining the slopeof the squashing/mappingfunction. We de-ri ve a bound on the control parameter for the first stageof the detector. Wealsoproposean annealingscheme.Weshow empirically the performanceof theproposeddetector, which follows closelythe optimum detector implementedbyexhaustive search, i.e. has gains of several dB’s compared to standard multi-user detectors lik e linear-MMSE, linear-MMSE first stagefollowed by inter-ferencecancellation,and multi-stage interfer encecancellationwith clipped softdecision,the lastbeingthe closestto our proposeddetector. Theproposeddetec-tor gainsup to

�dB compared to the Clipped Soft DecisionMultistage detector

at no addedcomplexity.

Keywords—CDMA, Multistage Multi-User Detection, Interfer enceCancel-lation, Anomalous ����� -curves.

I . INTRODUCTION

Direct-SequenceSpread-Spectrumcodedivision multiple ac-cess(DS-CDMA) systemshave desirablepropertiesto achieve ahighspectralefficientsystem.Thesearerobustnessagainstchan-nel impairments,suchasdispersionandfading,having gracefuldegradation,and easeof cellular planningby dynamicchannelsharingat variousdatarates[1]; all resultingin smoothhandlingof differentQuality of Services(QoS)anda highoverall networkcapacity. In orderto achieve this high capacitythe detectorhasto implementMaximumLikelihood(ML) detectionwhichis wellknown to haveexponentialcomplexity in thenumberof usersandthe coherencetime of the channel. Basedon this fact muchat-tention hasbeenusedin deriving nearoptimal low complexityreceivers [2]. One of the most promisingsuggestionsfor sucha detectoris multistageInterferenceCancellation,either imple-mentedby successive or parallelstructureresultingin the (SIC)or (PIC) [3], [4]. In thesedetectorsa non-linearityis usedto sep-arateout the transmittedsymbolsfrom varioususers.In caseofbinary signallingthe mostusednon-linearitiesaresign, tangenthyperbolicor clippedsoft decision. In thefirst stagesof thede-tectorlinear functionsareoftenusedinsteadof thenon-linearity[2] i.e. implementingiteratively adecorrelatingor LinearMMSEdetectorin thefirst stages[5]. Themultistagedetectorwith signsquashingfunctioncanbeseenasa local searchbasedoptimiserfor theML solution,whichhasconvergenceto a(local)minimumof thenegative log-likelihood. Convergenceto a local minimumis alsoobserved for the softernon-linearitieshyperbolictangentandclippedsoftdecision.In thiscontributionwill weempiricallyshow that theselocal minimaof thenegative log-likelihoodleadto a Bit Error Rate( ��� ) curve that is in excessof the inherent��� behaviour of the optimal detector. This excess ��� be-haviour shouldbecontrastedto the inherent���� -floor reportedin [6] thoughrelated. We will explain this relation. The local

minima leadto an excessBER for the local searchbasedmulti-stagereceivercomparedto thatusingexactenumeration.Thisfactleadsus to the main contribution, namelya multistagedetectorstructurethat implementsAnnealing.Annealingis a well known[7] generalapplicableheuristicto avoid local minima in searchbasedNP-hardoptimisationlike thetravelling salesmanproblem,graphpartitioningetc.Annealingoriginatesfrom condensedmat-terphysics[8]. Hereannealingis theprocessin whichasolid in aheatbathis heatedup to a level whereall theparticleshasa totalrandomordering,thentheheatbathis cooleddown slowly sotheparticlescanarrangethemself in a highly orderedlattice struc-ture, the temperatureat which the solid goesfrom dominatingrandomorderto dominatinglattice orderis thecritical tempera-ture.Thekind of annealingwewill useis Mean-FieldAnnealing.We will suggestanannealingschemethatavoidsmostof the lo-cal minima, maintainingthe polynomial law complexity of theconventionalmultistagereceivers. We show empirically that thisschemeobtainsa similar performanceto exactenumeration.Thepaperis organisedasfollows: In sectionII we describethemod-ulationandchannelmodel,in III we review optimaldetectors,insectionIV we explain theconceptof equilibriumdistributions,insectionV wederivethemean-fielddetector, in VI wedescribetheannealingheuristicandderive a conservative estimateof the(in-verse)startingtemperature,thenthis is followedby MonteCarlosimulationsin sectionVII, andfinally we concludeVIII.

I I . � USERS CHIP SYNCHRONOUS CDMA MODEL

AssumingastationaryAWGN channel,andBPSKmodulationthereceivedbasebandCDMA signalcanbemodelledas ���������� (1)

after chip waveform matchedfiltering, assumingthe chip wave-form to fulfil the Nyquist criterion. Where ������ is the re-ceived signal, ����� �"!$#%!%&(' are the transmittedbits for the �users, �)�*�,+.-/ 0 #�-/ 0 & 021 ' are the spreadingcodesfor the �userswith 3 chipsandunit energy, and �4�5��� is zeromeanwhite Gaussiannoisewith variance6.7 � 0987 .

We processthe receivedsignal by a bankof filters matchedto thespreadingcodesobtaining:;���9<=���4�>��<9�?�A@B�4�DC (2)

wherewe have definedthecorrelationmatrix of thespreadingcodes@E�F� < �G�H�JIK$I andthe transformednoise CD�F� < �whichnow hascovariance6 7 @ . All togetherthisdefinesthelike-lihoodof thetransmittedbitsassumingeverythingelsefor knownLNMO:�PQ�R��SS TVU 6 7 @HSS +XWY[Z + WYQ\ Y^] _ +�`acb(d,`[e,f ] _ +g`acb (3)

I I I . ML AND MEAN POSTERIOR DETECTOR

This sectionjust serves asa review of optimal detectors[2].ThedetectorthatminimisestheBit Error Rate( ��� ) is thesignmeanposteriorestimatorh�i�5jlk$m nco aqpqr +9-ls -ut v �gLNMO:�PQ�wRo aqpqr +9-ls -ut v LNMO:�Px�wR�y�5jlk$m nco aqpqr +9-ls -ut v � Z +,zq{ ] a$bo aqpqr +9-ls -ut v Z +gzq{ ] acb y zq|GW\ Y (4)

underequalapriori probabilityof thetransmittedbits,andwherewe have defined} M~�wR�� !T�� �w<2Mu@��H�%Rx�������=<9:c�;# (5)� beingtheidentitymatrix. Thedetectorthatminimisestheprob-ability of erroris themaximumlikelihoodreceiverh�i���V��k?�J���=LNMO:�Px�wRa�p�r +.-ls -ut v ���V��k��B��m } M~�Raqpqr +.-ls -ut v�������zq��� o aqpqr +9-ls -ut v � Z +,zq{ ] a$bo aqpqr +9-ls -ut v Z +gzq{ ] acb�� (6)

As seenbothdetectorscanbeimplementedby thesamesumma-tion structure,but with different � , � actsasa controlparameterwhich in physicswould correspondto the inversetemperature.We alsodefined

} M~�R which canbeviewedasthecostfunctionof the problemor the energy. It shouldbe noticedthat both ofthesedetectorshasexponentialcomplexity.

IV. EQUIL IBRIUM DISTRIBUTION

Before we start with the derivation of the detectorwe firstlook into the conceptof equilibrium distributions via the un-normalisedKL-divergence,in physicsdenotedthe free energy.We formulatethis for general� . Theergodic equilibrium distri-bution � z M~�R ataspecified� is thedistributionthatminimisesthefreeenergy definedas� z � �?� } M~�wRl� z � � ���$k2� z M~�Rl� z (7)

wherewedefine�Q� � z astheaveragewith respectto � z M~�R . Weseethat the minimum is obtainedfor the � z M~�wR thatalsominimizestheKL-divergence� �¢¡ � z M~�R£# Z +,zq{ ] a$b¤ z ¥ � �?� } M~�wRl� z � � ���$k � z M~�Rl� z ���¦�§k ¤ z

(8)since

¤ z � o aqpqr +�¨Vs ¨£t�© Z +,zq{ ] a$b the normalisingconstantisindependentof � z M~�R . Weseefrom this thatwhenwecanchoose� z M~�R freely to globally minimisetheKL-divergencewe get theergodicequilibriumdistribution� z M~�R� Z +,zq{ ] a$b¤ z (9)

i.e. theposteriorat thefixed � , with zeroKL-divergence.The two optimal detectorsconsideredin the last sectionwere

found by knowing the posteriormean � �� z at � � -ª Y and�D«­¬ respectively when � z M~�R is theergodicequilibriumdis-tribution. Until now we have rewritten the posteriordistribution

astheergodicequilibriumdistribution,althoughnicewe haven’tgainedanything in termsof complexity. For that reasonwe con-strain the family of � z M~�R so that averageslike � } M~�wRl� z and� ���§k � z M~�Rl� z becometractable. Sucha constraintimplies thatwe cannot minimize the KL-divergenceto zero,insteadwe de-fineaconstrained equilibrium distribution � z M~�R asadistributionthatwithin theconstrainedfamily is a(local)minimumof thefreeenergy, whichapproximatestheergodicequilibriumdistribution.

V. CONSTRAINING THE EQUIL IBRIUM DISTRIBUTION,MEAN-FIELD DETECTOR

In this sectionwill we constrainthe distribution � z M~�wR to thefactorisedfamily of distributionsover �®�¯���"!§#O!%& ' with °O± �� �"!§#O!%& , ² �³� !$# � & beingthe ²q´~µ -elementin thevector �� z M~�R� '¶± |=- � ] ± bz M °O± R

� '¶± |=- n !?��· ] ± bzT y W¹¸,ºu»Y n !¼�i· ] ± bzT y W e ºu»Y # (10)

where · z M ² R for ² �H� !$# � & parameterisesthedistributionat thespecified� . We have �~°O± � z �½· z M ² R andhence � �� z �*¾ zwhere ¾ z is thevectorof the · ] ± bz ’s. We now calculatethefreeenergy (7), with respectto this distribution� z � � !T�� ¾¿<z Mu@F�¿�%Rl¾ z �H�,¾¿<z : �� 'À± |w- !?�D· z M ² RT ���$k !2�D· z M ² RT� !X�³· z M ² RT ���$k !¼�i· z M ² RT

(11)

wherewe usedthat the distribution factorises.The constrainedequilibrium distribution is the distribution that is a minimum oftheconstrainedfreeenergy i.e.Á � zÁ · ] ± bz �� ² �H� !$# � & � (12)

We haveÁ � zÁ · ] ± bz � � ��MlMu@F�¿�ÃRl¾ z R ± �>M~:�R ± &q� !T ���§k !2��· ] ± bz![�³· ] ± bz �AÂ(13)

isolating · ] ± bz we getthemean-fieldequations· ] ± bz �5ÄÅ�Vm,Æ4Ç � M~:"�GMu@F�¿�ÃRl¾ z R ±OÈ ² �H� !$# � & � (14)

Iteratingthemean-fieldequationsataspecified� for thedifferent² ��� !$# � & is our multistagemean-fielddetector, eachiterationcorrespondingto onestage.Whenconvergedwe have obtainedtheconstrainedequilibriumdistribution. We seethat in the limit�³«É¬ wehave· ] ± b� �5jlk$m Ç M~:"�>Mu@��H�ORl¾ � R ± È ² �¿� !§# � & � (15)

correspondingto local ML optimisation,alsoknown ashardin-terferencecancellation.If weset� �Ê-ª Y wehavethedistributionin the family thatapproximatesthemeanposteriordetector, alsoknown assoft interferencecancellation.If we updateone · ] ± bzat a time, we have Successive InterferenceCancellation(SIC),

PSfragreplacements

ÄÅ�§m�Æ.M � R·ÌË +.-±

·ÌË±Í old Í newZOÎ£Ï Ð± � Ë @ ±

Fig. 1. Mean-FieldSuccessiveInterferenceCancellationUnit.

if we updateall we have parallel(PIC). In this contribution willwe restrictour selvesto successive updates,sincethis guarantiesthat we don’t increasethe free energy (7) in any iteration/stage.On figure (1) we have drawn a Mean-FieldSuccessive Interfer-enceCancellationUnit (MF-SICU), correspondingto theupdateof equation(14) for one ² , bold lines meansvectorsignalsandthin lines arescalarsignals, ² indicatesthe userand Ñ indicatesthe stage/iteration.If we wire together� MF-SICU’s we haveonestage,on figure (2) a typical Ò -stagemean-fielddetectorisshown.

PSfragreplacements :MF-SICU MF-SICU MF-SICU

MF-SICU MF-SICU MF-SICU

MF-SICUMF-SICUMF-SICU

· ]�Ó b- · ]�Ó b7 · ]�Ó b'

· ]¦Ô b- · ]�Ô b7 · ]¦Ô b'

Í ] -Õb'Í ] 7 b'Í ]�Ô b'

Fig. 2. D-stageMean-FieldSuccessive InterferenceCancellationdetector.

VI. MEAN-FIELD ANNEALING

Thereasonfor introducingannealingis dueto thenon-convexstructureof theenergy (5). Theexistenceof local minimais dueto thecorrelationmatrix of thespreadingcodes@ beingclosetosingularor singular. This is alsodescribedin [6], in theextremecasewhere �X@ is exactsingular. This impliesequalenergy of alocalminimumandthetrueminimumimplying thateventheML-detectorimplementedby exhaustive searchwill for somefractionof the possiblebit vectors � reportan erroneousminimum, thisleadsto theinherent��� -floor of theexhaustive searchML de-tector. Whatwe areconsideringhereis similar, namelyclosetosingularbehaviour which leadsto localminima,thecloserto sin-gular thecloserthecorrespondingenergy is to theglobalenergyminimum. In this case,local searchfor theglobalminimumwillhavedifficultiesin avoidingthelocalminimaandhencewill expe-riencea ��� -curve in excessof theexhaustive ML ��� -curve.

All theseeffectsof coursehave to be comparedwith the noiselevel 6 7 � 0�87 . If the effectsaresmall comparedto the noiselevel, i.e. have low probability of leadingto a erroneousdetec-tion comparedto theprobabilityof thenoisemakinganerroneousdetection,the ��� -curve will follow thenormal’waterfall’ be-haviour. At someÖq×Ø theeffectof thelocalminimacanbeseenasabifurcationaway from theexhaustive ML ���� -curve. At aneven higher Öq×Ø wherethesingulareffect becomesdominatingtheML ��� -curve will divergefrom the ’waterfall’ behaviour,becauseof the erroneousminima with the sameenergy. Thesethreedomainsarein full accordancewith large systemlimit re-sults[9], [10].

To avoid local minima we useannealing.The problemto besolvedis thatof finding theglobalminimumamongstmany localminima in the objective function. The idea is to weight a non-convex partsof the objective function relative to a convex partof the objective function. Here the free energy of the previouschaptercanbeviewedastheobjective functionand � astherel-ative weight. When � �ÙÂ we only have a convex part in theobjective function, i.e. oneminimum in the correspondingfreeenergy

� Ó . The convex part is believed to be dominatingfor �up to somecritical �=Ú , so below �.Ú the objective function stillonly posesoneminimum.Justabove �=Ú morethanoneminimumexists,amongstwhich theglobalone,if unique,hasto befound.The closer � comesto � Ú thecloserthe locationof thesolutioncomesto theglobalminimumat thedesired� Ô . If theminimumfrom just below �=Ú is usedasaninitial guessto thesolutionjustabove �.Ú the obtainedminimum amongstall the local minimawill betheonethatliesclosest.Whichwehopeis theglobalone.Sincethe global minimum hasthe lowest energy it also domi-natesthemost,sothehopecanbefulfilled with high probability.The increasein � from below to above �=Ú hasto be carriedoutslowly in order to get ascloseto the global minimum aspossi-ble beforecrossing� Ú . The reasonis the way we solve for theminimum at the specified� , becauseif we take a too large stepin onedirection,thenthis influencesthe otherdirections. Largestepsobviously occurif we change� in largestepsover � Ú .

Mean-Fieldannealingcannow beimplementedby choosingaserieof accenting� ’s: � -JÛ � 7 Û �Ã�O� Û � Ô , andsolving theequations(14) for eachof the � ’s,usingthesolution ¾³Ü +�¨z foundat theprevious � Ë +.- asinitial guess¾¿Üz on theactual � Ë . At � -somearbitraryinitial guesson ¾ ¨z canbeused,sinceif � -XÝ � Úonly onesolutionexists.A multistageSICthatimplementsMean-Field Annealingcan be constructedbe reusingthe Mean-Fieldmultistagedetectoron figure (2) by increasing� in someof thestages.

Fromtheabove discussiontwo considerationshasto bemade:Þ Thechoiceof thevalueof � - thatis below � Ú .Þ How fast � shouldbeincreasedtowardsthedesired� Ô .

The choicemadeon the two above considerationsdeterminestheAnnealing Scheme.

The critical �.Ú can, in the CDMA setting,be lower boundedby consideringthefix-point equations(14). We seetheargumentof ÄÅ�Vm,Æ9M � R in equation(14) is linear in ¾ z the maximal slopein any directionof ¾ z is determinedby themaximaleigenvalueof � � Mu@��³�OR . If the slopeis lessthan one, independentlyof: , the equations(14) only posesone solution, since ÄÅ�Vm,Æ�M � R isantisymmetricsigmoidwith slope ! in zero. We now let ßgà ˦ádenotetheminimumeigenvalueof @ , thenthemaximalslopeis� MQ!,� ß à ˦á R , thisslopeis equalto onefor � � --l+^â�ãäæå Û �=Ú i.e.

lower boundsthecritical �=Ú . Since @ is thecorrelationmatrixoftheunit energy spreadingcodes,@ is positive semi-definite.Thisimplies thattheworstcaseßgà ˦á over all setsof spreadingcodesequalszerohence� ÚÌç --l+�âVãä¦å ç ! . Using � - �è! ensuresthat only oneuniquesolutionexists to theequations(14) for allspreadingcodes.

Theboundon � Ú canalsoexplain why themultistagedetectorusedin [6] with clippedsoft decision(CSD)squashing/mappingfunction works so well. It simply operatesin the region whereonly onesolutionexists, sinceif an � was introducedit equalsone,andfor theCSDthis is enough,for thesamereasonsasforÄÅ�§m�Æ.M � R , to ensuretheexistenceof oneuniquesolution. In mod-erateloadedsystems'0 ��é Ý ! the energy differencesof thelocal minima to the global onearefor typical @ large,henceina typical transmissionthesolutionis not influencedthatmuchbythelocalminima.

Thesecondissueis morecomplex, andwehaven’t yetfoundananswer. Accordingto theoptimaldetectortheory, thedesired� Ôthaton averageminimisesthe �?�� is � Ô � -ª Y . In caseof nolocalminimaat � Ô for any setof spreadingsequenceswe wouldjust solve theequations(14) at thedesired� Ô . Sowe know thestarting� - andtheterminating� Ô , but thegraduationin betweenyet still hasto be determined.Onenotion is the distribution oftheminimumeigenvalue ß à Ëæá of @ whichis distributedbetweenzero and one skewed to one side dependingon the load of thenumberof users� to the numberof chips 3 denotedéA� '0 .In an actualsystemthis distribution canserve asa guideto de-terminingthegradingof the � ’s. Herewe assumetheminimumeigenvalueof Mu@4R to bedistributeduniformly betweenzeroandone,thatmeans!ê� ßgà ˦á alsois distributeduniformly betweenzeroandone. So the inverse� shouldbespacedequallyfrom !down to � +9-Ô � 6 7 thisgives� Ë � !!2��M Ñ �ë!�R�-l+ ª Y-l+ Ô (16)

whereÒ is thenumberof different � ’s.

VI I . MONTE CARLO SIMULATIONS

To have a referencefor the performancewe choosethe sameparametersettingsasin [6]’s secondsimulation. Thenumberofusersis � �¢ì andthespreadingfactoris 3 ��!Oí i.e. a moder-ateloadof é³�Ù-7 . Beforewe go to theactual ��� -simulations,we show an examplewherelocal minima exists in conjunctionwith theglobalminimum. We generatea setof spreadingcodes� so @ hasonesingularvalue.Wetransmitavector � thathasanuniquemaximumof theenergy (5). Soexhaustivemaximumlike-lihood searchfindstheright solutionwith noerrorsin the � �Aìbits. If we usethedetectorusedin [6] i.e. a multistagedetectorwith clippedsoftdecisionmapping/squashingfunctionand� �î!initialised in zero,we getoneerrorout of eight in this particularcase,i.e. the uniquesolution,seethe previous section,is dom-inatedby at leastonemoreminimum thanthe global minimumof theenergy. Usinga multistagedetectorsimilar to thepreviousbut with ÄÅ�Vm,Æ�M � R assquashingfunction,i.e. solvingtheequations(14) with � ��! initialised in zero,alsointroducedoneerror, bythe samereasons.For this particulartransmissionwe now wantto find all the solutions ¾ z of the equations(14) asa functionof � . We randomlyinitialised thestarting ¾ z from theuniformdistribution on the values ï^ð P��¢! Ý ð Ý !Vñ§' in orderto getdifferentsolutions.We iteratedtheequationsatfixed � until con-vergence.The solutionsoverlap-distanceto the transmittedbits

100

101

102

103

104

105

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

α

Ove

rlap

Fig. 3. Differentsolutionsto theMean-Fieldequations(14) at various ò ’s for oneparticulartransmission.� , definedhereas ó M � RÌ��' +�a <gôØõ7 is plotted on figure (3).

This definitionof theoverlap-distancetells how far in numberofsoft bits thefoundsolution ¾ z is to thetransmittedbits � . If westartat low � we seethatonly onesolutionexists in accordancewith the theory of annealing,whenwe increase� the overlap-distancebecomessmallerandcloserto one. At � �½! still onlyonesolutionexists in accordancewith our conservative estimateof � Ú . Between! and T anew solutionstartstocoexists,wehaveabifurcation,whichmeansthatthe’most’ critical �=Ú for this trans-missionliesbetween! and T . This indicatesthattheboundon �=Úis tight. The bestof the two solutionscontinuoustowardszero,whereasthebadonelies furtherandfurtheraway. At �Göî÷ thebadsolutionbifurcatesagain.Around �ëö ì it againbifurcates.At � �ø-ª Y we have drawn a vertical line indicatingthedesired� Ô . We solved at the desired� Ô the equations(14) startinginzero.Thesolutionis markedwith a cross,which certainlymarksa wrong solution. At the desired� Ô only ì out of the T$ù í ran-dom initialisationsconvergedto the goodsolution. We now didT§ù í randominitialisationsat � - �ú! i.e. below �=Ú andchose� Ô � � - Ó � -ª Y andusedthe annealingschemedescribedinequation(16),at each� we only usedtwo iterations,soa total ofT  iterations/stagesper randomstart. All the T§ù í initialisationsconverged to the true solutionwith no errors!, the samedid theoneinitialisedin zero.

We alsodid a ��� -simulationof thesystemwith � �¯ì and3 �û!Oí asin [6]. Herewe usedthreedifferentdetectors.Thefirst beingtheexhaustivesearchML detector;thesecondthemul-tistagedetectorusedin [6] with clippedsoftdecisioninitialisedinzeroanda � �î! for T  stages;thethird aMean-FieldAnnealingmultistagedetectorwith !à� ’s, � - �¯! and � Ô � � - Ó �ü-ª Y , ateach� we usedtwo iterationsi.e. also T  stages.Theresultscanbeseenonfigure(4).

The inherent ���� -floor is seencloseto the prediction ù � ÷$÷��!ÃÂq+�ý found in [6]. The threedomainsdescribedin section(VI)caneasilybe seen.Above bit error ratesof þ�� !ÃÂq+ 7 all the de-tectorsfollows closelytheexhaustive searchML detectorwith atypical waterfall behaviour. FromheretheClippedSoftDecision(CSD)detectorbifurcatesaway from theexhaustive ML-detectorindicatingtheinfluenceof localminimain theenergy. TheMean-Field Annealingdetectoravoidsthis influencefrom localminima

0 5 10 15 20 2510

−5

10−4

10−3

10−2

10−1

Eb/N

0 (dB)

Ave

rage

d B

ER

Fig. 4. �^��� performanceof exhaustivesearchML (solid line), ClippedSoftDeci-sion multistagedetector(dashed),andtheproposedMean-FieldAnnealingde-tector(solid with stars)all lower boundedby thesingleuserperformance(alsosolid). Theparametersare ÿ�� � and ������� .

down to ��� ��í � !à+ at í dB, this ���� is reachedby theCSDat � dB.ThelocalminimainfluenceontheMean-FieldAnnealingdetectoris fairly small,andthedetectoris closeto theexhaustiveML detector. But still thereexistssomeexcesserrorscomparedtoexhaustive ML searchmethod,indicatingthata betterannealingschemecouldbeinvented.

We also did simulationsfor a larger system � � T  and3 � T þ , i.e. a fairly loadedsystemé�� '0 � -- � 7 , but becauseof thelongerspreadingsequencestheprobabilityof a singular@is lower thanin theaboveexampleeventhoughtheloadis higher.The resultscanbe seenon figure (5). The two of the detectors

0 1 2 3 4 5 6 7 8 9 1010

−4

10−3

10−2

10−1

Eb/N

0 (dB)

Ave

rage

d B

ER

Fig. 5. �^��� performanceof ClippedSoft Decisionmultistagedetector(dashed),the proposedMean-FieldAnnealingdetector(solid with stars),Linear MMSE(solid with smalldots),Multistagedetectorwith first stageLinearMMSE (solidwith diamonds),all lower boundedby thesingleuserperformance(alsosolid).Theparametersare ÿ�� ���

and ��� ���.

are identical to thosein the previous simulation. The two oth-ersare linear MMSE, anda multistagedetectorwith first stagelinear MMSE followed by T Â stagesof Mean-Fielddetectionat� � -ª Y . The first thing that shouldbe notedis the complexity

of thetwo lastdetectorswhicharehigherthanthetwo first duetothematrix inversion.Oneinterpretationof themultistagedetectorwith first stagelinearMMSE, is asannealingwith two � ’sthefirstbeingvery closeto zerothe secondwith � 7 � � Ô � -ª Y . Thereasonfor this is thatthelinearMMSE detectorcanbederivedbyexpandingtheequations(14) to secondorderin � around� �A ,i.e. is valid for very small � ’s, but in thedomainwhereonly onesolutionexists. Thenwe decrease� to thedesired� Ô � -ª Y inonelargestep,which we know from thetheoryof annealingcangoarbitrarywrong.Theresultsonfigure(5) follow thosefrom thefirst example.AgaintheMean-FieldAnnealingdetectorperformsbetterthantheCSDdetectorby around ! dB. The two detectorsbasedon the linearMMSE detectorhave an increasinglyperfor-mancegapto theMean-FieldAnnealingdetectorof morethan ùdB at a ��� ��!à+ . Again theCSDbifurcatesaway from theMean-FieldAnnealingdetectorat ÷ dB. Whereasit seemslike abifurcationof theMean-FieldAnnealingdetectorat � dB but witha very smallexcesserrorlevel.

VI I I . CONCLUSION

In thiscontributionweproposedamultistagemulti-userdetec-tor implementingMean-FieldAnnealing. It hasno addedcom-plexity comparedto existing multistagedetectors.Performancegains of several dB’s are obtainedin simulations,closely ap-proachingtheexhaustive searchML detector. We alsoderivedaconservativeestimateonthefirst stage’scontrolparameter, whichhowever seemsto befairly tight. Theproposedannealingschemeis generalapplicable,andcanbeusedfor any choiceof spreadingcodes,sinceit is notoptimisedfor any particulareigenvaluespec-trum of the codecorrelationmatrix. Thoughsimulationsshowthat further improvementscanbe achieved in orderto reachtheperformanceof theexhaustivesearchML detector. Wealso,in thelight of Mean-FieldAnnealing,gave someexplanationsto whythe ClippedSoft Decisionmultistagedetectorperformsso well,andwhy a multistagedetectorwith aninitial linearMMSE stageperformslesswell.

ACKNOWLEDGEMENTS

Thefirst authorwould like to thankresearchersat Nokia Mo-bile PhonesRetDin Denmarkfor encouragingandsupportinghiswork onCDMA detection.A specialthankto Dr. Ole Nørklit forhelpful discussions.

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