An Efficient code-timing estimator for DS-CDMA systems over resolvable multipath channels

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An Efficient code-timing estimator for DS-CDMA systems over resolvable multipath channels
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EURASIP Journal on Applied Signal Processing
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Liu, Jianhua
Li, Jian
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We consider the problem of training-based code-timing estimation for the asynchronous direct-sequence code-division multipleaccess (DS-CDMA) system. We propose a modified large-sample maximum-likelihood (MLSML) estimator that can be used for the code-timing estimation for the DS-CDMA systems over the resolvable multipath channels in closed form. Simulation results show that MLSML can be used to provide a high correct acquisition probability and a high estimation accuracy. Simulation results also show that MLSML can have very good near-far resistant capability due to employing a data model similar to that for adaptive array processing where strong interferences can be suppressed.
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Publication of this article was funded in part by the University of Florida Open-Access publishing Fund. In addition, requestors receiving funding through the UFOAP project are expected to submit a post-review, final draft of the article to UF's institutional repository, IR@UF, (www.uflib.ufl.edu/ufir) at the time of funding. The Institutional Repository at the University of Florida (IR@UF) is the digital archive for the intellectual output of the University of Florida community, with research, news, outreach and educational materials

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EURASIPJournalonAppliedSignalProcessing2005:5,670c 2005HindawiPublishingCorporationAnEfcientCode-TimingEstimatorforDS-CDMASystemsoverResolvableMultipathChannelsJianhuaLiuCollegeofEngineering,Embry-RiddleAeronauticalUniversity,600S.ClydeMorrisBoulevard,DaytonaBeach,FL32114,USAEmail:jianhua.liu@erau.eduJianLiDepartmentofElectricalandComputerEngineering,UniversityofFlorida,P.O.Box116130,Gainesville,FL32611-6130,USAEmail:li@dsp.u.eduReceived31July2003;Revised26March2004Weconsidertheproblemoftraining-basedcode-timingestimationfortheasynchronousdirect-sequencecode-divisionmultiple-access(DS-CDMA)system.Weproposeamodiedlarge-samplemaximum-likelihood(MLSML)estimatorthatcanbeusedforthecode-timingestimationfortheDS-CDMAsystemsovertheresolvablemultipathchannelsinclosedform.SimulationresultsshowthatMLSMLcanbeusedtoprovideahighcorrectacquisitionprobabilityandahighestimationaccuracy.SimulationresultsalsoshowthatMLSMLcanhaveverygoodnear-farresistantcapabilityduetoemployingadatamodelsimilartothatforadaptivearrayprocessingwherestronginterferencescanbesuppressed.Keywordsandphrases:DS-CDMA,synchronization,codetiming,multipathchannels.1.INTRODUCTIONDirect-sequencecode-divisionmultipleaccess(DS-CDMA)isoneofthemostpromisingmultiple-accesstechnologiesforthenext-generationwirelesscommunicationservices.Fortime-dispersivefadingchannels,DS-CDMAcanoutperformothermultiple-accessschemes,suchasFDMA(frequency-divisionmultipleaccess)andTDMAtime-divisionmultipleaccess),duetoitscapabilityofexploitingtheRAKEcombi-nation[1 ]tocombatthetime-dispersivefadingproblemef-fectively.ThestructureoftheRAKEreceiverisdeterminedbythetypeofthetime-dispersivemultipathchanneloverwhichtheDS-CDMAsystemworks.Therearemainlytwokindsoftime-dispersivemultipathchannelsfortheDS-CDMAsys-tems.Therstoneisthe(nearly)continuouschannel,whichcanberepresentedbyaniteimpulseresponse(FIR)l-terchannelmodel;thesecondoneisthediscretechannelwheretheseparationsoftheresolvablebunchesofmulti-paths(referredtoasresolvablepathsinthesequel)arelargerthanthechipdurationofthespreadingcode.Thelattercanberepresentedbyaresolvabletime-dispersivechannelmodel.Whiletherstmodelisquitee ectiveindescrib-ingchannelsinurbanareas,thesecondmodelissuitablefordescribingchannelsinruralormountainareasorinthecaseofsofthando [ 2 ],whereafewresolvablepathsexist.Inthispaper,wefocusonthechannelsrepresentedbythesecondmodelandwerefertothemastheresolvablemulti-pathchannels. Fortheresolvablemultipathchannelsconsideredherein,theRAKEreceiverassumestheknowledgeofthechannelpa-rametersincludingthecodetiming,signalpower,aswellasthecarrierphaseforeachresolvablepathofeachuser.Theseparametersaretypicallyunknowninpracticeandhenceneedtobeestimated.Inthispaper,weconsidertheproblemofcode-timingestimationfortheasynchronousDS-CDMAsystem.Givenanaccuratecode-timingestimate,thereisawealthofgoodmethodsforestimatingtheotherparame-ters[3 ]. Astandardtechniqueforcode-timingestimationisthetraining-basedcorrelator(ormatcheder)[4 ]whichneedsaknowntrainingsequence.Itiswellknownthatthecorre-latorcoincideswiththeoptimalmaximum-likelihood(ML)methodinthesingle-usercasefortheGaussianchannel.Thecorrelatorcanalsobeusedfortheresolvablemultipathchannelsinthesingle-usercase.However,inthemultiusercase,itsperformancedegradesconsiderablybecauseoftheso-callednear-farproblem.Anear-farresistantcode-timingestimatoristheMUSICalgorithm,whichwasproposedindependentlyin[5 6 ].Un-liketheaforementionedcorrelator,theMUSICalgorithmisablindapproach,thatis,itneedsnotrainingforthepurpose

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ADS-CDMACode-TimingEstimatorforMultipathChannels671 ofcode-timingestimation.Hence,theMUSICalgorithmcanbeusedtoperformnotonlycode-timingestimation(acqui-sition),butalsotracking.YetMUSICsu ersfromlowcorrectacquisitionprobabilityandpoorestimationaccuracy,aswellaspoorsubscribercapacity.Anothernear-farresistantcode-timingestimator,re-ferredtoasthelarge-sampleML(LSML)algorithm,waspro-posedin[7 ].LSMLisalsotraining-basedandcanbeusedtoperformcode-timingestimationfortheasynchronousDS-CDMAsystemovertime-invariantfadingchannels.UnlikeconventionalMLestimators,whichemploycompu-tationallyextensivemultidimensionalsearch,LSMLusesaclosed-formsolution,whichiscomputationallyverye cient.TheunderlyingideaofLSMLwasextendedin[8 ]toderiveatraining-basedMLmethodtoestimatethecodetim-ingforresolvablemultipathchannels.However,themethodin[8 ]needstoperformamultidimensionalsearchtoobtainthecode-timingestimationforeachresolvablepath,whichiscomputationallyveryheavy.Inthispaper,weproposeanewtraining-basedcode-timingestimationmethod,referredtoasthemodiedLSML(MLSML)algorithm.TheMLSMLalgorithmcanbeusedtoavoidthemultidimensionalsearchproblemofcode-timingestimationof[8 ]byusingaclosed-formsolutionsimilartoLSMLandhenceiscomputationallyverye cient.Moreover,weshowthattheMLSMLalgorithmcanbeusedinthecaseofarbitraryresolvabledelays,thatis,theresolvabledelayscanbeindi erentprocessingwindowsandthereforerelaxtheconstraintsin[7 8 ].Asaresult,theMLSMLalgorithmiseasiertouseinthepracticalmultipathchannelscenarios.SimulationsshowthatMLSMLcanbeusedtoprovideahighcorrectacquisitionprobabilityandahighestimationaccu-racy.SimulationsalsoshowthatMLSMLcanhaveverygoodnear-farresistantcapability,duetoemployingadatamodelsimilartothatforadaptivearrayprocessingwherestrongin-terferencescanbesuppressed.Theremainderofthispaperisorganizedasfollows.InSection2,weprovidethedatamodelforcode-timingestima-tion.InSection3,wederivetheMLSMLalgorithmindetail.NumericalexamplesarepresentedinSection4todemon-stratethee ectivenessoftheproposedMLSMLalgorithm.Finally,weendourpaperwithcommentsandconclusionsinSection5. 2.DATAMODELConsideranasynchronousDS-CDMAsystememployingBPSKmodulationitcanbereadilygeneralizedtootherPSKmodulations),operatingovertime-invariantresolvablemul-tipathchannels.Weassumethatalltheusersusethesamecarrierfrequencyandthereisnofrequencyo setbetweenthetransmittersandthereceivers.Weconsiderthecaseofusingshortcodes.ThespreadingcodeforeachuserisapseudonoisePN)sequencewithlengthN Letc k t = N 1 n = 0 c k n P T C t nT C (1) beoneperiodofthespreadingwaveformforuserk ,whichhasaperiodofT = NT C ,whereT isthedurationoftheinformationcarryingdatasymbol,T C isthechipduration,and P T C t denotestheunitrectangularpulseon[0,T C ).Thek thuserstransmittedsignalhastheformx k t = 2 P k s k t )cos 0 t + k ,(2whereP k isthetransmissionpowerofuserk 0 isthecom-moncarrierfrequency, k isthecarrierphase,ands k t = M m = 0 d k m c k t mT 3isthebasebandspreadspectrumsignalwithd k m 1, +1 } m = 0,1,... M ,beingtheknowntrainingdatasymbol.Theresolvablemultipathchannelforuserk canbede-scribedash k t = L k l = 1 k l t k l ,(4where t istheDiracdelta, k l isthecomplex-valuedfad-ingcoe cientofthel thpath,and k l isthetimedelaybe-tweenthetransmitterandthereceiveronthel thpathwith k l +1 k l T C l = 1,2,... L k 1.Thenoise-freereceivedbasebandanalogsignalcanbewrittenasr A t = K k = 1 L k l = 1 k l s k t k l ,(5where k l P k k l e j k 0 k l (6) Weemployareceiverwiththefrontendconsistingofanin-phase/quadrature(I/Q)mixerfollowedbyanintegrate-and-dumperwithintegrationtimeT C .Then,thenoise-freeoutputdigitalsequencefortheanalogsignalisr i = 1 T C K k = 1 L k l = 1 k l iT C i 1) T C s k t k l dt. (7) Weconsiderthecasethat k L k k ,1 < N 1) T C .Itisassumedin[7 8 ]thatallthetimedelaysarewithin[0,T ), thatis, k l [0, T )forallk and l .Whilethisassumptioncanbereasonableforthefadingcase,itisnolongervalidinthecaseofresolvablemultipathchannels.Thereasonisthatifthedelayoftherstpathiswithin[0,T ),thepathswithlongerdelayscanbewithin[T ,2 T ).Henceweassumethatthetimedelaysarewithin[0,2T inthispaper.Fornotationalsimplicity,inthederivationofthedatamodel,werstas-sumethatthetimedelaysarewithin[0,T ).Thenwewillad-dressthecasethatthetimedelaysarewithin[0,2T )lateron.

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672EURASIPJournalonAppliedSignalProcessing For k l [0, T ),weconsideritasthesummationofmultiplechipdurationsandaresidue,thatis,let k l = p k l T C + k l ,wherep k l 0,1,... N 1 } and k l [0, T C ). Thek l )thtermoftheintegrationin7 isthengivenbyr k l i 1 T C iT C i 1) T C s k t k l dt = 1 T C M m = 0 d k m N 1 n = 0 c k n iT C i 1) T C P T C t mN + n + p k l T C k l dt = T C k l T C M m = 0 d k m N 1 n = 0 c k n i mN + n + p k l 1 + k l T C M m = 0 d k m N 1 n = 0 c k n i mN + n + p k l 2 = 1 k l T C c k i m 1 N p k l 1 d k m 1 + k l T C c k i m 2 N p k l 2 d k m 2 (8) whichisafunctionofp k l and k l ,theparameterstobeesti-mated.Herem 1 and m 2 aretheindexesforthetrainingdatasymbolssuchthat0 i m 1 N p k l 1 N 1and0 i m 2 N p k l 2 N 1,respectively.Inwhatfollows,wedeterminem 1 and m 2 foreachi From8 weknowthatfori p k l 2 = mN + N 1,thatis, i = m +1N + p k l +1,wehavem 2 = m and m 1 = m +1,whichleadstor k l mN + N + p k l +1 = c k (0) d k m +1)+ k N 1) d k m ), (9) where k l /T C and 1 .Similarlyfori = mN + N + 1, ... mN + N + p k l ,wehaver k l i = c k i mN p k l 1 + k i mN p k l 2 d k m ), (10) andfori = mN + N + p k l +2,... mN +2 N ,wehaver k l i = c k i mN N p k l 1 + k i mN N p k l 2 d k m +1. (11) Stacking r k l of9 ),11 ),and10 )inavector,wehave(bylettingp = p k l fornotationalconvenience)r k l m +1 r k l mN +2 N r k l mM + N +1 = J 1 p c k + J 1 p +1c k J 2 p c k + J 2 p +1c k d k m +1,(12) wherec k c k N 1) c k N 2) c k (0) T (13) with T denotingthetranspose,J 1 p 00 I p 0 J 2 p 0I N p 00 ,14)withI p beingthep p identitymatrixand0 beingazeromatrixwithrequireddimensions,andd k m +1 d k m d k m +1 T (15) Bydenotingz k m +1 1 2 d k m +1)+d k m d k m +1 d k m ,16)wehaver k l m +1= a 1 k l a 2 k l z k m +1),(17)wherea 1 k l J 2 p +1)+J 1 p +1 + J 2 p )+ J 1 p c k a 2 k l J 2 p +1 J 1 p +1 + J 2 p J 1 p c k (18) Hence,wehaver k m +1 L k l = 1 r k l m +1= L k l = 1 k l a 1 k l a 2 k l z k m +1= A k ,1 k A k ,2 k z k m +1,(19) whereA k ,1 a 1 k ,1 a 1 k ,2 a 1 k L k A k ,2 a 2 k ,1 a 2 k ,2 a 2 k L k k k ,1 k ,2 k L k T (20) Notethatinsteadofz k m +1in17 wecouldhaveusedd k m +1= [ d k m d k m +1] T .Yet,theautocorrelationoftheformer,E d k m +1d H k m = 1 4 E d k m d k m 1) E d k m 2 E d k m +1d k m 1) E d k m +1d k m = 01 00 (21)

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ADS-CDMACode-TimingEstimatorforMultipathChannels673 isnotasscatteredasthatofthelatter,E z k m +1z H k m = 1 4 E d k m 2 E d k m 2 E d k m 2 E d k m 2 = 1 4 11 1 1 (22) whichisbeingpreferred.Byexpressingthemultiple-accessinterferenceMAI)inasimilarwayandlumpingtheMAIwiththeadditivenoiseintow m ),wehavethedatamodelforthek thuserasr m = D k z k m )+ w m ), m = 1,2,... M ,23)where D k A k ,1 k A k ,2 k (24) Thevectorw m isassumedtobeindependentofthede-siredsignalandtobeawhitecircularlysymmetriccomplexGaussianrandomvectorwithzero-meanandcovariancema-trixQ Theproblemofinteresthereinistoestimate k l l = 1,2,... L k ,oftheinterestedk thuserfromr m basedontheassumptionssummarizedbelow:(A1) L k isknownand k l +1 k l T C for l = 1,2,... L k 1 with k L k k ,1 < N 1) T C ; (A2) z k m ), m = 1,2,... M ,isknown;(A3) c k isknownandhaslowautocorrelation;(A4) Q isarbitrarydeterminedbyotherusersandnoise.3.THEMLSMLALGORITHMWepresenttheMLSMLalgorithminthissection.Fornota-tionalconvenience,wewilldropthenotationaldependenceon k inthesequel.Thelog-likelihoodfunctionofthereceiveroutputvectorr m ), m = 1,2,... M ,isproportionaltoF ln | Q | tr Q 1 1 M M m = 1 r m Dz m r m Dz m H (25) whichisafunctionofunknownQ and D k ,withthelatterbe-ingdeterminedbyunknowncode-timingandcomplexam-plitude(cf.6 ),20 ),and24 )).Here denotesthedeter-minantofamatrixand H denotestheconjugatetranspose.Thefocusofthispaperisonestimatingtheunknowncode-timingbyminimizingthecostfunctionF Bytakingthederivative[9 ],wehavebF bQ i tr Q 1 Q bQ i +tr Q 1 Q bQ i Q 1 1 M M m = 1 r m Dz m r m Dz m H (26) whereQ i isanelementofQ .Theaboveequationmeansthatthecostfunctionof25 ismaximizedby Q = 1 M M m = 1 r m Dz m r m Dz m H ,27)whichleadstothefollowingcostfunctiontobeminimized:F 1 1 M M m = 1 r m Dz m r m Dz m H (28) AnunstructuredestimateofD isobtainedas[10 ] D = R H zr R 1 zz ,29)where R zr 1 M M m = 1 z m r H m ), R zz 1 M M m = 1 z m z H m (30) Byplugging29 into27 ), Q canberewrittenas Q = R rr R H zr R 1 zz R zr ,31)where R rr 1 M M m = 1 r m r H m (32) Ithasbeenshownin[10 ]thatminimizingF 1 isasymp-toticallyforlargeM equivalenttominimizingF 2 tr R zz D D H Q 1 D D (33) NotethatR zz isadiagonalmatrixwithequaldiagonalele-mentsduetothefactthatd m canbeselectedtobeaninde-pendentlyandidenticallydistributedsequenceandthefactthat E z 1 m z 2 m = 1 4 E d m +1 2 d m 2 = 0,(34)

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674EURASIPJournalonAppliedSignalProcessing where standsforthecomplexconjugate.Then,mini-mizing33 isequivalenttominimizingthefollowingcostfunction:F 3 d 1 A 1 H Q 1 d 1 A 1 + d 2 A 2 H Q 1 d 2 A 2 (35) where d 1 and d 2 denotetherstandthesecondcolumnsof D ,respectively.rstglanceat(35 seemsthatweneedamultidimen-sionalsearch,asin[8 ],toestimate l l = 1,2,... L .Now,weshowthatbyutilizingthepropertyoflowautocorrelationofthespreadingcodec ,wecandecouplethemultidimen-sionalsearchintoL one-dimensionalsearches.Furthermore,weshowthattheseL one-dimensionalsearchescanbecom-binedintooneone-dimensionalsearch,whichcanbesolvedinaclosedform.Fornotationalsimplicity,weonlyconsidertherstpartof35 inthefollowingderivation.WehaveF 4 d 1 L l = 1 l a 1 l H Q 1 d 1 L l = 1 l a 1 l = d H 1 Q 1 d 1 2Re d H 1 Q 1 L l = 1 l a 1 l + L l = 1 l 2 a H 1 l Q 1 a 1 l + L l 1 = 1 L l 2 = 1, l 2 = l 1 l 1 l 2 a H 1 l 1 Q 1 a 1 l 2 (36) Forc withlowautocorrelation,wehavea H 1 l 1 a 1 l 2 = N l 1 = l 2 O (1), l 1 l 2 T C (37) Thislowautocorrelationpropertyofc canbeexploitedtoapproximatelyrepresenttheeighteautocorrelationasa H 1 l 1 Q 1 a 1 l 2 N l 1 = l 2 O (1), l 1 l 2 T C (38) forarbitraryeightingmatrixQ .Notethattheeighteautocorrelationof38 isnotaslowastheautocorrelationof 37 );yet,undertheassumption(A1),theformercanstillbeusedtoyieldthefollowingsimpliedcostfunction,bydrop-pingthecrossitemsin36 ): F 4 d H 1 d 1 2Re d H 1 L l = 1 l a 1 l + L l = 1 l 2 a H 1 l a 1 l = L l = 1 d 1 l a 1 l H d 1 l a 1 l L 1) d H 1 d 1 = L l = 1 d 1 l a 1 l 2 L 1) d 1 2 (39) where d 1 Q 1 / 2 d 1 a 1 l Q 1 / 2 a 1 l ,40)and denotestheEuclideannorm.Usingthesametech-niqueandnotationforthesecondpartof35 ),wehaveF 3 L l = 1 d 1 l a 1 l 2 + d l 2 a 2 l 2 +const. (41) Notethat41 isactuallyadecoupledcostfunction,whichmeansthatwecanobtainthecode-timingestimatesinL onedimensionalsearches.Thatis,forl = 1,2,... L ,wehave l l = argmin l l F l ,42)whereF l d 1 l a 1 l 2 + d 2 l a 2 l 2 (43) Inourestimationproblem, l l = 1,2,... L ,isanui-sanceparameterwhichneedstobeconcentratedouttosim-plifytheoptimizationproblem.Byletting d = [ d T 1 d T 2 ] T and a l = a T 1 l a T 2 l T ,wehaveF l = d l a l 2 = d 2 l d H a l l a H l d + l 2 a l 2 = a l 2 l a H l d a l 2 2 + d 2 a H l b 2 a l 2 (44) Let l = a H l d / a l 2 and J l = a H l b 2 a l 2 (45) Then,theestimateof l l = 1,2,... L ,canbedeterminedby l = argmax l J l (46) Weremarkthat,dueto46 ),wecanperformthesameone-dimensionalsearchforeach l l = 1,2,... L .Combin-ingtheL one-dimensionalsearchesintoone,wehavethenewMLSMLestimator,whichobtainstheestimateof l l = 1,2,... L ,byndingtheargumentscorrespondingtothe L largestlocalmaximumsofJ ). Theaforementionedone-dimensionalsearchovertheparameterspacefortheL largestpeaksisstillcompu-tationallyheavy.Toavoidthisburdensometask,weusetheapproachof[7 ]toobtainaclosed-formsolutionforeachp 0,1,... N 1 } ,asdetailedinthefollowing.

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ADS-CDMACode-TimingEstimatorforMultipathChannels675 Fromthedenitionsofa 1 l )anda 2 l ),wehave a 1 l = Q 1 / 2 a 1 pT C a 1 p +1T C = A 1 pT C a 2 l = Q 1 / 2 a 2 pT C a 2 p +1T C = A 2 pT C (47) where [ ] T = [ 1 ] T A 1 pT C Q 1 / 2 a 1 pT C a 1 p +1T C A 2 pT C Q 1 / 2 a 2 pT C a 2 p +1T C (48) Fornotationalsimplicity,wewilldroppT C from A 1 and A 2 aswellas{ k l } from .Plugging47 into46 ),wehaveJ = T A H 1 d 1 + A H 2 d 2 A H 1 d 1 + A H 2 d 2 H T A H 1 A 1 + A H 2 A 2 = T G T H N D (49) whereG A H 1 d 1 + A H 2 d 2 A H 1 d 1 + A H 2 d 2 H H A H 1 A 1 + A H 2 A 2 (50) Weknowfrom49 thatJ isarationalfunctionoftwosecond-orderpolynomials.Anyextremepoint inthedif-ferentiableregionofJ mustsatisfytheequationS = N D N D = 0 (51) NotethatbothG and H areHermitian.Lettingg ij and h ij be the ij thelementsofG and H ,respectively,wehaveN = g 11 + g 22 2 g 12 2 +2 g 12 g 11 + g 11 n 2 2 + n 1 + n 0 D = h 11 + h 22 2 h 12 2 +2 h 12 h 11 + h 11 d 2 2 + d 1 + d 0 (52) PluggingN = 2 n 2 + n 1 D = 2 d 2 + d 1 (53) into51 yieldsS = n 2 d 1 n 1 d 2 2 +2 n 2 d 0 n 0 d 2 + n 1 d 0 n 0 d 1 s 2 2 + s 1 + s 0 (54) whichhasthefollowingroots: = s 1 s 2 1 4 s 0 s 2 2 s 2 (55) TheMLSMLalgorithmforcode-timingestimationfortheuserofinterest,thek thuser,inmultipathchannelswiththetimedelayswithin[0,T canbesummarizedasfollows.Step1.Compute D and Q using29 )and31 ),respectively.Step2.Calculate using55 )foreachp 0,1,... N 1 } Step3.CalculatethecostfunctionJ )in(49 fortherootsobtainedinthepreviousstep.Inaddition,calculatethecostfunctionJ foreachin-di erentiablepoint,thatis, = 0, T C ... ,( N 1) T C Step4.FindtheL argumentscorrespondingtotheL largest peaksastheestimatesoftheL resolvabledelays.Now,werelaxtheconstraintthatallthetimedelaysarewithin[0,T ).AmongtheL resolvabledelays,weassumeL (0) ofthemarewithin[0,T andtheothersarewithin[T ,2 T ). FortheformerL (0) paths,themodelderivedintheprevioussectionremainsthesame.Asforthelaterpaths,from8 ),weknowthatifN p l 2 N 1,wehave,similarlyto12 ), r l m +1= J 1 p c + J 1 p +1c J 2 p c + J 2 p +1c d m 1) d m (56) wherep = p l N and17 )becomesr l m +1= a 1 l T a 2 l T z m (57) BylettingD (0) = A (0) 1 (0) A (0) 2 (0) D (1) = A (1) 1 (1) A (1) 2 (1) (58) whereA (0) 1 = a 1 1 a 1 2 a 1 L (0) A (0) 2 = a 2 1 a 2 2 a 2 L (0) A (1) 1 = a 1 L (0) +1 T a 1 L T A (1) 2 = a 2 L (0) +1 T a 2 L T (0) = 1 2 L (0) T (1) = L (0) +1 L (0) +2 L T (59) thedatamodelof23 )becomesr m = D (0) z m )+ D (1) z m 1)+w m ), m = 2,3,... M. (60) Itseemsthatestimatingcodetimingbasedon60 )ismuchmorecomplicatedthanthatbasedon23 sincez m and z m 1)arenotindependentwitheachother(cf.22 )). ThiscanbetrueifwewereusinganexactMLapproach.ForMLSML,anapproximateMLapproach(cf.38 )),thecom-putationsareonlyreasonablyincreasedSteps1 2 ,and3 in theMLSMLalgorithmareperformedtwicebeforegoingtoStep4.Fortherstroundcalculation,wecalculatebasedon

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676EURASIPJournalonAppliedSignalProcessing D (0) and Q (0) Q (0) correspondsto D (1) z m 1)plusw m withrespecttoz m ),andforthesecondround,wecalculatethecode-timingbasedon D (1) and Q (1) Q (1) correspondsto D (0) z m )plusw m withrespecttoz m 1). Weremarkthatforagivenresolvablemultipathchannel,theperformanceforthe[0,2T casecanbebetterthanthatforthe[0,T casethesetwocasesareduetodi erentinitialconditions).Thereasonisasfollows.Fortheformercase,thecross-correlationbetweentherstL (0) pathsin[0,T andtheremainingL L (0) pathsin[T ,2 T canbeseenas(cf.32 and22 D (0) E z m z H m 1) D (1) = 1 4 D (0) 11 1 1 D (1) ,61)whereasforthelattercase,thecross-correlationbetweentherst L (0) andtheremainingL L (0) paths(allin[0,T canbeseenas(ifwestilluseD (0) and D (1) D (0) E z m z H m D (1) = D (0) D (1) (62) Thecross-correlationof61 islowerthanthatof62 sincethecorrelationbetweenz m )andz m 1)islowerthanthecorrelationofz m itself.Thisperformancedi erencecanbeseenclearlyinthesimulationexamplesgiveninthenextsection.4.SIMULATIONRESULTSInthissection,weprovidenumericalexamplestoshowtheperformanceoftheproposedMLSMLcode-timingestimatorovertheresolvablemultipathchannels.NotethatMLSMLisanextensionofLSMLwhichis,inthefadingcase,supe-riortotheotherestimatorssuchasthecorrelator[4 ],theminimummean-square-error-basedcode-timingestimator[ 11 ],andtheMUSICcode-timingestimator[5 6 ](see[7 ]). Here,weonlyprovidesimulationexamplesshowingtheper-formanceofMLSMLintheresolvablemultipathchannelcase,alongwithacomparisonwiththecorrelator,whichcanalsobeusedintheresolvablemultipathchannelcase,when-everapplicable.Wedonotcomparewiththemethodin[ 8 ]sincea)thelattercanworkonlyforthe[0,T )caseand(b)thelatterhasordersofmagnitudeslowerthanthefor-mer.Thesimulationconditionsareasfollows.Eachuserisas-signedaGoldsequenceofN = 31.Allusershavetwopaths,thatis,L k = L = 2, k = 1,2,... K .Fortheuserofinterest,say,userone,thetimedelaysarexedto 1,1 = 20 5 T C and 1,2 = 23 5 T C ,respectively,inthe[0,T casereferredtoasCase1inthesequel),and 1,1 = 29 5 T C and 1,2 = 32 5 T C respectively,inthe[0,2T casereferredtoasCase2inthesequel),withthereceivedpowersbeing| 1,1 | 2 3dBand| 1,2 | 2 = 0dB,respectively.Theotherusershaverandomre-ceivedpowerswithlog-normaldistribution.Theaveragere-ceivedpowerofeachpathoftheinterferingsignalsisd dB (tobespeciedinvariousexamples,refereedtoasthenear-farratio(NFR))abovethedesireduserwithastandardde-viationbeing10dB,thatis,| k l | 2 = 10 k l / 10 ,where k l 15 10 5 0 051015202530TC J 1,1 1,2 (a) 15 10 5 0 051015202530TC J 1,1 1,2 [0, T [ T ,2 T (b) Figure1:IllustrationofJ )fora 1,1 1,2 [0, T Case1and(b) 1,1 [0, T ), 1,2 [ T ,2 T Case2whenK = 10, d = 2, SNR = 10dB,andM = 100. N d ,100),k = 2,3,... K l = 1,2.Theadditivenoiseforr i in7 iswhitecircularlysymmetriccomplexGaussianwithzero-meanandvariance 2 n = NN 0 /E s ,whereE s /N 0 isthesignal-to-noiseratio(SNR)withE s beingtheenergypersymbolforuserone.Inthesimulations,weassumeE s = N | 1,2 | 2 .) BeforeprovidingthesimulationexamplestoshowtheperformanceofthenewMLSMLcode-timingestimator,welookattwoguresillustratingJ )of49 inCase1andCase2,respectively.Thespecicsimulationparametersareasfollows:K = 10, d = 2,SNR= 10dB,andM = 100. TheresultsareshowninFigures1a ,and1b ,respectively.

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ADS-CDMACode-TimingEstimatorforMultipathChannels677 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 2030405060708090100M Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML0dBpath,correlator 3dBpath,correlator (a) 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 2030405060708090100M Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (b) 10 0 10 1 10 2 2030405060708090100M RMSE 3dBpath,correlator0dBpath,correlator 3dBpath,MLSML0dBpath,MLSML (c) 10 0 10 1 10 2 2030405060708090100M RMSE 3dBpath,MLSML0dBpath,MLSML (d) Figure2:ProbabilityofcorrectacquisitionandRMSEversusM fora)andc)Case1,andb)andd)Case2whenK = 10, d = 20,andSNR = 10dB.Wecanseethatwehavethecorrectacquisitioninbothcases.(Acode-timingestimate k l providesacorrectacquisitionif| k l k l |
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678EURASIPJournalonAppliedSignalProcessing 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 4681012141618202224K Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML0dBpath,correlator 3dBpath,correlator (a) 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 4681012141618202224K Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (b) 10 0 10 1 10 2 4681012141618202224K RMSE 3dBpath,correlator0dBpath,correlator 3dBpath,MLSML0dBpath,MLSML (c) 10 0 10 1 10 2 4681012141618202224K RMSE 3dBpath,MLSML0dBpath,MLSML (d) Figure3:ProbabilityofcorrectacquisitionandRMSEversusK fora)andc)Case1,andb)andd)Case2whenM = 60, d = 20,andSNR = 10dB.inthatthetwopaths,especiallytheweaksignalpath,havebetterperformance.WecanalsoseethatM = 60 2 N isathresholdfortheperformance,whichconformswiththeanalysisof[12 ]showingthatwhenthesamplenumberistwicethenumberofsensors,theperformanceofthear-rayusingtheestimatedcovariancematrixiswithin3dBoftheperformanceofthearrayusingtheexactcovariancema-trix.ThedatamodelemployedinthispaperisequivalenttoanarraywithN sensors.)WealsoprovidecorrespondingcurvesofthecorrelatorforCase1(itdoesnotworkforCase2).WeobservethatMLSMLsignicantlyoutperformsthecorrelator.Example2(performanceversusK ).Theparametersaresim-ilartothoseinthepreviousexamplewiththeexceptionthat M = 60andK changesfrom4to24.TheresultsareshowninFigures3a 3b 3c ,and3d ,respectively.Wecanseefromtheguresthat,similartothepreviousexample,

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ADS-CDMACode-TimingEstimatorforMultipathChannels679 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 10 505101520SNR Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (a) 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 10 505101520SNR Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (b) 10 0 10 1 10 2 10 505101520SNR RMSE 3dBpath,MLSML0dBpath,MLSML (c) 10 0 10 1 10 2 10 505101520SNR RMSE 3dBpath,MLSML0dBpath,MLSML (d) Figure4:ProbabilityofcorrectacquisitionandRMSEversusSNRfora)andc)Case1,andb)andd)Case2whenM = 60, K = 10,andd = 20. theperformanceofCase2isbetterthanthatofCase1.NotethatwhenK changesfrom12to16,thereissignicantper-formancedegradation.ThereasonisthatwhenK = 16,theoverallsignalnumber,LK = 32,isgreaterthanN 1,thedegreesoffreedom.Still,thedatamodelemployedhereinisequivalenttoanarraywithN sensorswhichhasthede-greesoffreedomN 1.)WealsoprovidethecorrespondingcurvesofthecorrelatorforCase1.Again,weobservethatMLSMLsignicantlyoutperformsthecorrelator.DuetothehugeperformancegapbetweenMLSMLandthecorrelator,wedonotneedtocomparethemanyfurther.Example3(performanceversusSNR.Theparametersaresimilartothoseinthepreviousexamplewiththeexceptionthat K = 10andSNRvariesfrom 10dBto20dB.Theper-formanceoftheMLSMLestimatorisshowninFigures4a 4b 4c ,and4d ,respectively.Still,theperformanceforCase2isbetterthanthatforCase1.

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680EURASIPJournalonAppliedSignalProcessing 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0102030405060NFR Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (a) 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0102030405060NFR Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (b) 10 0 10 1 10 2 0102030405060NFR RMSE Path1 3dBPath20dB (c) 10 0 10 1 10 2 0102030405060NFR RMSE 3dBpath,MLSML0dBpath,MLSML (d) Figure5:ProbabilityofcorrectacquisitionandRMSEversusd fora)andc)Case1,andb)andd)Case2whenM = 60, K = 10,andSNR = 10dB.Example4(performanceversusd ).Thisexampleshowsthenear-farresistantcapabilityofthenewcode-timingestima-tor.Figures5a 5b 5c ,and5d ,andFigures6a 6b 6c ,and6d showtheperformancewithK = 10andK = 15,respectively.Theotherparametersaresimilartothoseinthepreviousex-amplewiththeexceptionthatd variesfrom0to60.Wecanseefromtheguresthat,forK = 10,wheretheinterferencenumberL K 1) = 18ismuchlessthanN ,weexperi-encealmostnoperformancedegradationwhend increases(onlyaminordegradationfortheweakpathinCase1).However,forK = 15,wheretheinterferencenumberL K 1) = 28isonlyslightlylessthanN 1,weexperiencesigni-cantperformancedegradationwhend increases.Thesesim-ulationssuggestthatwiththereasonableselectionofparam-eters,suchas K k = 1 L k
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ADS-CDMACode-TimingEstimatorforMultipathChannels681 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0102030405060NFR Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (a) 1 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 1 0 0102030405060NFR Probabilityofcorrectacquisition0dBpath,MLSML 3dBpath,MLSML (b) 10 0 10 1 10 2 0102030405060NFR RMSE 3dBpath,MLSML0dBpath,MLSML (c) 10 0 10 1 10 2 0102030405060NFR RMSE 3dBpath,MLSML0dBpath,MLSML (d) Figure6:ProbabilityofcorrectacquisitionandRMSEversusd fora)andc)Case1,andb)andd)Case2whenM = 60, K = 15,andSNR = 10dB.5.CONCLUDINGREMARKSWehavepresentedanMLSMLestimatorthatcanbeusedtoperformthecode-timingestimationfortheDS-CDMAsys-temsovertheresolvablemultipathchannelsinaclosedform.SimulationexampleshaveshownthatMLSMLcanbeusedtoprovideahighcorrectacquisitionprobabilityandahighestimationaccuracy.SimulationexampleshavealsoshownthatMLSMLcanhaveverygoodnear-farresistantcapabil-ity,duetoemployingadatamodelsimilartothatforadap-tivearrayprocessingwherestronginterferencescanbesup-pressedwithinthecapabilityofthedegreesoffreedomofthearray.ACKNOWLEDGMENTThisworkwassupportedinpartbytheNationalScienceFoundationGrantCCR-0097114.

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682EURASIPJournalonAppliedSignalProcessing REFERENCES [1]J.G.Proakis,DigitalCommunications,McGraw-Hill,NewYork,NY,USA,3rdedition,1995.[2]J.C.LibertiJr.andT.S.Rappaport,SmartAntennasforWire-lessCommunications:IS-95andThirdGenerationCDMAAp-plications,Prentice-Hall,UpperSaddleRiver,NJ,USA,1999.[3]H.V.Poor,OnparameterestimationinDS/CDMAformats,in AdvancesinCommunicationsandSignalProcessing,W.A.PorterandS.C.Kak,Eds.,pp.5970,Springer-Verlag,NewYork,NY,USA,1989.[4]R.L.Peterson,R.E.Ziemer,andD.E.Borth,IntroductiontoSpreadSpectrumCommunications,Prentice-Hall,EnglewoodCli s,NJ,USA,1995.[5]E.G.Strom,S.Parkvall,S.L.Miller,andB.E.Otter-sten,Propagationdelayestimationinasynchronousdirect-sequencecode-divisionmultipleaccesssystems,IEEETrans.Commun.,vol.44,no.1,pp.84,1996.[6]S.E.BensleyandB.Aazhang,Subspace-basedchannelesti-mationforcodedivisionmultipleaccesscommunicationsys-tems,IEEETrans.Commun.,vol.44,no.8,pp.1009,1996. [7]D.Zheng,J.Li,S.L.Miller,andE.G.storm,Ane cient code-timingestimatorforDS-CDMAsignals,IEEETrans.SignalProcessing,vol.45,no.1,pp.82,1997.[8]G.YeandG.Bi,CodetimingestimatorforDS-CDMAsig-nalsinslowfadingmultipathchannels,ElectronicsLetters, vol.35,no.19,pp.1604,1999.[9]P.StoicaandR.L.Moses,IntroductiontoSpectralAnalysis, Prentice-Hall,EnglewoodCli s,NJ,USA,1997.[10]J.Li,B.Halder,P.Stoica,andM.Viberg,Computationallye cientangleestimationforsignalswithknownwaveforms,IEEETrans.SignalProcessing,vol.43,no.9,pp.2154,1995. [11]R.F.SmithandS.L.Miller,Codetimingestimationinanear-farenvironmentfordirect-sequencecode-divisionmultiple-access,inProc.IEEEMilitaryCommunicationsCon-ferenceMILCOM,pp.47,FortMonmouth,NJ,USA,October1994.[12]I.S.Reed,J.D.Mallett,andL.E.Brennan,Rapidconver-gencerateinadaptivearrays,IEEETrans.Aerosp.Electron.Syst.,vol.10,no.6,pp.853,1974. JianhuaLiureceivedtheB.S.degreeinelec-tricalengineeringfromDalianMaritimeUniversity,Dalian,China,in1984,theM.S.degreeinelectricalengineeringfromtheUniversityofElectronicScienceandTech-nologyofChina,Chengdu,China,in1987,thePh.D.degreeinelectronicengineeringfromTsinghuaUniversity,Beijing,China,in1998,andthePh.D.degreemajoringinelectricalengineeringandminoringinstatisticsfromUniversityofFloridain2004.FromMarch1987toFebruary1999,heworkedattheCommunication,TelemetryandTelecontrolResearchInstitute,Shijiazhuang,China,wherehewasanAssistantEngineer,Engineer,SeniorEngineer,andFellowEngineer.FromMarch1995toAugust1998,hewasalsoaRe-searchAssistantatTsinghuaUniversity.FromFebruary1999toJune2000,heworkedatNanyangTechnologicalUniversity,Sin-gapore,asaResearchFellow.FromJuly2000toAugust2004,hewasaResearchAssistantattheUniversityofFlorida.SinceAugust2004,hehasbeenanAssistantProfessorofelectricalengineeringatEmbry-RiddleAeronauticalUniversity,DaytonaBeachcampus.Hisresearchinterestsincludewirelesscommunications,signalpro-cessing,andavionics. JianLireceivedtheM.S.andPh.D.degreesinelectricalengineeringfromTheOhioStateUniversity,Columbus,in1987and1991,respectively.FromApril1991toJune1991,shewasanAdjunctAssistantProfes-sorintheDepartmentofElectricalEngi-neering,TheOhioStateUniversity,Colum-bus.FromJuly1991toJune1993,shewasanAssistantProfessorintheDepartmentofElectricalEngineering,UniversityofKen-tucky,Lexington.SinceAugust1993,shehasbeenwiththeDe-partmentofElectricalandComputerEngineering,UniversityofFlorida,Gainesville,wheresheiscurrentlyaProfessor.Hercurrentresearchinterestsincludespectralestimation,arraysignalprocess-ing,andtheirapplications.Dr.LiisaMemberofSigmaXiandPhiKappaPhi.Shereceivedthe1994NationalScienceFoundationYoungInvestigatorAwardandthe1996O ceofNavalResearchYoungInvestigatorAward.ShewasanExecutiveCommitteeMem-berofthe2002InternationalConferenceonAcoustics,Speech,andSignalProcessing,Orlando,Florida,May2002.ShehasbeenanAs-sociateEditoroftheIEEETransactionsonSignalProcessingsince1999andanAssociateEditoroftheIEEESignalProcessingMag-azinesince2003.SheispresentlyaMemberoftheSignalProcess-ingTheoryandMethods(SPTM)TechnicalCommitteeoftheIEEESignalProcessingSociety.