Compressive Sensing Based High-Resolution Channel Estimation for OFDM System

CompressiveSensingBasedHigh-ResolutionChannelEstimationforOFDMSystem

Jia(Jasmine)Meng,StudentMember,IEEE,WotaoYin,YingyingLi,NamTuanNguyen,StudentMember,IEEE,

andZhuHan,SeniorMember,IEEE

Abstract—Orthogonalfrequencydivisionmultiplexing(OFDM)isatechniquethatwillprevailinthenext-generationwirelesscommunication.ChannelestimationisoneofthekeychallengesinOFDM,sincehigh-resolutionchannelestimationcansignifi-cantlyimprovetheequalizationatthereceiverandconsequentlyenhancethecommunicationperformances.Inthispaper,weproposeasystemwithanasymmetricdigital-to-analogcon-verter/analog-to-digitalconverter(DAC/ADC)pairandformulateOFDMchannelestimationasacompressivesensingproblem.Byskillfullydesigningpilotsandtakingadvantagesofthesparsityofthechannelimpulseresponse,theproposedsystemrealizeshigh-resolutionchannelestimationatalowcost.Thepilotdesign,theuseofahigh-speedDACandaregular-speedADC,andtheestimationalgorithmtailoredforchannelestimationdistinguishtheproposedapproachfromtheexistingestimationapproaches.Wetheoreticallyshowthatintheproposedsystem,a-resolu-tionchannelcanbefaithfullyobtainedwithanADCspeedat

=(2log()),whereisalsotheDACspeedandisthechannelimpulseresponsesparsity.Sinceissmalland

isrelativelycheap,weincreasingtheDACspeedto

obtainahigh-resolutionchannelatalowcost.Wealsopresentanovelestimatorthatisbothfasterandmoreaccuratethanthetypical1minimization.Inthenumericalexperiments,wesimulatedvariousnumbersofmultipathsanddifferentSNRsandletthetransmitterDACrunat16timesthespeedofthereceiverADCforestimatingchannelsatthe16resolution.Whilethereisnosimilarapproaches(forasymmetricDAC/ADCpairs)tocomparewith,wederivetheCramér–Raolowerbound.

IndexTerms—Channelestimation,compressivesensing,orthog-onalfrequencydivisionmultiplexing(OFDM).

I.INTRODUCTION

I

ManuscriptreceivedMarch28,2011;revisedAugust06,2011;acceptedAugust31,2011.DateofpublicationSeptember26,2011;dateofcurrentversionJanuary18,2012.TheworkofZ.HanwassupportedinpartbyNSFCNS-0910461,NSFCNS-0901425,NSFECCS-1028782,andNSFCAREERAwardCNS-0953377.TheworkofW.YinwassupportedinpartbyNSFECCS-1028790,NSFCAREERAwardDMS-07-48839,andONRGrantN00014-08-1-1101.TheassociateeditorcoordinatingthereviewofthismanuscriptandapprovingitforpublicationwasProf.PhilippeCiblat.

J.MengiswiththeCGGVeritas,LLC,Houston,TX77072USA(e-mail:jmeng2@uh.edu).

W.YiniswiththeDepartmentofComputationalandAppliedMathematics,RiceUniversity,Houston,TX77005USA(e-mail:wotao.yin@rice.edu).

Y.LiiswiththeDepartmentofComputationalandAppliedMathematics,RiceUniversity,Houston,TX77005USA,andalsowiththeDepartmentofElectricalandComputerEngineering,UniversityofHouston,Houston,TX77204USA(e-mail:yingyingli1985@gmail.com).

N.T.NguyeniswiththeDepartmentofElectricalandComputerEngineering,UniversityofHouston,Houston,TX77004USA(e-mail:tuannam79@gmail.com).

Z.HaniswiththeDepartmentofElectricalandComputerEngineering,Uni-versityofHouston,Houston,TX77004USA,andalsowiththeDepartmentofElectronicsandRadioEngineering,KyungHeeUniversity,Seoul130-701,Korea(e-mail:zhan2@mail.uh.edu).

Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineathttp://ieeexplore.ieee.org.

DigitalObjectIdentifier10.1109/JSTSP.2011.21699

NATYPICALwirelessscenario,thetransmittedsignalarrivesatthereceiverviavariouspathsofdifferentlengths.Thisleadstointersymbolinterference(ISI)andpostsamajordifficultytoinformationdecoding,forexample,inorthogonalfrequencydivisionmultiplexing(OFDM).OFDMhasbeenwidelyappliedinwirelesscommunicationsystemsbecauseittransmitsatahighrate,achieveshighbandwidthefficiency,andisrelativelyrobusttomultipathfadinganddelay[1].OFDMapplicationscanbefoundindigitalaudiobroadcasting(DAB),HDTV-digitalvideobroadcasting(DVB),wirelessLANnet-work,3GPPLongTermEvolution(LTE),andIEEE802.16broadbandwirelessaccesssystem,etc.CurrentOFDM-basedWLANstandards(suchasIEEE802.11a/g)requireacoherentdetectionattheOFDMreceiver.Thisrequirementneedsanaccuratemultipathchannelestimationofchannelstateinfor-mation(CSI),whichcomeswithcomputationandbandwidthoverheads.ThereisrichliteratureonOFDMchannelestima-tion.Below,weprovideabriefoverview.

Therearetwomajorclassesofchannelestimationschemes.Onedoesnotusepilotsymbolsandiscalleddecision-directed,andtheotherusespilotsymbols[13].Theapproachesintheformerclasscanbedeployedwherethesendingpilotsignalsisnotapplicable(e.g.,passivelisteninginamilitarycontext)[14],[15].Ontheotherhand,theyrequirealargeamountofdatatoconvergeduetothereceiverbeing“blind.”Theapproachesinthelatterclasscantakeadvantagesofthepilotsinthetrans-mitteddata,whicharethetrainingsequencesknownbyboththetransmitterandreceiver,andtherefore,theyachievemoreaccuratechannelestimationandarefaster.Theapproachdevel-opedinthispaperbelongstothisclass.

Thedesignofapilot-assistedapproachincludesboththepi-lotsandtheestimationalgorithm.Thegoalistoachieveanop-timalcombinationofspectrumefficiencyandestimationaccu-racy[16]–[20].AmongtheexistingOFDMchannelestimationapproaches,somearebasedonthetime-multiplexedpilot,fre-quency-multiplexedpilot,andscatteredpilot[21].Theyachieverelativelyhigherestimationaccuracyyetuserelativelymorepi-lots.TherehavebeenattemptstoreducethenumberofpilotssuchasJ.Byunetal.[22],whichsendsoutasmallnumberofpre-estimationpilotstoestimatethenumberofpilotsneededinthemainestimation.Thereisnoguaranteedoverallreduc-tionofpilotsthough.Anotherapproachistheadaptivechannelestimationproposedin[23],whichusesalogiccontrollertochooseamongseveralavailabletrainingpatterns.Thecontroller

1932-4553/$26.00©2011IEEE

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choiceisbasedonthecross-correlationbetweenthepilotsym-bolsovertwoconsecutivetimeinstants,aswellasthedeviationfromthedesiredbiterrorrate(BER).Comparedwiththetra-ditionalleast-squareschannelestimator,thisadaptivechannelestimationhastheadvantagesofalowBERandhighdatarate.Unliketheaforementionedapproacheswithpilotsymbolsonregularlattices,therecentworkofFertlandMatz[24]pro-posesirregularpilotarrangementsandnonuniformsamplingtechniquesalongwithaconjugate-gradient-basedchannelestimator.Theirproposedsystemfeaturesalowcomputationalcomplexitywhilemaintainingasimilarchannelestimationaccuracyasthemean-squared-error-minimization(MMSE)channelestimator.

Webelievethatasasensingproblem,OFDMchannelesti-mationcanbenefitfromcompressivesensing(CS),whichac-quiresasparsesignalfromfewersamplesthanwhatisdic-tatedbytheNyquist–Shannonsamplingtheorem(cf.asurveyofthistopicinthesettingofwirelesscommunication[27]).CSencodesasparsesignalbytakingits“incoherent”linearpro-jectionsandsubsequentlydecodesthesignalusingsparseopti-mizationsuchasminimization.TomaximizethebenefitsofCSforOFDMchannelestimation,oneshallcarefullydesignitsencodinganddecodingsteps.Theycorrespondtothetwofo-cusesofthispaper:thedesignsofthepilotsandtheestimator,respectively.WeshallnotethatCShasbeenappliedtochannelestimationin[28]–[31],andsomepreliminaryresultswithlittleproofandanalysishasbeenpublishedin[39].

ComparedtotheexistingCS-basedwork[2]–[5],ourap-proachisuniqueinvariouswaysasfollows.Weusepilotswithuniformrandomphasesandofferanoveltheoreticalguaranteeforfaithfulestimation.Itsproofisbasedonfirstshowingaconcentration-of-measurephenomenonforacertainsubsam-pledcirculantmatrix,subsequentlyshowingitsrestrictedisom-etryproperty(RIP),andapplyingtheexistingRIP-basedresultstoestablishtherecoveryguarantee.Theresultshowsthatonecanobtainhigh-resolutionchannelbyjustincreasingthetrans-mitterDACspeedwhilekeepingthereceiverADCunchanged.Inaddition,anovelestimatoristailoredforOFDMchannelre-sponse;insteadofusingthegenericminimization,wemodifyittotakeadvantagesofthecharacteristicsofchannelresponse,byusingiterativesupportdetection(ISD)[6]andalimited-sup-portleast-squaressubproblem.Theresultingalgorithmisverysimpleandperformsnoticeablybetterthangenericminimiza-tion.Furthermore,wederiveaCramér–Raolowerboundofthemeansquareerror,whichiscomparedtotheactualperformanceoftheestimator.

Wedemonstratetheefficiencyandeffectivenessofthepro-posedapproach.Wehopethattheresultsofthispapercon-vincethereaderwiththepotentialoftheproposedapproachasalow-costandhigh-performancechannelestimator.

Therestofthispaperisorganizedasfollows.SectionIIreviewsthegeneralOFDMsystemmodel.SectionIIIrelateschannelestimationtoCSandpresentstheproposedpilotdesignwithitstheoreticalproperties.InSectionIV,introducesourOFDM-tailoredestimator,analyzesitscomplexity,andde-rivesaCramér-Raolowerboundforperformancecomparison.

SectionVpresentsthesimulationresults.Finally,SectionVIconcludesthisworkanddiscussessomefuturework.

II.OFDMSYSTEMMODEL

AbasebandOFDMsystemisshowninFig.1.Inthissystem,themodulatedsignalinthefrequencydomain,representedby

,isinsertedwithpilotsignal,andthenan-point

IDFTtransformsthesignalintothetimedomain,denotedby

,whereacyclicextensionoftimelengthisadded

toavoidinter-symbolandinter-subcarrierinterferences.There-sultingtimeseriesdataareconvertedbyadigital-to-analogcon-verter(DAC)withaclockspeedof

Hzintoananalogsignalfortransmission.Weassumethatthechannelresponse

comprises

propagationpaths,whichcanbemodeledbyatime-domaincomplex-basebandvectorwithtaps:

(1)

whereisacomplexmultipathcomponent,standsfortheDiracdelta,andisthemultipathdelay.

Since

isshorterthantheOFDMsymbolduration,thenonzerochannelresponseconcentratesatthebeginning,which

translateto

,i.e.,onlythefirst

componentsofcanpossiblytakenonzerovaluesand

.Assumingthatinterferencesareeliminated,whatarrivesatthereceiveristheconvolutionofthetransmittedsignalandthechannelresponseplusnoise,denotedbygivenby

(2)

wheredenotesconvolutionanddenotestheAWGNnoise.Passingthroughtheanalog-to-digitalconverter(ADC),

issampledas,andthecyclicprefix(CP)isremoved.TraditionalOFDMchannelestimationschemesas-sume

.If,thenisadownsampleof.An-pointDFTconvertsto,wherethepilotsignalwillberemoved.Thegoalistorecoverthechannelvectorfromthemeasurements(or,equivalently),giventhepilots(or,equivalently).Throughoutthepaper,weusecapitallet-tersforfrequencydomainsignalsandlowercaselettersfortimedomainsignals.

III.COMPRESSIVESENSINGANDPILOTDESIGN

Inthissection,wepresentanovelCS-basedOFDMchannelestimationarchitecture.Wefirstprovidethemotivation,aswellastheCSbackground.Next,weproposetodesignpilotswithuniformrandomphasesandgivethereasonsbehind.Alongwithatheoreticalguarantee,wepresentnumericalevidenceshowingthatourdesignachievesanoptimalencodingperformance.Fi-nally,wecompareourproposedapproachwiththerelatedex-istingresults.A.Motivation

CSallowssparsesignalstoberecoveredfromveryfewmea-surements,whichoftentranslatestofewersamplesandshorter

MENGetal.:COMPRESSIVESENSINGBASEDHIGH-RESOLUTIONCHANNELESTIMATIONFOROFDMSYSTEM17

Fig.1.BasebandOFDMsystem.

Fig.2LogicBlockDiagramoftheproposedCS-OFDM.

sensingtimes.Becausethechannelimpulseresponseisverysparse(especiallyintheoutdoorcase),wearemotivatedtoapplyCStorecoverahigh-dimensionalfromasmallnumberofsamples.Sinceinchannelestimation,thesamplenumberisdeterminedbythereceiverADCspeedandthedimensionofbythetransmitterDACspeed,weproposetoobtainahigh-di-mensional(thushigh-resolution)byemployingapairofhigh-speedDACandregular-speedADC.Hereregular-speedmeansthespeedforgeneraldatatransmission.Intoday’smarket,thepriceforDACismuchlowerthanthatofADC.SincetheADCrunsataregularspeed,weconsiderourhigh-speed-DACap-proachaninexpensivewaytoobtainhigh-resolutionchannelestimation.B.CSBackground

CStheories[7],[8],[25]statethatan-sparsesignal1canbestablyrecoveredfromlinearmeasurements,

rowsandcolumns,whereisacertainmatrixwith

,andisnoise,byminimizingthe-normof.ClassicCSoftenassumesthatthesensingmatrix,afterscaling,satisfiestherestrictedisometryproperty(RIP)

(3)

istheRIPparameter.forall-sparse,where

Theworksin[36],[37],and[43]alsostudythestablerecoveryoffromnoisyobservationsbasedonconditionson.TheRIPissatisfiedwithhighprobabilitybyalargeclassofrandom

1In

matricessuchasthosewithentriesindependentlysampledfromasubgaussiandistribution.

However,theclassicrandomsensingmatricesarenotadmis-sibleinOFDMchannelestimationbecausethechannelresponseisnotdirectlymultipliedbyarandommatrix;instead,asde-scribedinSectionII,isconvolutedwith,followedbynoisecontaminationanduniformdownsampling.Becauseconvolu-tionisacirculantlinearoperator,wecanpresentthisprocessby

(4)

whererepresentsafullcirculant(convolution)matrixdeter-denotestheuniformdown-samplingfrompointsminedby,

toitssubset,isnoise.Asiswidelyperceived,CSfavorsfullyrandomand

matrices,whichenjoyRIPsandthusadmitstablerecoveryfromfewestmeasurements(intermsoforderofmagnitude),butboth

andinourcasearenotas“random.”Thesefactorsseem-wouldbeunlikelytoworkwellforCS.inglysuggestthat

Nevertheless,carefullydesignedcirculantmatricescandeliverthesameoptimalCSperformance.C.PilotsWithRandomPhases

Todesignthesensingmatrix,weproposetogeneratepilotsineitheroneofthefollowingtwoways:1)therealandimag-aresampledindependentlyfromthestan-inarypartsof

;2)(sameas[30])dardGaussiandistribution,

,,haveindependentrandomphasesbutauniformamplitude.NotethatsinceistheinversediscreteFouriertransformof,theentriesoftheresultingoftype1)areindependentandidenticallydistributed(i.i.d.)standardGaussian.Furthermore,oftype1)haveindependentrandomamplitudes,sotype2)ismorerestrictivethantype1).Onthe

ofbothtypeshaverandomphases.Letde-otherhand,

notethediscreteFouriertransform.Fromtheconvolutionthe-and,wehaveorem

ourcase,SisequaltoP,thenumberofnonzerotapsin(1).

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,sothemeasurementscanbewritten

as

(5)

whichisillustratedinFig.2.

Notethattheproposedsamplingisverydifferentfrompartial

Fouriersampling

orwidelyusedincompressiveimaging(e.g.,MRI).Thelatterrequiresarandomtoavoidaliasingartifactsintherecoveredimage.Incontrast,thepro-posedschemepermitsarbitrarytypesofincludingtheonecorrespondingtouniformdownsampling,whichnaturallyoc-curswhentheADCrunsataspeedlowerthantheDAC.There-fore,theproposedschemeiseasytoimplementintheOFDMsystem.Inthenexttwosubsections,weshowtheencodingef-ficiencyofthisschemeboththeoreticallyandnumerically.Tokeepourexpositiongeneral,thediscussionsinthissectiondonotassumethatthenonzeroentriesofonlyoccurinitsfirst

positions.ThispropertyofOFDMchannelsshallbe

exploitedinthenextsectiontoimproveboththetheoreticalandnumericalperformances.

D.CSbyRandomConvolution

WefirstreviewtheexistingCSresultsofrandomconvolu-tion.In[28],Toeplitz2measurementmatricesareconstructedwithi.i.d.randomrow1[thesameastype1)]butwithonly1or{1,0,1};theirdownsamplingeffectivelytakesthefirstrows;andthenumberofmeasurementsneededforstablerecoveryisshownas.[29]usesa“partial”Toeplitzmatrix,withi.i.d.BernoulliorGaussianrow1,forsparsechannelestimationwherethedownsampling

effectivelyalsotakesthefirst

rows.Theirschemerequiresforstablerecovery.In[30],random

convolutionoftype2)witheitherrandomdownsamplingorrandomdemodulationisproposedandstudied.Itisshownthattheresultingmeasurementmatrixisincoherentwithany

givensparsebasiswithahighprobabilityand

recoveryisstablegiven

.Ourproposedtype2)ismotivatedby[30].Recentresultsin[38]showthatseveralrandomcirculantmatricessatisfytheRIPinexpectation

given

witharbitrarydownsampling.Therestofthissubsectionfocusesonprovingtherecoveryguaranteesfortheproposedtype-1)sensingscheme.Inshort,weshallestablishstablerecovery

underthecondition

,thatis,whenthechannelissparse,therecanbeuptoalogdifferencebetweentherecoveredchannelresolutionandthereceiverADCspeed.

Wenotethatonemighthopetoimprove

to,likeinthesameofi.i.d.Gaussiansensingmatrices,butitwillrequireanovelapproach.

Letthetype-1)CSsensingmatrixbedenotedby

(6)

2which

isslightlymoregeneralthancirculant.

whereisjustafactorforthenormalizationpurpose,isadownsamplingoperatorthatkeepstheentriesinanarbitrary

indexsetofcardinality

anddiscardstherest,and..

.

isacirculantmatrixwithcomplexstandardGaussianrandom

.

Theproofsketchisthefollowing.Themainstepisacon-centration(isometry)result:foranarbitrary-sparsevectorwith,isconcentratedarounditsmean,whichequals1.Theunit-normofgivestheunitmean;theyarenotessential.Theremainingstepsfollowtheargumentsin[40],withminorchangestosomeformulasandnumbers:roughly

speaking,wefixanarbitraryindexsetwith

,pickan-net—andusetheaboveconcentrationresultforasingletoestablish

theisometryfor

uniformlyover;then,basedonthe-nettrickandaunionbound,theisometryisex-tendedfromtoall

uniformly;andfinally,theunionboundisappliedagaintoextendtheisometrypropertyfrom

withafixedtothesetofall-sparsevectors.Thises-tablishestheRIPof,moreaccurately,withhighprobability

given

.QuotingexistingRIP-basedre-coveryresults,wethenobtainstablerecoveryguaranteesforall-sparsevectors.

Themajorworktoprovetheconcentrationresultisbasedon

reducing

toandapplyingthefollowingresultfrom[42]thatrelatestheconcentrationoftotheparameters.

Lemma1(Sec.4.1of[42]):Assumethat

fori.i.d.and.Let.Thefollowinginequalitiesholdforany:

(7)(8)

Therefore,weshallexpress

asandboundand.

1)ConcentrationResultofRandomCirculantMatrices:Letbesuchthatand.(Weshallremovetheunit-normassumptionlater.)Webreakthedevelop-mentintoafewsteps:

Step1)Basedonthesymmetryofconvolution,wecan

rewrite

MENGetal.:COMPRESSIVESENSINGBASEDHIGH-RESOLUTIONCHANNELESTIMATIONFOROFDMSYSTEM19

whereand

..

.

Step2)Letbethefull-sizesingularvalue

decomposition(SVD)ofmatrix

,andassume.Introduce.

Sinceisunitary,iscomplexstandardGuassianaswell.Forsimplicity,weassumethereal-valued

,whichcausesalossoffactorof

butdoesnotchangetheresultsbelowinanyessentialway.Hence,

(9)

ToapplyLemma1,weletand.Weshallbound

and.

Step3)Since

,wehave.

Step4)Sinceeveryroworcolumnofhasaunit2-norm

andatmostnonzeroentries,theroworcolumn

hasamaximal1-normof.Hence,wehaveand

.Therefore,

(10)(11)

andapplyingLemma1to

(12)

gives

(13)(14)

Let

andobtain

.Combining(13)and

(14)andnoting

(15)

wegetconcentrationinequality

(16)

where

.

Theorem1:Amatrixgeneratedby(6)satisfiestheconcen-trationinequality(16)forany-sparsevector.

2)FromConcentrationtoRIP:Inequality(16)letsusfollowtheargumentsof[40]andobtainthefollowingtworesults.

Lemma2:Foranygivenindexsetwith

and,amatrixgeneratedby(6)satisfiestheinequality

(17)

holdswithprobabilityatleast

From(17)totheRIPinequality(3),weshallapplyingtheunion

boundwiththemultiple

.Hence,(3)failstoholdwithprobabilityatmost

(18)

Ifwechoose

andlet,thenandtheright-handsideof(18)

.Hence,foreachwecanchoosesmallenoughtoen-sure

.Therefore,wegetthefollowing:Theorem2:Letmatrixbegeneratedby(6).If

,thensatisfiestheRIPwithaprescribed

withprobabilityatleast,wheretheconstantsin

dependonlyon.FromTheorem2andthefact[43]thatisasuffi-cientconditionfor-minimizationtorecoverall-sparsevec-torsuniversallyandrecoverallnearly-sparsevectorsstably,wecanconcludethatuniversalstablyrecoveryconditionforma-trixgeneratedby(6)is.E.IntuitiveExplanations

Letusexplainintuitivelywhy(5)isaneffectiveencoding

schemeforasparsevector.ThekeyofsuccessfulCSencodingisthatnomatterwherethenonzerosinare,eachmeasure-mentmustcontainaroughlyequalamountofinformationfromeachnonzeroin;inotherwords,theinformationinmustspreadoutinthemeasurements,andthespreadingmustnotde-pendonwheretheinformationislocalizedin.Itiscommonly

knownthataslongasissparse,

isnon-sparse(theuncer-taintyprinciple)andthusitsinformationisspreadoverallits

components.Thechallengesaretoavoid

fromde-spreading

.Therandomphasesofbydesignareofcrit-icalimportance.They“scramble”thecomponentsof

andbreakthe\"delicaterelations\"amongthesecomponentsinaway

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Fig.3.Meansquareerrorversusnumberofmultipath(SNR=30dB).

that,contrarytobeingsparse,isnotsparseatall.Onecanseethisbyrecallingthatthephases

of

encodethelocationoftheinformationin.Whenissparse,itsinformationishighlylocalized.Randomly“scram-bling”thephasescausestheinformationtospreadover.Duetoaphenomenoncalledconcentrationofmeasures,theinforma-tioninspreadsoverthecomponents

inawaythat,withhighprobability,thesizesofall-sparseareuniformlypre-served(scaledbyafactor

)bywithofasizeessentiallylinearin

and.Preservingsizemeanspreservingpair-wisedistances,sothosedownsampledmeasure-mentsperformstableembedding,whichsubsequentlyallowsminimizationtoobtainastablerecoveryof.

F.NumericalEvidenceofEffectiveRandomConvolutionCSperformanceismeasuredbythenumberofmeasurementsrequiredforstablerecovery.Tocomparetheproposedsensingschemeswiththewell-establishedGaussianrandomsensing,weconductnumericalsimulationsandshowitsresultsinFig.3.WecomparethreetypesofCSencodingmatrices:thei.i.d.Gaussianrandomcomplexmatrix,andthetwocirculantrandomcomplexmatricescorrespondingtotypes1)andtype2)above.Inaddi-tion,thestandardminimizationiscomparedtoourproposedalgorithmCS-OFDM,whichisdetailedinthenextsection.ThesimulationsresultsshowthattherandomconvolutionsofbothtypesperformjustaswellastheGaussianrandomsensingma-trix,andouralgorithmCS-OFDMfurtherimprovestheperfor-mancebyhalfofamagnitude.

G.RelationshiptoExistingCS-BasedChannelEstimationOurworkiscloselyrelatedto[29]and[31].In[29],i.i.d.BernoulliorGaussianvectorisusedastrainingsequence,anddownsampleiscarriedoutbytakingonlythefirstrows,whilechannelestimationisobtainedasasolutiontotheDantzigse-lector.In[31],MIMOchannelsareestimatedbyactivatingallsourcessimultaneously.Thereceiversmeasurethecumulative

response,whichconsistsofrandomconvolutionsbetweenmul-tiplepairsofsourcesignalsandchannelresponses.Theirgoalistoreducethechannelestimationtime.minimizationisusedtorecoverthechannelresponse.

Ourcurrentworkislimitedtoestimatingasingle-vector.Althoughourworkisbasedonsimilarrandomconvolutiontech-niques,wehaveproposedtouseapairofhigh-speedDACtransmitterandregular-speedADCreceiverforthenovelgoalofhigh-resolutionchannelestimation.Furthermore,wederivetheoreticalguaranteesandapplyanovelalgorithmtailoredfortheOFDMchannel,whichisdescribedindetailsinSectionIVbelow.

IV.OFDMCHANNELESTIMATOR

Inthissection,wefirstformulatetheproblemfortheOFDMchannelestimator.Then,wepresentthenumericalalgorithm,aswellasitscomplexityanalysis.Finally,anestimatedperfor-mancelowerboundisgiventoevaluatetheproposedalgorithm.A.ProblemFormulation

Asaresultofrapiddecayingofwirelesschannels,—thenumberofsignificantmultipathcomponents—issmall,sothechannelresponseishighlysparse.Recallthatthenonzero

componentsofonlyappearinthefirst

components.3Weshallrecoverasparsehigh-resolutionsignalwithaconstraintfromthemeasurementsatalowerresolutionof.Wede-fineastheamplitudeofacomplexnumber,asthetotal

numberofnonzerosof

,and.Thecorre-spondingmodelis

(19)

wheredenotes

in(5).Generallyspeaking,problem(19)isNP-hardandisimpossibletosolveevenformoderate.Acommonalternativeisitsrelaxationmodelwiththesameconstraints:

(20)

whichisconvexandhaspolynomialtimealgorithms.Ifhasnonoise,both(19)and(20)canrecoverexactlygivenenoughmeasurements,but(20)requiresmoremeasurementsthan(19).B.Algorithm

Insteadofusingagenericalgorithmfor(20),wedesignanalgorithmspeciallytoexploittheOFDMsystemfeatures,in-cludingthespecialstructureofandnoisymeasurements.Atthesametime,wemaintainalgorithmsimplicitytoachievelowcomplexityandmatchwitheasyhardwareimplementation.

3fordataN~

isandknown.0.8󰀂Comparedsforcyclicwithprefix).N,theEvenratioforis0.81/5󰀂ins,thetheWiFinumbersystemofmultipath(3.2󰀂sisstillrelativelysmallespeciallyintheoutdoorenvironment.Therefore,thechanneltapsarestillsparse.

MENGetal.:COMPRESSIVESENSINGBASEDHIGH-RESOLUTIONCHANNELESTIMATIONFOROFDMSYSTEM21

Firstofall,wecansimplycombinetwoconstraintsintoone

bylettingthevariablesbe

anddrop-pingtherestcomponentsof.Letbethematrixformed

bythefirst

columnsof.Hence,theonlyconstraintsare.Sincethesolutionsparsityremainstobemuch

smallerthan,thesparseoptimizationisstillneeded.TheRIPresultinthelastsectiontellsusthenumberofrequiredmeasure-mentsis

,whereforOFDM,insteadof.Since,withthesamenumberofmea-surements(receiverADCspeed)onecanestimatethechannel

withalargeandthusanevenlarger.Moveover,fromthecomputationalpointofview,itreducesthesizeandcomplexityofourproblemandthusmakesthealgorithmfaster.

WealsodevelopouralgorithmCS-OFDMforthepurposeofhandlingnoisymeasurements.Theiterativesupportdetection(ISD)schemeproposedin[6]hasaverygoodperformanceforsolving(20)evenwithnoisymeasurements.OuralgorithmusestheISD,aswellasafinaldenoisingstep.Inthemainloop,itestimatesasupportsetfromthecurrentreconstructionandre-constructsanewcandidatesolutionbysolvingtheminimization

problem

,andititeratesthesetwostepsforasmallnumberofiterations.Theideaofiterativelyup-datingtheindexsethelpscatchmissingspikesanderasefakespikes.Thisisan-basedmethodbutoutperformsthestan-dardminimization.Becausethemeasurementshavenoise,thereconstructionisneverexact.Ouralgorithmusesafinalde-noisingstep,whichsolvesleast-squaresoverthefinalsupport,toeliminatetinyspikeslikelyduetonoise.ThepseudocodeoftheproposedalgorithmislistedinAlgorithm1.Algorithm1CS-OFDMInput:;

Initalize:

thefirst

columnsof.and

whilethestoppingconditionisnotmet,do

Subproblem:

(21)

Supportdetection:,

where

.

Weightsupdate:;

,

otherwise.endwhile

Support-restrictedleast-squares:

threshold;solve

and

(22)

}Return

and

.

InAlgorithm1,ateachiteration,(21)solvesaweightedproblem,andthesolutionisusedforsupportdetectiontogenerateanew.Afterthemainloopisdone,asupportisestimatedaboveathreshold,whichisselectedbasedonem-piricalexperiences.Ifthesupportdetectionisexecutedsuccess-fully,wouldbethesetofallchannelmultipathdelay.Finally,isconstructedbysolvingasmallleast-squaresproblem,and

,falltozero.C.ComplexityAnalysis

Thisalgorithmisefficientsinceeverystepissimpleandthetotalnumberofiterationsneededissmall.Thesubproblemisastandardweightedminimizationproblem,whichcanbesolvedbyvarioussolvers.Asisaconvolutionoperator,wechooseYALL1[11]since1)itallowsustocustomizetheoper-atorsinvolvinganditsadjointtotakeadvantagesoftheFFT,makingiteasiertoimplementthealgorithmonhardware,2)YALL1isasymptoticallygeometricallyconvergentandefficientevenwhenthemeasurementsarenoisy.Withourcustomization,allYALL1operationsareeitherFFTsorone-dimensionalvector

operations,sotheoverallcomplexityis

.Moreover,forsupportdetection,werunYALL1withamoreforgivingstoppingtoleranceandalwaysrestartitfromthelaststepsolu-tion.Furthermore,YALL1convergesfasterastheindexgetsclosertothetruesupport.ThetotalnumberofYALL1callsisalsosmallsincethedetectsupportthresholddecaysexponen-tiallyandboundedbelowbyapositivenumber.Numericalex-perienceshowsthatthetotalnumberofYALL1callsneverex-ceeds,whichisthenumberoftaps.

Thecomputationalcostofthefinalleast-squaresstepis

negligiblebecausetheassociatedmatrix

hasitsnumberofcolumnsapproximatelyequalto,namely,theassociated

matrixforleast-squareshassize

.Generallyspeaking,thecomplexityforthisleast-squaresis.Sinceandaremuchsmallerthan,thecomplexityoftheentirealgorithmisdominatedbythatofYALL1,whichis

.D.Cramér-RaoLowerBound

TheCramér–RaoLowerBound(CRLB)isanindicatoroftheperformanceofanyunbiasedestimator,whichhasbeenusedinmanyapplications[12].Inthissubsection,wederiveaCRLBundertheassumptionthatthetaplocations(thesupportof)areknown.Wearenotawareofwaystodropthesupportassump-tion.Sinceourestimatordoesnotknowthesupport,thesupportawareCRLBderivedispessimistic.Ithasavaluelowerthanthe

22IEEEJOURNALOFSELECTEDTOPICSINSIGNALPROCESSING,VOL.6,NO.1,FEBRUARY2012

CRLBwithanunknownsupport.Nevertheless,thepessimisticCRLBdoesservethecomparisonpurpose.

TheCRLBforeachentryofis

,whereistheFisherinformationmatrix,writtenas

,where

denotesexpectationandistheconditionalprobabilitydensityfunction(pdf)ofgiven.Withknown,thechannelestimationmodelcanbewrittenas

(23)

where

,denotesisthesub-matrixofwithcolumnscorrespondingtotheindicesin,andistheAWGNnoisewithdistribution.Following(23),wecanderivetheconditionalpdfofgiven:

(24)

ItisastandardexercisetoobtaintheoverallCRLB:

(25)

TheaboveCRLBiscomparedtotheactualperformanceinthenumericalstudyinthenextsection.

V.NUMERICALSIMULATIONS

Inthissection,wepresentnumericalsimulationstoillus-tratetheperformanceoftheproposedCS-OFDMalgorithmforhigh-resolutionOFDMchannelestimation.Ourevaluationsarebasedonthemeansquareerror(MSE)ofchannelestimationandtherateofsuccessfulmultipathdelaydetectionswithrespecttodifferentchannelprofilesandsignal-to-noiseratios(SNRs).A.SimulationSettings

WeconsideranOFDMsystemwith-pointIDFT

atthetransmitterand-pointDFTatthereceiver.Thisgivesacompressionratioof16.Thenumberofsilentsub-carrierthatactsasguardbandis256among1024sub-carriers.Thechannelisestimatedbasedon768pilottoneswithuniformlyrandomphasesandaunitamplitude(recallthattheunitamplitudedoesnotchangeestimationresultsbutmakesouralgorithmfaster),withmeasurementSNRsrangingfrom10to30dB.Weassumetheusageofcyclicprefixandthattheimpulseresponseofthechannelisshorterthancyclicprefix,i.e.,thereisnointer-symbolinterference.Forallsimulations,wevarythetotalnumberofmultipathfrom5to15.Wedonotconsiderthecompensationofinphase/quadraturephase(I/Q)imbalanceandcarrierfrequencyoffset(CFO),andleavethemforfuturework.B.MSEPerformance

Fig.4isasnapshotofonechannelestimationsimulation.ItshowsthattheproposedpilotarrangementandCS-OFDMsuccessfullydetectanOFDMchannelwithmultipathand

Fig.4.Exampleofreconstructedmultipathdelayprofile.

Fig.5.MSEperformanceversusnumberofmultipath.

SNRdB.Ourmethodnotonlyexactlyestimatesthemul-tipathdelaysbutalsocorrectlyestimatesthevaluesofthecor-respondingmultipathcomponents.

Fig.5depictstheMSEperformanceonOFDMchannelswiththenumbersofmultipathvaryingfrom5to15andSNRlevelsfrom10to30dB.Asthenumberofmultipathgrows,theMSEincreases.WhenthereareonlyamoderatenumberofmultipathontheOFDMchannel,theMSEisverylow.Inaddition,theincreaseofSNRalsoreducestheMSEforabout10timesper20dB.

Fig.6showsthereconstructedSNRsversusthenumberofmultipathatdifferentinputSNRs.WecanseethatCS-OFDMachievesagaininSNR.Forexample,whentheinputSNRis10dB,weobtainareconstructedSNRhigherthan20dBfor5multipath.Asthenumberofmultipathincreases,theSNRgaindecreases.However,evenwhenthenumberofmultipathis10,westillhavea5dBgain,e.g.,thereconstructedSNRis15dBwhentheinputsignalSNRis10dB.ThesimilarSNR

gainappearsforinputSNR

dBandSNRdBcases.MENGetal.:COMPRESSIVESENSINGBASEDHIGH-RESOLUTIONCHANNELESTIMATIONFOROFDMSYSTEM23

Fig.6.ReconstructedSNRversusnumberofmultipath.

Fig.7.CSrecoveredchannelvarianceversusCRLB.

OverthesetofSNRsandmultipathnumbersinourtested,thereisanaveragegainof6dBfromtheinputSNRtotherecoveredSNR.

C.CRLBPerformance

Fig.7depictstheestimatedchannelvarianceversusthesup-port-knownCRLB,correspondingtodifferentSNRsandmul-tipathnumbers.SincethealgorithmdoesnotknowthesupportwhiletheCRLBdoes,webelievethatthesmallgapsindicateastrongperformanceofthealgorithm.D.MultipathDelayDetectionPerformance

Figs.8and9depicttheprobabilityofcorrectdetection(POD)andthefalsealarmrate(FAR)ofmultipathdelayscorrespondingtodifferentSNRsandmultipathnumbers.WhentheSNRisabove10dB,simulationshows100%PODfornomorethan12multipath.Forthelargenumberofmultipath15,theprobabilityofcorrectmultipathdelaydetectionishigherthan95%forSNRdB.EvenwhenSNRisaslowas10dB,aslongasthenumberofmultipathdoesnotexceed10,westillhaveaPODofgreaterthan95%.TheFARperformance

Fig.8.Probabilityofdetectionversusnumberofmultipath.

Fig.9.Probabilityoffalsealarmversusnumberofmultipath.

showstheconsistantresults:astheSNRdecreasesandthenumberofmultipathincreases,theperformancedecreases.For

SNR

dBandthenumberofmultipath,weobtainnearlyzeroFAR.

VI.CONCLUSION

EfficientOFDMchannelestimationwilldriveOFDMtocarrythefutureofwirelessnetworking.Agreatopportunityforhigh-efficiencyOFDMchannelestimationislentbythesparsenatureofchannelresponse.RidingontherecentdevelopmentofCS,weproposeadesignofprobingpilotswithrandomphases,whichpreservestheinformationofchannelresponseduringtheconvolutionanddown-samplingprocesses,andasparserecoveryalgorithm,whichreturnsthechannelresponseinhighSNR.Thesebenefitstranslatetothehigh-resolutionofchannelestimation,aswellasshorterprobingtimes.Inthispaper,thepresentationislimitedtoanidealizedOFDMmodelandsimulatedexperiments.Inthefuture,wewillfusethemintomorerealisticOFDMframeworks.Theresultspresentedherehintahigh-efficiencyimprovementforOFDMinpractice.

24IEEEJOURNALOFSELECTEDTOPICSINSIGNALPROCESSING,VOL.6,NO.1,FEBRUARY2012

REFERENCES

[1]O.Edfors,M.Sandell,J.-J.VandeBeek,D.Landström,andF.Sjöberg,

AnintroductiontoorthogonalfrequencydivisionmultiplexingLuleåTekniskaUniv..Luleå,Sweden,pp.1–58,Sep.1996.

[2]C.R.Berger,S.Zhou,P.Willett,B.Demissie,andJ.Heckenbach,

“CompressedsensingforOFDM/MIMOradar,”inProc.42ndAnnu.AsilomarConf.Signals,Syst.,Comput.,Asilomar,CA,Oct.2008,pp.213–217.

[3]C.R.Berger,S.Zhou,andP.Willett,“Signalextractionusingcom-pressedsensingforpassiveradarwithOFDMsignals,”inProc.11thInt.Conf.Inf.Fusion,Cologne,Germany,Jul.2008.

[4]G.TauböckandF.Hlawatsch,“Acompressedsensingtechniquefor

OFDMchannelestimationinmobileenvironments:Exploitingchannelsparsityforreducingpilots,”inProc.IEEEInt.Conf.Acoust.,Speech,SignalProcess.(ICASSP),LasVegas,NV,Apr.2008,pp.2885–2888.[5]C.R.Berger,S.Zhou,W.Chen,andP.Willett,“Sparsechanneles-timationforOFDM:Over-completedictionariesandsuper-resolutionmethods,”inProc.IEEEInt.WorkshopSignalProcess.Adv.WirelessCommun.,Perugia,Italy,Jun.2009,pp.196–200.

[6]Y.WangandW.Yin,“Sparsesignalreconstructionviaiterativesupport

detection,”SIAMJ.ImagingSci.,no.3,pp.462–491,Aug.2010,no.3.[7]E.Candes,J.Romberg,andT.Tao,“Robustuncertaintyprinciples:

Exactsignalreconstructionfromhighlyincompletefrequencyinfor-mation,”IEEETrans.Inf.Theory,vol.52,no.2,pp.4–509,Feb.2006.

[8]E.CandesandT.Tao,“Nearoptimalsignalrecoveryfromrandompro-jections:Universalencodingstrategies,”IEEETrans.Inf.Theory,vol.52,no.12,pp.06–25,Dec.2006.

[9]M.Ledoux,“Theconcentrationofmeasurephenomenon,”Amer.Math.

Soc.,2001,082182.

[10]W.GuoandW.Yin,“EdgeCS:Anedgeguidedcompressivesensing

reconstruction,”inProc.Vis.Commun.ImageProcess.(VCIP),HuangShan,AnHui,China,Jul.2010.

[11]J.YangandY.Zhang,Alternatingdirectionalgorithmsfor`-problems

incompressivessensingRiceCAAMRep.TR09-37.

[12]H.Zayyani,M.Babaie-Zadeh,andC.Jutten,“BayesianCramer–Rao

boundfornoisy.non-blindandblindcompressedsensing,”ComputingResearchRepository(CoRR),abs/1005.4316,2010.

[13]Y.Li,“Pilot-symbol-aidedchannelestimationforOFDMinwireless

systems,”IEEETrans.Veh.Technol.,vol.49,no.4,pp.1207–1215,Jul.2000.

[14]P.CiblatandL.Vandendorpe,“Non-dataaidedcarrierfrequencyoffset

estimationforOFDManddownlinkDS-CDMAsystems,”inProc.IEEEthVeh.Technol.Conf.,AtlanticCity,NJ,Oct.2001,vol.4,pp.2618–2622.

[15]P.CiblatandL.Vandendorpe,“Blind,carrier,frequencyoffsetesti-mationfornoncircularconstellation-basedtransmissions,”IEEETrans.SignalProcess.,vol.51,no.5,pp.1378–13,May2003.

[16]P.CiblatandL.Vandendorpe,“Onthemaximum-likelihoodbased

data-aidedfrequencyoffsetandchannelestimates,”inProc.Eur.SignalProcess.Conf.,Toulouse,France,Sep.2002,vol.1,pp.627–630.

[17]S.Coleri,M.Ergen,A.Puri,andA.Bahai,“Channelestimationtech-niquesbasedonpilotarrangementinOFDMsystems,”IEEETrans.Broadcasting,vol.48,no.3,pp.223–229,Sep.2002.

[18]H.WuandX.Huang,“Jointphase/amplitudeestimationandsymbol

detectionforwirelessICIself-cancellationcodedOFDMsystems,”IEEETrans.Broadcasting,vol.50,no.1,pp.49–55,Mar.2004.

[19]A.Punchihewa,Q.Zhang,O.Dobre,C.Spooner,S.Rajan,andR.

Inkol,“OnthecyclcostationarityofOFDMandsinglecarrierlinearlydigitallymodulatedsignalsintimedispersivechannels:Theoreticalde-velopmentsandapplication,”IEEETrans.WirelessCommun.,vol.9,no.8,pp.2588–2599,Aug.2010.

[20]R.Carrasco-Alvarez,R.Parra-Michel,A.Orozco-Lugo,andJ.Tugnait,

“Enhancedchannelestimationusingsuperimposedtrainingbasedonuniversalbasisexpansion,”IEEETrans.SignalProcess.,vol.57,no.3,pp.1217–1222,Mar.2009.

[21]S.TakaokaandF.Adachi,“Pilot-assistedadaptiveinterpolation

channelestimationforOFDMsignalreception,”inProc.Veh.Technol.Conf.,Milan,Italy,May2004,vol.3,pp.1777–1781.

[22]J.ByunandN.P.Natarajan,“AdaptivepilotutilizationforOFDM

channelestimationinatimevaryingchannel,”inProc.WirelessMi-crow.Technol.Conf.,Clearwater,FL,Apr.2009,pp.1–5.

[23]W.M.AfifiandH.M.Elkamchouchi,“Anewadaptivechannelesti-mationforfrequencyselectivetimevaryingfadingOFDMchannels,”inProc.Int.Conf.Comput.Eng.Syst.,Cairo,Egypt,Dec.2009.

[24]P.FertlandG.Matz,“ChannelestimationinwirelessOFDMsystems

withirregularpilotdistribution,”IEEETrans.SignalProcess.,vol.58,no.6,pp.3180–3194,Jun.2010.

[25]D.Donoho,“Compressedsensing,”IEEETrans.Inf.Theory,vol.52,

no.4,pp.12–1306,Apr.2006.

[26]E.Candes,J.Romberg,andT.Tao,“Stablesignalrecoveryfromin-completeandinaccuratemeasurements,”Commun.PureAppl.Math.,vol.59,no.8,pp.1207–1223,Aug.2006.

[27]Y.Li,Z.Han,H.Li,andW.Yin,CompressiveSensingforWireless

Networks.Cambridge,U.K.:CambridgeUniv.Press,2012.

[28]W.U.Bajwa,J.D.Haupt,G.M.Raz,S.J.Wright,andR.D.Nowak,

“Toeplitz-structurescompressedsensingmatrices,”inProc.IEEE/SP14thWorkshopStatist.SignalProcess.,Madison,WI,Aug.2007,pp.294–298.

[29]J.D.Haupt,W.U.Bajwa,G.M.Raz,andR.D.Nowak,“Toeplitz

compressedsensingmatriceswithapplicationstosparsechannelesti-mation,”IEEETrans.Inf.Theory,vol.56,no.11,pp.5862–5875,Nov.2010.

[30]J.Romberg,“Compressivesensingbyrandomconvolution,”SIAMJ.

Imag.Sci.,vol.2,no.4,pp.1098–1128,Nov.2009.

[31]J.K.RombergandR.Neelamani,“Sparsechannelseparationusing

randomprobes,”InverseProblems,vol.26,115015.

[32]Z.TianandG.Giannakis,“Compressedsensingforwidebandcogni-tiveradios,”inProc.IEEEInt.Conf.Acoust.,Speech,SignalProcess.(ICASSP),Apr.2007,pp.1357–1360.

[33]J.Meng,W.Yin,H.Li,E.Houssain,andZ.Han,“Collaborativespec-trumsensingfromsparseobservationsincognitiveradionetworks,”IEEEJ.Sel.AreasCommun.SpecialIss.CognitiveRadioNetworkingCommun.,vol.29,no.2,pp.327–337,Feb.2011.

[34]J.Paredes,G.Arce,andZ.Wang,“Ultra-widebandcompressed

sensing:Channelestimation,”IEEEJ.Sel.TopicsSignalProcess.,vol.1,no.3,pp.383–395,Oct.2007.

[35]J.Meng,J.Ahmadi-Shokouh,H.Li,Z.Han,S.Noghanian,andE.Hos-sain,“Samplingratereductionfor60GHzUWBcommunicationusingcompressivesensing,”inProc.AsilomarConf.Signals,Syst.,Comput.,Asilomar,CA,Nov.2009.

[36]D.L.Donoho,M.Elad,andV.Temlyakov,“Stablerecoveryofsparse

overcompleterepresentationsinthepresenceofnoise,”IEEETrans.Inf.Theory,vol.52,no.1,pp.6–18,Jan.2006.

[37]J.Tropp,“Justrelax:Convexprogrammingmethodsforidentifying

sparsesignalsinnoise,”IEEETrans.Inf.Theory,vol.52,no.3,pp.1030–1051,Mar.2006.

[38]H.Rauhut,J.Romberg,andJ.A.Tropp,“Restrictedisometriesfor

partialrandomcirculantmatrices,”arXiv:1010.1847,2010.

[39]J.Meng,Y.Li,N.Nguyen,W.Yin,andZ.Han,“Highresolution

OFDMchannelestimationwithlowspeedADCusingcompressivesensing,”inProc.IEEEInt.Conf.Commun.,Kyoto,Japan,Jun.2011.[40]R.Baraniuk,M.Davenport,R.DeVore,andM.Wakin,“Asimpleproof

oftherestrictedisometrypropertyforrandommatrices,”Construct.Approx.,vol.28,no.3,pp.253–263,2008.

[41]E.CandesandT.Tao,“Decodingbylinearprogramming,”IEEETrans.

Inf.Theory,vol.51,no.12,pp.4203–4215,Dec.2005.

[42]B.LaurentandP.Massart,“Adaptiveestimationofaquadraticfunc-tionalbymodelselection,”Ann.Statist.,pp.1302–1338,2000.

[43]Q.MoandS.Li,“Newboundsontherestrictedisometryconstant󰀂k,”

Appl.Comput.Harmon.Anal.,vol.31,no.3,pp.460–468,2011.

Jia(Jasmine)Meng(S’10)receivedtheB.S.andM.S.degreesinelectricalengineeringfromtheSouthwestPetroleumUniversity,Chengdu,China,in2004and2007,respectively,andthePh.D.degreeinelectricalengineeringfromtheUniversityofHouston,Houston,TX,in2010.

Herresearchinterestsaretheframeworkofcom-pressivesensingandimplementationforcommuni-cationandsignalprocessing.

MENGetal.:COMPRESSIVESENSINGBASEDHIGH-RESOLUTIONCHANNELESTIMATIONFOROFDMSYSTEM25

WotaoYinreceivedtheB.S.degreeinmathematicsfromNanjingUniversity,Nanjing,China,in2001,andtheM.S.andPh.D.degreesinoperationsresearchfromColumbiaUniversity,NewYork,in2003and2006,respectively.

Since2006,hehasbeenwiththefacultyoftheDepartmentofComputationalandAppliedMathe-matics,RiceUniversity,Houston,TX.Hisresearchinterestsincludeoptimization,aswellasitsapplica-tionsininverseproblems,compressedsensing,signalprocessing,andvariationalimageprocessing.

Dr.YinwontheNSFCAREERAwardin2008andtheAlfredP.SloanRe-searchFellowshipin2009.

NamTuanNguyen(S’11)receivedtheB.E.degreeinelectricalandcomputerengineeringfromHanoiUniversityofTechnology,Hanoi,Vietnam,in2002andtheM.E.degreeinelectricalandcomputerengineeringfromtheSouthernIllinoisUniversity,Edwardsville,in2009.HeiscurrentlypursuingthePh.D.degreeattheUniversityofHouston,Houston,TX.

Hisresearchinterestsincludeapplicationsofma-chinelearningandinformationtheoryincognitiveradio,wirelesssecurity,wirelessnetworks,andwire-lesscommunicationsystems.

YingyingLireceivedtheB.S.degreeinmathematicsfromPekingUniversity,Beijing,China,in2006andthePh.D.degreeinappliedmathematicsfromtheUniversityofCalifornia,LosAngeles.

SheiscurrentlyapostdocjointlyattheUniver-sityofHouston,Houston,TX,andRiceUniversity,Houston.Herresearchinterestsincludesparseopti-mization,machinelearning,andalgorithmdesign.

ZhuHan(S’01–M’04–SM’09)receivedtheB.S.de-greeinelectronicengineeringfromTsinghuaUniver-sity,Beijing,China,in1997,andtheM.S.andPh.D.degreesinelectricalengineeringfromtheUniversityofMaryland,CollegePark,in1999and2003,respec-tively.

From2000to2002,hewasanR&DEngineerwithJDSU,Germantown,MD.From2003to2006,hewasaResearchAssociateattheUniversityofMaryland.From2006to2008,hewasanAssistantProfessoratBoiseStateUniversity,Boise,ID.Currently,heisan

AssistantProfessorintheElectricalandComputerEngineeringDepartment,UniversityofHouston,Houston,TX.Hisresearchinterestsincludewirelessresourceallocationandmanagement,wirelesscommunicationsandnetworking,gametheory,wirelessmultimedia,security,andsmartgridcommunication.Dr.HanhasbeenanAssociateEditoroftheIEEETRANSACTIONSONWIRELESSCOMMUNICATIONSsince2010.HeisthewinnerofFredW.EllersickPrizein2011.HeisanNSFCAREERawardrecipientin2010.HeisthecoauthorforthepapersthatwonthebestpaperawardsattheIEEEInternationalConferenceonCommunications2009andthe7thInternationalSymposiumonModelingandOptimizationinMobile,AdHoc,andWirelessNetworks(WiOpt09).