Radar er

A Clutt er Suppression Scheme for Hi gh Frequency (HF) Radar

By

@Mar ti n Wai Yee POOh,B. Eng .

A thesissubmittedto the Schoolof Graduate Studiesin partialfu lfillm en t ofthe

requi rem entsforthe degre eof Masterof Eng ineer ing

Faculty ofEngineeringand Applied Science MemorialUniversityof Newfound land

December , 1991

St.John's Newfou ndland Canada

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IS Bn 0-3 15- 73 33 2·2

Canada

Abstract

Aclurter suppressionschemeforhighfrequency(IIF) radaris present edinthi~

thesis. TheHrradar has beendevelopedforcOiL~talsurvelllaneeandtheremote sensing of theocean.WhentheIIFelect romagneticwavespropagate over the ocean,thebackscatter from the ocean surfacehaswell definedfreque ncies, known

1\.9"Bragg" frequencies, shiftedfrom theradarfrequency.Oneof the characteristic s ofthe HF backscatteris thehighlevel of oceanclut ter which ham perstarget detection. Itis desirableto suppress the ocean clutter beforeta rge tdetection operation.The proposedscheme is developedLased on therecogni tionof the time- varying behaviourofthe ocean clutter Lhatcanbe simply characte rizedhytwo narrowband Crequency-modulatcdsinusoidal signalswith theircentre Ircquenclea equalto the~Bragg"freque ncies. The schemeconsists oftwo part'!. First,alime- varyinglechniquereferred to asHan kelrankreduct ion met hod is used toes tim at e theinstantaneousfreq uenciesoftheclutt ersignals.Themet hodstalesthat a Hankel matrixofalime seriesdata consistingofafinite number of sinusoidscan beapproximatedto, viaSingularValue Decomposition(SVD), alowerrankmatri x defined by the finitenumber of theprincipal singularvalues,even ifthefrc'(lllencic's ofthe einuacidaarevaryingslowly with lime.The instantan co uefrequencies are estimate d from those princip alsingular values.TheuscofSVD isto decompose the Hankel mat rixinto usignaland anoisevectorsubspace.The signalsubspaceis identifiedby the largestsingular val ues.Second,aprocess inwhich the frequency componentof theclutte r signalsisremovedfromthe reducedrank Hankelmatri x instan ta neously is developedto suppresstheocean clutte r. Subsequently,another reducedrank Hankelmatr ix is constructedfrom whichthe targetsignalcanbe extracted.

Theperformanceof thescheme ha3 beenevaluatedoncomputer-synt hesized

dab.andonecme real datacollected fromarecently developed HF radar.The results from bothCMelII_bowedthat the instantaneous frequencies ortheocean clutt ersignals andthe target signal were properlytrACkedby theHankelrank reductionmethodand that^&substantiallevel oftheocean clutter.intherange of 2{)to 50 dB.couldbe suppressedbythescheme proposed.

iii

Acknowl edgement s

Iwould liketoexpress my deepgratitudetomy twolupervison.Or.R.II.Kh"l1 andDr. S.LeNgoc,(or&11their~idance,edvicesIU\~usi,wcrduring the("oll r!Pf' of mygradua te stu dies and thewritingup ofthethaie.

Iamthankr ulto Nort hernRad ar SyslcmsLtd.forthloirpermissio ntoUM!the..ir HFdata forthetests.

Thefinancial &S3istancereceivedfromtheSchoolofGrad uateStud;~,till"FA(·

uh yof Engineer ing ande-CORE(Cent re for ColdOCUliResourceEngilUocri ng) is gratefully acknowledged .

Last ,but110tleast , the completionof the thesiscannot he done without tIlt' supporthum mywife,Vivian.

Contents

List of Figures

Lis tofTables

Listof Principal Symbols 1 Introduction

1.1 ProblemStatement.. 1.2 DriefBackground Review . 1.3 Scopeofthethesis ..

1.4 Organization of theThesis .

vii

xi

2 Introdu ct iontoHigh Frequency (H F ) Rad ar 2.1 DriefReviewofOperatingTheory of IIF Radar 2.2 Overview

o r

^SomeHF Radar Systems.. .

2.3 Ocean ClutterinIIF Radar ... ... . ..•. ...•... ... . 11 2.3.1 Time-Doma inModelforOcean ClutterSignals. 13 3 Som e'I'ime-varyingFreq uenc y Trac king Techn iques 16 3.1 AdaptiveLinearPredictionFilter... . . .. . ... • . . . .. 16 3.2 EigcnstructureUpdating Meth od . . . . .. ... . . 19

3.3 Hankel Rank ReductionMethod.. 22

.. FrequencyEstimationofStationarySineWaves 4.1 NolselesaCMe..

4.2 Noisy Case.

:) The ClutterSuppressionScheme 5.1 HankelRank ReductionMethod. 5.2 Suppressionof Ocean Clutter Signals.

6 Computer Analysisand Simulation 6.1 Comp uterSimulationsand Results

6.1.1 TestI

6.1.2 TestII .

6.1.3 Tl,.'StIII 6.2 RealHFRadar Data Testing.

6.2.1 TestCase I 6.2.2 TestCaseII. 6.3 Discussion .

7 Conclusio nsandRecommend at ions

Refe r ences ADop ple r Effect

2.

25

38 :!!I

.7

·17 ,19

52

no

86

80

o.

BEige nvalu es, Eigenvectors and SingularValueDecomposition ofa

Ma trix 97

BJ Eigenvaluesand Eigenvectors"Ifa Matri x. 91 B.2 Singular ValueDecompositiono.ta Matrix 98

C TheComputer Program ofthe Clu tterSuppressionScheme 100

,;

List of Figures

2.1 Typical IIF radar powerspectrum showingthecharact erist icsof lirst·( F)and second-order(S)scattering peaks (FromRef.(7])

...

15 2.2 Time-var ying behaviorofthe twodomina nt narro wband frequ ency

compo nentsinIIFrada roceanclutter (From Ref.[7]). 15 4.1 (a) anAftprocess,(b)a linearpred iction-err or filler 26

6.1 IlIs~alltanCO\lSfrequencies ofSI(t)in noiselesscase. 55

6.2 Instantaneous frequenciesofSI(t)at SNR

=

20 dB 55

6.3 Insta ntaneous frequenciesof 31(t)at SNR=10dB 56 6.4 Power spectrumof81(t)

+

82(t)+53{l )

. .

56 6.5 Suppr ession ofoceanclutte rin noiseless case.. . 51

6.6 Sup pression ofocean dut terat SNR=20dB 51

6.1 Suppression ofocean clutterat SNR

=

10dB

. . . ... . ...

58 6.8 Compa risonofthe Doppler spect ra of the HFradarsignal before

and after theclutter suppressio nprocess at range of 70.4 km:(a) datasample length==128 (1:128);(b)(129:156);(c)(257:384); (d ) (385'512). . • .. . . .. . . ... . . . ..•. . . . 64 6.9 Comparisonof theDoppler spectraof theHFradarsigna l before

andelte rthecluttersuppress ionprocess atrange of 70.8km: [a]

dala sample length=128 (1,,28);(b) (129,256);(0) (257,384);(d)

(385,512) ... ..• •.. ... 66

vii

6.10 Comparisonofthe Dopplerspectra of theIIFradar signal before andafterthe clutt er suppressionprocessat range of 71.2kill: (a) data sample lengt h=128 (1:128);(b)(129:256);(c)(257:384);(tI) (385,512).. . . .. .... ... • .. ..

r.s

6.11Comparisonof the timeseries of theshipsignal atrangeof70.·' km:(a)original lIFradarsignal;(b) shipsignal aft!'rsuppression

of oceanclutter , .

6.12 Comparisonofthetimeseriesoftheshipsignal at range of70.8 km:(a) originalll F radar signal;(b)shipsignalaftersuppression

of oceanclutter .

6.13 Comparisonof the timeseriesofthe shipsignalatrange of 71.2 km: (a)original IIFradarsignal;(b) ship signal ...ncrsuppression ofoceanclutter

6.14 Ccmperiscnof the Dopplerspectraofthe IIFradarsignal before and aftertheclutt er suppressionprocessatrangeof88..1kill: (a l data samplelength""128 (1:128);(b)(129:256);(c)(257:334);(II) (385'512).

6.15 Comparisonofthe Dopplerspectr aof theIIF radarsignal hefore and aftertheclutt ersuppression process atrange of 88.8krn:(a)

70

71

72

77

data samplelengthe 128 (1:128);(b) (129:256);(c)(2.'>7:384):(tI)

(385,512). .•.. . ... . . .. ... 79

6.16Comparisonof theDopplerspectra oftheIIFradarsignalbefore and aflertheclutt ersuppression processatrange of 89.2 kill:(a) dat a samplelengrhe 128 (1:128);(b)(129:256): (c) (2.'>7:384);(d)

(385,512) . 81

viii

6.17 Comparison ofthe time series of the ship slgnalatran ge of 88.4 km:(a) originalHFradarsignal;(b)shipsignal aftersuppression ofcceenclutter ..

6.18 Comparisonofthetime series of the shipsigna.!atrange of88.8 km:(a)originalHFradarsignal;(b)shipsignal aftersuppression ofoceanclutte r... ..•.. .•. .... . ... .... . .. 6.19Comparison of the time series of the shipsignalatrange of89.2 km:(a)originalHFradarsignal;(b)ship signalafter suppression

83

8'

ofocean clutter .. •. . 85

A.1Targetgeometryand transmittedand receivedwaveformsfor Doppler effect derivation(J.L.EavesandE.K.Reedy: Principlesof Modern

Radar, 1987) 95

List of Tables

6.1 Thesingular values of }{consistingofstationarysinewaves at SNR

=00,20 dBand10dB:(a)L

=

3, (b)L=5 60

6.2 The singularvaluesofHconsistingof time-varyingsine waves at SNR~00,20 dBand 10 dli:(a)L~2,(b)L

=

6 51 6.3 Asummaryofthe relativequantificationofthecluttersuppression

(indB)inthera nge cells;70.4km,10.8km and 11.2km at fourdif- ferent periodsofthe observation time (where±/""arcthe"BrAgg"

frequencies)_{' "} li2

6,4 Asummaryoftherelative magnitudesof theret urn energy from theship (indB) in therangecells: 7004km, 70.8kmand 71.2km at four differentperiods oftheobservat iontime 63 6.6 Asummaryoftherelative quantification ofthedull ersuppression

(indB)in the range cells:88.4km, 88.8km and 89.2km atfour dif·

ferent periodsoftheobservationtime (where±focarethe"Bragg"

frequencies)

. . ... .. ... ... ... ..

74 6.6 Asummaryoftherelat ive magnitudesorthe returnenergyIrom

theship(in dB)in the rangecells:88,4 km,88.8 km and89.2 km atfourdifferent periodsor the observat iontime

.. ... .. . . . .

76

Li st of Principal Symb ols

Coefficients ofthe autoregressive (AR)process A(z ) Charactcristi r. polynomial ofH(z )

Speed of light c;

D ,(k ) E{"lk ))

s ;

!.

t.

F'

Amplitu deofithcom plexsinusoid Diagonalma.tri x containingtheeigenvaluesofR.

Oulput of the predictionerrorfilter (PEF) Mean -squareerror

Instantaneousfrequencies oftheoceandullersignals

"Bragg"frequencyof the oceanduttersignal Carrierfreque ncy of theradarsignal Dopp ler freqe ncy ofa target Stalefeed bac kmatrix Statematrix /I Hank eldatamatrix llllctll Reduc ed rank Hankelmatrix 1/(: ) a-dom ain trans fer{unction ofthe AR process

Acceleralionduetothe gravity Orderofthe AR proce ss jI! Tota lnumbe r ofdata.samples

Q Orthegonalmatrixcont aining eigenvectorsofR"

Q"'(,,') Modifiedpowerspectrum orPEF

R Covarian cedata matrix

RI< Time- varyingestimate or kth covariance dat a matrix

tiR Range resolu tion ortheHF radar 81(1),,,,( 1) Simulatedoceancluttersignals

S Diagonal matrixcontainingthe singularvalues of a matrix S~(w) Powerspectr umof PEF

U Unitary matrix

Velocityof the ocean wave v(k) State inp ut vector

V Unitarymatr ix

V. Radialvelocity ofatarget W BandwidthoftheFMCWwaveform y(k) Data sam ples

X Stalevectormatrix

YI Datamat rix (armedfromaforward linear predi ction )) Datamat rixformed fromabackwardlinear prediction

eigenvaluesofa matri x ..\~ Oceanwavelength ..\. Wavelengtho{the radarsignal

e

Observabilit y matrix 8new New observabilitymatrix

Wi Angular frequencyof ith complexsinusoid 11'; Singularvalu esof theHankel matrix,I/(z) 4Ji Phase ofit hsinusoidalsign&!.

Memory fact.or Scalar porportionality constant

xil

Chapter 1 Introduction

1.1 Proble m Statemen t

High frequen cy(HF)radar system s havebee n developed forcoas ta lsu rveillance and the remote sensingof theocean,e.g. detect ion ofshipsand monitoringthe sea conditions . The HF elect romagn eticwavesin thefrequ ency band of 3·30MHz propagateover the oceansurfaceingro und wavemode. The highcondu ct ivit y of the sea.wa terreeulte in lowpropa gati onlOllSofthegro u nd wave andhen ce allows

long·ran~c(overthe hori7.on) ta rget detect ion. Echoes from the sea surface and

forany sur face targetsarereceiv edby the radarsyste m.One of thecha rac t eristi cs ofthesea echo, known as theocean clutter,is itshigh level ene rgy. Although these ocean dullersignals havebeen realized to beII.goodsou rceofinfo rma t io n on theoceanconditions,suchMwaveheight ,wind directio n and etc"they become the unwantedsignalsasfar as targetdet ectionis concern ed becausetheywill ob- scure the required targetsignals ,particularlywhenthe target' sDopplerfrequency (afrequencyshifted fromthetransmittedfrequencyof theradar) fallsdoseto the dutter'sDopplerfrequency.Dopplerfrequen cy discriminat iontoseparatethe target from tile ocean dutterwillnot be poeelbleexceptby additionalsignalpro- cesslng.Thus,it is desirabletosuppressthe clutteras much&IIpossiblebefore the ta.rgetdetectionoperation . Theob jective of thisthesisis to developaclutter

suppressionscheme for HF radar.

1.2 BriefBackground Review

In orderto suppressthe oceanclut ter inthe HF radar,it is import antto understand thebehaviour of the clutte rsignal fint a.ndthenspeclficelgnalprocesairtg techni q ue can be developedballed ontheclutter's characteristics.In 1955, Crombie[I)nrllt observedtbat the two dominant peaks displayedby theoceandutterin theDoppler spectrumwere due to thescatteringfrom twosets of ocean waveswhose wavdf'ngth equals half the radarwavelengt happroaching and recedingfro mthe radar . The scatteringfro m these two sets of waves is similar to that from a diffraction graling which is sometimes describeda.,the Bragg-scatler,by analogyto the Bragg-scatter mode for the X-raydiffract ion by crystals(21.Thus,the Do pp ler freq uencies of this ocean eluucrMealsoknownasthe "Bragg"freq uencies.Inaddit iontothose two dominant firs t-orderpeaks, the rearesomesmallerand more complexhigh- ordercomponents. The theoreticalexpreseionscfthe ocean duUerwere derived by ot hersin morerecentyears [3,4,5,61.Anaverage Dopplerspectrumof the firsL- and second-orderscattering canbe determinedfro mthe theoretical expressions. Re cent work by Khan (7Jhas demonstra tedthat the ocean duller hall a time- varyingbehavloue thatcan be modelled bytwo narrowbandtime-va ry ingsignals andtracked by tim e-varyingsignal processin g techn iques.This time-var yingmodel treatstheoceandull er as twomovin g targetsin additionto anyotherpotentia l targets duringta rgetdetection.Thus, thatpolICSaprobleminsupp ressingtwo moving targetsamongtheother targets. The trad itio nalwa.yof supp ressing rad ar clutter,referredto as the moving targetindicator(MTI) filter,is not applicableto this problembeca usethe oceanclut terhallnonzeroDoppler frequencywhile the MTI ASsumesthattheclutter is stationaryandhas zeroDopp ler frequ ency18J.

As the radarclutterin reality isra ther non-stationary,adaptive filtertheory providesapopularway of dealing withthe problem. GibsonandHaykin[91pro- posed anadaptive clutter supp ressiontechniq uein 1983.Thetec hnique is based on theuse of anadaptive autoregressive(AR) modelling ofthe radar clu t teralong wit hthe LeastMeanSquare(LMS)adaptation algorithm.Itis assumed that the cluttersigna lscan be modelledquite closelyby a relative lylow order ARprocess.

That means thecluttersignalismodelle d as theoutputof an all- polefilter having awhitenoise source to its inp ut.Thetechnique also assumesth atthe target and the clutterhave generally diffe rent Doppler spectrawherethe clutter'sspectrum tendsto be diffused as comparedto that of thetarget.This techniquework ed well whendealing with theclutter such as weather, groundandicepellet encounte red by airtrafficsurveillanceradar . Allthe se clutter phenome naappearto havea wide spread Dopp ler spectrum.However,theoceanclutter inOFradar is observedto have a similarDopple rspectrumasthatof the target,i.e. ofnarrow spectral widthe.In thiscase,thecluttercomponentsarenotlikelyto be suppressed.

Asimilarapproach has beenused by Hou[101in1984.In thiscluttersuppres - siontechnique, MaximumEntropyMet hod (MEM) which isa. spectralanalysis methodof theAR prOCCS8 isused to model theclutter of interest. A predeter - mine dmodel or theclutteris assumed to he availableandadap tivelyupdated to obtainan optimumestimate of the clutier.The updatingprocess ofthe clutter StOPlIwhen atarget is detect ed.The model ofthe clutte ris thensubtrac tedfrom the received signal.

Another type of clutt ersuppressionscheme wasprese nted by Zhangand Haykin [ll]in the seme year.Thisscheme makesuseof the ideaofnoisecancellers in which the datasamples fromtwo a.d.jacentrange cellsare used asre ferences when the datasamples fromthe range cell ofinterestis processed.Here,itis assumedthat notarget ispresent inthose twoadjacent cellsexcept clutter.Thesupp ression of

therad ar dut te ris eeh levedby aubtrac t.ingthe data of the twoadj~ellLcells from theone being processed.

Acommo n point is observedamong the method presen ted by Zh a ngan d llaykin [Illandthe ARmodellingclut tersupp ression methods.They eeaurncthatapre- deter min ed clutt ermodel can be obtainedfromthe rangecell which hasnothing butonly the clutter, andthen themodel isada plively updated. lIowev er,the ocea nclutte ris notstat ionary.The characteris ticoftheocean cluuc e may vary from one rangecellto anothe r. Itwillbediffic ult to have apred e terminedOCCAn clutter modelto begin with.'T'herefore , an ult imatesolu tionto the clu ttersup- pressio nprob lemwou ldutilize modelsfor theclutt er an dtheta rget ineach range cellsim ultaneouslyan dthen identitytheclutter by itsproper t.les. Theclutter issu p pressedby subt r acti ng effectsattributed toclutter by themodelfromthe receivedsign a l. Sincethe oceanclutter is round to have tlmc-vary lngfre q uencies , thecl ut tersu ppressionschemepropos ed inth isthesisis based on theII""ofa.

time-varyingfrequency tracki ngtechniquetotrack thefrequen ci esoftheocean clutteras wellas thoseof anyothertargets. The oceanclutt er isthen eupprnesed by re mo ving it s corre s p o nding frequen cy compo nenh fromthere ceived signal.

1.3 Scope of the thesis

Anewtechn iq ue,base d onth elise or atime-v a rying Frequencytracking method referredtoas the Hankel rank reductionmeth od , to selectively removethe ocean dutte rcompo nents fromthe radarda.taispro posedinthisthee ie. Themethod [12]Hhows that a Hankelmat r i xoftime series data cont a ining a finitenumberor nerrewbendttrne-vaey ing sinusoid,canhe app ro ximate dbya ma.trixwhoserank is eq u altothe finitenumberor the pri ncipalsin gular values give n bytheSingular ValueDecomp osition(SVD).Thefreq u e nciesof theairrnsoidsare cstim ated from

thosepr-incipa l singularvalues.Thisattributelinkedwiththe time-varyingocean dullermodel sllggests thatforanocean range cellconta iningasingletarget,the Hankelmatrixwouldcontainonlythree principalsingularvalues- onecorr espond- ingto thetarget signal andthe othertwocorresponding tothewell known"Bragg~

cluttersign3111.Thisassumptionis validfor oceanrangecellswithdimensio nsun- dcr one kilome te r.(The reis a commonpracticethat the ships usuallykeep a certain distanceaway from each otheronthe sea.}For larger range cells,onejust needs to increase the numberof thesing ula r valueswithout affectingtheanalyela orsign i ficantly Inereaslng the computation al burd en.ThereasonwhytheHankel matrix is utilized inthe methodis thatthelinearpredictionporpe r ty is foundin the stru cture of the matrix.Each entryof thematrixcan be expressed aa theslim ofaweighted linearcomb inatlon ofthe restofthe dataalong thero w ofthe matrix.

Thelin ea r predicti onis a basisforthe Ieequency es t imationofthe sinusoidal signal.

How the linearpredictionis used toestim a tethe sinusoidal freque n c y isdiscussed inCha pter 4. TheSVDis usedtodecom p osetheHa nkelmatrixintoasign a l and a noisevectorsubepece. The sig n alsubs p ace isassociatedwiththefinitenumber of the dominantsingula r values.By approximatingtheHankel matrix toa matrix definedby the signal sub space, the effect of noiseca n 00 substantiallyreduced.

By uscoftheHa nke l rankreductionmethod, the frequenciesoftheclutter signalscan be trackcd or the signal parameters of the oceanclutte rsignals can be estima ted.A processisthe ndevelopedto suppressthe cluttersign alsbyremoving theirestimated frequenc y components fromthereceived data. Theper{orm a nce of theproposed schemeis test edon both thecomputer-synt hesizedandthereal HF data which is collectedfromII;recently developed HF groundwav e radarloc ated at CapeRace,Newfcuncljand,Canada. Itshouldbepoin te dout that theabove uvaluat.ic nisan off-linetCliting.

1.4 Organiz ationofthe Thesis

Thethesisisorganized inthe following way:

• Chapler2presen t s a brief descriptionofthebackgrou lIIIofIII-' radarandthe tirne-dorn ninmod elof theocean dullersigna ls.

• Chapter3reviewssomete chniquesoftracking time-v arying Iecqucn c tesof thesinueo lds.

• Chaplt r4 isaback ground reviewofthefrequen cy cstima t ionof s.tation- arysine wavesbeca use itformsthebasisofthemet hodLodeal withnon- sta t ionary(time- va rying)sinewaves.

• Chapter5deecelbesdetailsontheproposed clutte r-suppres sionmet hodin thisthesis includi ng themethodusedtoest i mate th einstan ta.neouaFrequcn- des of theoceancluueesignalsandtheprocedurestorcrri o vc theocean clutte rfro mthere ceived sig nal.

• Chaplcrt)presen tstheresult s obtai ned byusi ng bothrhecomp uter-syntheslzcd andtherealHFrad ardat a.

• Chapter7cont ai n s conclusion s andsome recommendationsforfutu re work.

Chapter 2

Introduction to High Frequency (HF) Radar

Highfrequency(HF)radar,usingthegroundwav emodeof propagation,haa been est a.bliehcdas aremotesenairag;method ology(ortheoceanenviro nment[13 , 14,15 ].

Thehigh conducti vityofsea -water ac count s for thelow prop agation louofthe ground wa v e mode andallo w slongranges(overthehoriaon] to beachievedwith modest transmitter power. Thedetectio n<)ftargets such as sh ips,icebergs and sea- tee, and ocean environmen talmonitoringofwaves,currents andwindsareso rn e oftheapplicationsfortheHFgroundwaveradar.Mo reover, thistechnology ca n lillsome gapsin the radarcoverage presently",valla.b lewit hmicrowa ve rada rs.

Forexample,surfa ce based microwave radar islimit e dtoline -of-sight detection end cannotdetect targetsoverthehorizon.Also,due to themullipath reflection effcch,det ectionof low altitu detargetsis verydifficultwithmicrowaveradar.In the followin g sections ,otheraspectsof HF radararediscussed.The discussion incl udes:abriefrev ie w or th eoperati ngtheory ofUF radar; an overvi e wof some IIFradar systems;theoceanclutter in liP radarandthetime-domain model of oceanc1uttersigna.ls .

2 .1 Brief Review of Operating T heo ry of H F

Radar

In the bec kseat ter HFrada rsystem ,thelramlmiUerandthereceivercaneither be at the samesite (monOlltal ic )Ofsepar atrd by somesmAlldistanrn(bi!lt a Lic).Gem- erally , a vertically polarized AIItenna.isutilizedlor&diA~l"!cdromagnelicWll\'(., offrequenciesintheband ofJ.30MHz tha.tpropAgat C!llove rtbe eee ll;urfuc.Ifi\

tar getlies along the path,areflected signalwillbecapturedby the receivervia. til..

sa mepathtakenbythetra nsmittedsignal.The hackscattcrfromthetar get hi\.'1i\

fre q uencyshift proportionaltoitsradialvelocity,Suc hfrequencyshift, knownil.~

the Doppler frequency,prov idesamea nsfortArgetdiscrimin a t ionandillgivenby (",I)

wh ere11istheDo pplerfre q uency;V,isthe radial velocityoftheta rget ;and>'.

isthewavclcn~hof the tra nsmitted sigu!.ThederivationorEq .(2. 1)is sho wn in AppendixA.lnadditiontothedetectionofa targcl,thehou:h cattcrIIFrad Ar canalsoprovideinf ormation00thetar!:et ' srangefrom the time delaybetween thetransmi tted andthereceived signals.

ForanHF groundwavetedar,long-range target detection requiresmaxim u m dut.ycycleofthetransmitter wavefo r m (16).Thus,the frequency modu lated co n- tinuouswave(FMCWIwaveformis commonlyusedinifFradar.In anFMCW ra dar , a continuous frequency-swep t signa.!withIbandwidth,W,ist ransmittcd.

This sweepbandwidthdeterminesthe desiredrangeresoluti on118t::.R=c/ (2W ).

Thetarget rangeillmeasuredlUIth e iMlan t an eousfrequen cydifferencebetween thetransm ittedandreceivedwavefor ms.TheFMCWhas atoopercentdutycycle anditis ideal forthe bistaticconfigurationbut notfor themonostaticoperat ion.

On e problem.ith theFMCWwav e forminthemo mtatic operation isthedir·

ficultyin iso latingtht·recei ver fromthetransmitter.Thereceiver sufTef1from

the transmitter inducednoise.Thus,an interr u pted FMCW waveform,namely FMICW,is implementedinorde r to overcome theproble m. The FMCWwave' formis simplygated on and nITwith awell definedsequence.The sequence disables the transmi ssion whilethereceiveris on.

2.2 Overv iew of So me HFRadarSystems

In recognizingthe uniqueadvllntagesof HF radarucom pared tothemicrowave radar-, count riesliketheUnited Slates, the United Kingdom andCanad a have been doingexte nsive workon HF radar. using both groundwaveandsky wave propagatio n,for the past 30 years.In the United St ates, research Anddevelo p ment of IIF radarbeg anin the early1940s.NavalResearch La b oratory(NRL) was one of the pioneerinstitutions10designandcond uc t experiments with HP radar.

MADREwasan experimentalHF skywa ve rada r designedby NRL and was first putinto operation in 1961 [141.It hada target-detect ioncapabilityupto 4000 km.

ltsantenna had a dimensionof 98 mete rs (m)wide by 43m highandconsi st edof twentycorner reflectorelements arrangedin tworowsof tenelemen ts eac h. The radargenerallyoperatedwithan average power from 5 to 50 KW.In 1970,the Office of Naval Resear ch/SRIInternatio n a l develop edanotherUF skywaveover the horizon(OTH)radar, namely theWide Aperture Research Facility (W A RF).

for detect ingand tracking ships atrangesof 2000 km or more at sea1171.WARF employe da linear frequency-mod ulate d co ntinuo us-wave (FMCW)waveform and transmittedI MW aver age effect iverad iat ion power.Reflectionsignal!from the oceanwere receivedbya2.55km broadside arrayof vertical monopole element pairs. The saidsystern aha.vea eonaide rablyhuge physical size and highcost.

A sm al l transp or table system withabroad beamscillm ing characteristic.called CODA R(Coas t a l OceanDyna mi c Applica tions Rad....rl. was developedla ter in

the70's. The conceptsofeODAR were origi n at ed fromthe NationalOn'auk and Atmospheric Admi nstrat ion 's(NO A A) WavePropagationLaboratory forthe measu re ment of ocean surfacecurrents from thecoastorthe offshore plat forma.

In the United Kingdom , the Univeeettyol Birmingham in association with oth..r organ iz a tionshavealso donesig nificant workonIIF OTIlradar.Ane:{!'l'rillll'nlal HFgro undwa verada rwasde sig nedto transm i tIK\Vpeak powerIrc qucncy- mod ula tedinterruptedcontinuo us-wave (FMICW) signa l vie a6-30 I\IlIzvervirnlly polarizedloga rith micperiodicdipole array.Ther~eivingantermaco nsi~t ~of two nested brcadstde arrays with15vortlcalIoopclementseach [18J.The sys te mha.'1 been us ed forre mote se nsingof oceanwevcsandcurrents.

In Canada,theCO DA Rs ysle m hasbeenused<Illremotesensorforthe northern oceans whereexploration andtra nsport a.t ionactt vltieslake plan'[191.keh ergsarc consid eredasasignifica nt haza rd to th eships andcons t ructionsuch as offshore platfo r msin those areas. Theabilityoflongra nge(ove rthe hori7.on ),J.!trd ioll demo ns t rated by HFradar wouldreducethech a nceof possible collisione between the icebergsand the shipsorany othe rconst ructions.'1tsea.

Recently,an F'MICW groundwave radarsystemhaa heenbuilt at Cape!tace.

Newfo u ndland,by Nor t hern Radar Syste msLtd.in aesociafiou with theCe ntre for ColdOceanResourceEngineering (C-C O RE), MemorialUniversityof Newfound- land,for thepur poseof oceansur veillanceincludingthe detectionandtho tracking ofvesse ls andicebergs,plusthemeasurementofsea-sla t eandcurrents/131.The radar operate s at acentrefreq ue ncyof 6.75MHz alongwithasweep bandwidthof 375kHz andisdesigned to detectand trac kshipsat adletance upto100 km, with II.range resolution of400m.The trans m itterand thereceiverarc locat ed ILtthe samesi te (amono:tatic configuration).Thetransmit antenna ill anoff-the-shelf log periodicarraywithanavera ge transmitpoweror 2,.')KW and coversovera.

120 degreesec to rof theocean with an averagenominalbeam width of 3.5degree.

10

The receiving antennacons ists of an array of 40 quarter wavelengthheightbroad- bandelements which are equallyspaceda distanceof half wavelength.The to t a l distencetakenby the receivingantennais about 880 m. The cluttersuppression scheme proposedin thistheais will betest ed usingthe data collectedfrom this rad ar.

2.3 Ocean Clut te r in HF Radar

InflF radar,the backscatter from theocean surface,namelythe ocean clutter, appeanl at welldefinedfrequen ciesshiftedfro m the t.ransmit.tedfrequency ofthe radar.Crombie(I]firstobserved th iseffec t. andattributedthe dominan t cern- ponent of the returnenergytothe back-scatteringFromthe oceanwaves ha.v i n g a wavelengthhalftheradar wavelength.Twosetsofsuch ocean waves,moving radiallyto and away(rom the rada.r elte,behaveas diffra.ction grating s and ca uee co nst ruct i ve interferenceofthe~catteri n greturns. Asshow n in Fig. 2.1, the se

"Bragg"scatte ringret urns exhibittwodistin ct Dopplerfrequencieseoeeespcndfng 10the eharncterlat !c velocity ofpropagationofthe twosets ofocean ...aves.These frequenciesare givenby

(2.2) whereI""arethe Doppler frequenciesof the ocean clutter signals;I~is the radar carrierfre quency;9 ia the accelerationduetothe gravityandc is thespeed of lig ht.Equation(2.2) is deelvedfrom Eq. (2.1) withthe velocityofthe ocean wa ve,v=(gAo/2:r)1/1,whe rethe ocean wavelength,A.,equalsha lf therad a r wavelength,A..l.c. A.

=

A./2.TheDopplerspec t rum o£the oceanclutterin Fig.2.1is observedby an HF groundwaveradar operating at a frequencyof25.4 MHz.In additiontothe dominant "Bragg" scatteringreturns , asmalle r andmorc com plicated featureofthe HFrada rspectr u m isreferredlo as the"second-order"

11

sca tt ering. Asthe nameimplies,this port io nofthe Dopple rapccrrum hM been mo de lled asdouble scatteringfromtwo ocean wave components, whichmatchhalf the radarwavelength afte rvectoradditionaiong thereder beam direct ion [20.21].

Also, partof this sec ond-or de rcernesfromasin glescattcr fromsecond-o rde rore,'ln WAves prod u ced by nonlin"!a r wave-w a ve inte ract ion.

The Dop plerspectrumorthe ocean cluttercan qualitat ivelyshowsome prop.

ert ies oftheocean waves.Ifthewindblows towards the shore,the dominantpe ak atthe plu ssideof theDoppler spect rumwillexhib ita noticeable higherma g n i- tu de levelthan theone at themin usside.Dppcait e scenariowillbe seen ift.Iu- windblowsaway fromtheCO&'lt.Ingeneral,the direction ofthe windcan bede- ducedfrom thera tio ofthe amplitudeof the dominant pea ks .As the!Ip«d of the windincreaaee, theenergyin the oceanwaveapoetr-um increases.The- peak inthe spectrum movesto the lowerfrequencies, th e n theamplit ud e ofthe second-order Dopplerspectrumincreases which results in adose frequency separationbetween thefir~t-andthe second-o rde r~cattering(15 ].

Barrick [3,4] first derivedtheoreticalexp ressions for thefirst-andtheseco n d - order HF scatl erin gfromthe ocean surface using Rice's122]perturbationtech- nique which hubee n used to stud y theproblems ofscattering fromrandomand sligh tlyroughsurface .The ocean su rface ismodelled asII.th reedlmenslcne l ran- dom eurfecegover nedby aFourie r aeries expansionovertime aswe llas spACe.

Th esurfaceFourier coefficientsaretreated as rando m variabl es[31.Average first - an dsecond-order beckseae t ceed Dop p lerspectraco u ldthen bederivedfromEar - rick 's theoreticalmodelof HFsea.echo.Cromb ie'sexperimentalobserva tionWAS con firmedbyBarri ck t~theoreticalmodel for thefirst-orderscat tering.

Anothertheore ticalenalyeieofliPlICatteri ngfroman ocean surfaceW3.11carr ied ou tbySrivastava.[51u~i ngan alternateapp roach based onWal~b '8[21]gener al formulationforthe scatteri ngfrom atimeinv ariantrough euefece des cr ibed by a

12

setof twovectorintegralequat ions ina two-dimensionalspati al Fourier transform domain. Srivastavahasalso derived averagefirst- andsecond-orderbeckeceu ered Doppler spectraof the ocean clutter.BothBanick's andSrivastava's derivat ions had thesameresult in the first-orderscatte ring hut differedin the second-order wherethefirstautho r'sresult cont ained only one termwhilethe latter contained three.The firstterm,interpretedas theoccurrence ofa doubleocean wavein- teractionwiththe radar wave,is almosr the same in bot h cases [5J. The two additionalsecond-orderterms whichare significantforiceberg detectionwhen the radarisused on theship orplatform may beneglectedif theradarislocated on thecoast119). Both findingsestablis hedafundamentalmodelof ocean clutte r and the modelofthe second-orderscatteringenablesinformat ionon theocean, such as waveheight , to be extrac ted.

2.3.1 'I'ime-DomainModel for OceanClutter Signals Knowing the averagefirst-andsecond-order Doppler spectraof the oceanclutter may not be sufficientto distinguishtheoceanclutterfrom thepotent ialtargets.

This isbecause Doppler frequencydiscrimination maybecome ineffective when the targetexhibitsaDopplerfrequency close tothe~B ragg"frequencies. Also,target detectionisacontin uous processwhere itis preferable toknowtheinstan taneous behaviour ofthe ocean clutter.Therefore, thetime domain charact erist ic ofthe ocean clutter would possiblyprovidea meanswhichcould be usedto separ ate theoceanclutte rfromthe targets.Thespect rumof Fig. 2.1 suggeststhat fairly broadban d processes are responsible for theoceandutter. However ,Khan (7) has shown that the ocean clut tercanbe simply modelled by twonarrowband signals, withtime-varying frequencies, centered abutthe two"Bragg"frequencies described in Eq.(2.2).Thetimevariationofthetwo narrowbandclutt er signals, as shownin Fig. 2.2,can beinterpretedas twoindependent angle modulated

13

components.Iti,noticedthat the in,tantaneou' IreqoencieeoftheeluueesignAllI fluctuatearoundthe centrefrequencies of ±O.51HzASobtained fromEq.(2.2).

Itis demonst r...tedthat thespectrum ofsuch an angle modulatedsignAbAgr""

closely with thespectrum of HF rad..udata and accounts forthech.ua.cteri~ti"

corTdpondingtoboth thewellknownfirst· and second-ordersCAttering pt"aks, Thus,pre-proceu ingo~ration,on HF rM&!'dat&,beforetarget detection, ra.n simplybespecifiedas the estimatio n of theparametersorthe two narro wband

"Bragg"signal,centeredAbout theiraverageIrequcnciee given byEq,(2.2).Ttw clutter signals can be subsequentlysuppressedso lhl\tthecapability todetecta targetisenhanced.

14

"Or---,

F

I~ "\-,-.,,-,---.--;:,---,:----::----:---;;---! A . .

oomalUQUllCYlIZ

FiSUre2.1:TypiealHF radar powerspectrum .bowillltllIIa.,.den.\ic.oftint- (F)andsecond·order(S)Icatteri ns peak. (FromRef.!7])

01,---,

~ 0.1

i °

i

^-c.,

....

..

^.11»

- ^..

^{1»1.1.1_ •}

Fiaure 2.2:Time-varyin«behaviorof \betwo~awrow'b.Mfrequency componemeinOFradaroceanclutter(From~.(7)

15

Chapter 3

Some Time-varying Frequency Tracking Techniques

Itbasbeen showninChapter2 that the ocean clutte rhas atime-varyingclll\f/U::- teristicandcanbe modelledas two narrowbandfrequency-modulatedlIinUllOi(!,J signals which canbe trackedbytime-varying signal processingtechniques.The proposed clutt ersuppressionschemeinthisthesisis beeedon theuseofatime- varyingfrequencytrackingtechnique.Thus,inthefollowing !leCtio ns,At('nMal reviewofsome time-varyingIreqoeecy trackingmelh odtillpresented .One of them is selectedtobethebasis of theclut te r sup pressionscheme presentedinthi, thesis.

Thedecisionismade based on theanalyti cal comparisonofthemethod,rat her thantheactual experi ment al results.

3.1 Adaptive L inea r Pred ict ion Filter

In1975, Griffiths{23]presentedan adaptivetechniqueto trackthe instanla.nCOllS frequencies ofa signalwhichhasanarrow-band,time-varyingspectrum. The method makesuse ofa linearprediction-errorfilter(PEF)derivedfrom theprop- ertyoflinearprediction.Itisused toestimatethe instantancolls frequenciesof the signal by computingthepower spectrumfromthefiltercoefficientswhich are updated conti nuouslybythe LeastMean Square (LMS)gradient&dapl l.tionII.Igo-

16

rithmwhen anew datasampleis obtained .

In theprocess oflinear prediction-errorfiltering,apredicted value,i(k),of the inputat time kis givenasa linear combinatio nof the previousvalues.

L

irk)~~g;x(k-i) (3.1) where giarctheLth orderpredictionfilter coefficients.Anoutputerrorsequence, t(k),of the filteris producedby subtract ing thepredicted valuefromtheactual input.

,(k)~x(k) - ilk) (3.2)

Aminimum mean-squareerror,E(t'(k )J,is produced by a set of optimum filter coefficients,

gi

,9i,'"

,gi.

The powerspectrum of thePEF for a stationary process is given by

S.(w)=

:(" ~k)Jm;".,

^(3.3) 11 Ei...^19jexp()W;)j

Fora perfectlypredictableinputsignal,S,,(w)in Eq.(3.3) willbe equalto 0 for w:f:.Wiand% whenw=Wi.Amodified powerspectrumofPEFis definedby Griflilhstolocate thefrequencyof narrowbandinputsignals,

(3.4) Q,,(w) andS,,(w) differ by a numeratorscalefactor. The advanta ge of having Q,,(w)overS,,(w)istoreplace the% indeterminacy inherent withnarrowband spectra by thecomputat ionally tract able limit ofI/O.Ifthesignal is time-varying, the instantaneouspowerspectr umestimatewill be givenby

Q,,(w,k)= L_l • 1 .

11- Ei",Ogj(k)ex p[-)w('+II]l' (3.5) The PEF coefficientsaredetermineddirectlyfrom the data sampleby the followingrelationship that is derived fromthe LMSalgorithm,

G(k

+

I)~G(k)

+

p(x(k)X(k -1)-X(k_1)X T(k- !)G(k)1 (3.6) 17

(3.7)

(3.8) wherepis a scalarproportion alityconstantwhich regulatestheiterat ion step size, and

G(k)~ [::lZl ]

gdk)

X(k - l ) =

[~::~;l ]

z(k- L)

The above technique is basically aparametric estimationmethodinwhichthe signalparameters,thepredicti on filtercoefficients,MCestimatedinarantanccuely and thenusedto computethe instantaneous power spectrum. Uulikeothertech- niques,the data samples are directlyused in thistechnique to calculatethe filter coefficientsandno autocorrelatio n function isinvolved.

18

3.2 Eigenstructure Upda ting Method

In1988, DeGroat andRober ts (24) presentedAnothertechniqu etotradetime- va.ryint;Irequencleaofnarro w-bend,i~als.huedon the useofweiSht.cd linear predictionalongwithrank-oneupdatingoftheeigenvaluedecompos ition(EVD) of anestimated data covariancemat rix. Theideabehind thismethodisthat the eigen valuedecom pceitio n ofAllestimatedcovari ance matrixisto ident ify signal and noisesubspacesinthedata vectorapace, Eachsubspace is characterized byasetofeigenvector leigenvaluepain. In thecue of highSNR,the signal subspaceis distinguishedbythe lar ger eigenvaluesas comparedtothose associated withthenoisesubspace. However,asthe SNR decreases,thesignalsubspace becomes pert urbed.The eigenvalues ofthe signalsubspace do notappell.!' tobe quitedistinguishablefrom thenoise subspace.Asignalsubspace pee-processing i!int rod uced.The noiseinthe slgneli!ident ified endsup pre ssed byzero ingita corresponding eigenvelueebefore thetracking of the freq uenciesis done. Tbis pre-p recess ingtechnique analsoreduce thecomp utational loa.d by mean s of ran k reduct ion ofthecovar iancematrix.The meth od byDeGroatandRobertsi!briefty diK11!JSCdbelow.

1£the signalisBtationary. thecovariance ma.trixof a da.tasequence.r(n )which iscomposed of r sinusoidalcompon ents pluswhite noise, based on an L-by·L Toe plit zmatri x,canbewritt en as

(3.9)

where e,

=

[z (i).r(i +l)·· ..r(i+L- 1)}TendTdenotesthetrans poseofa.mat rix.

Inthecase ofa non-stationaryprocess,a time-varying estim ateof thekth covariancemat rixi!givenas

fu=(1-0)~O·-;.ri.rr

.

19

(3.10)

where ais a memory factor,0 5 a:51,and itisused to de-emphasizeolddata ,\11 new data is received.The estimatedcovariancematrixisrecursivelyupdated1\.'

(3.11)

with the recursioninitializedallR₁=%'IZ[ .The EVD isused to reducethc rank ofIlkand to sup press noiseallwell.EVDofR~is givenby

(:1.121 whereD

=

diag[Ah.\~ "' "

hl

containsthe eigenvaluesofR~intheorderof AI;::A2 ';::: "',h;andQisanorthogonalmatrix containing thecorresponding eigenvectors. However, anewR~ca n beapproximatedby set tingthe clgcovalues A.+1 through AT,to zeroinD.Thisnoisesuppression processis cloneeach time when thecovariancematrixis updated.

With each updatedcovariance matrix, theinstant aneousfrequen cies arceati- matedusingweightedlinear prediction (LP) filter .The weightedLP equationsat the kthupdatein the matrix formis given below.

(3.13)

(3.14)

where

gf

illa l-by-Lcolum nvectorof LPcoefficients;_R~illthe kth estimatedco- variancematrix;

rf

^illtheestimated covariancevector ;and_R~is the peeudo-lnversc ofR~.Thcinstantaneousfrequencyis then obtained(romthe angularlocet lons of the zeros(on the unitcircleon the zplane) ofthe denominat orpolynomialinthe transferfunctionof the LP filter,thatisIormthe rootsof

L G(z)=I

+

~g~(i)Z-i

=

0

20

(3.1')

The eigenstruct ureupdatin~met hodi,ab o&p&fametric eatimationtechnique

with&,i~n~pee-prceeeieg whichanreduceacertain levelofnoise.The inslan·

taneou, frequenciesof the.inusoid,areestimated from theLPcoefficients.

21

3.3 Hankel Rank Redu ct ion Method

DiMonte and Arun[121 presentedanother techniqueto tracktheinstant aneous frequenciesofsuperi mposed harmonics,referred toasthe Hankelrank reduction method .The technique utilizesthe property that a Hankelmat rix(data matrix) const ruct ed directly from a timeseries data containing a finitenumber of sinusoid, canbeapproximated by theSingularValue Decomposition(SVD )to a matrix of ran k equal to thenumber of principalsingular values, even whenthefrequencies ofthesinusoidsareslowly varyingwit htime;andthen the instantaneous frequen- cies ofthe sinusoidscanbeestimated fromthose principal singularvaluCll.The frequency estima tion isassociatedwiththeproperty oflinear predictionwhich is observedinthechar acteristic ofthe Hankelmatrix.Thedatain the matrixcan sim plybe expressed asthe sumof a weightedlinear combinationoftheothe r data along the row ofthe ma trix.The weighted coefficients are usedto esume te the signal'sfrequency.The discussionon how the signal'sfreque ncyis estimat ed from theweight edcoefficients is detailed in Chapter4.TheSVD playshereasimilar roleas the EVD.Itdecomposes the Hankelmatrixintoa signal andanoise vector subspace. Then theHankelmatrixis approximatedby the signalsubspacede- fined by the dominantsingularvalues. This met hodisimplementedintheclutter suppressionscheme proposedin thisthesis afterthe following considerat ion.(The derivationof the method isdescribedinChapter5).

Theeigenstruct ureupdating methodbyDeGroat and Robert sworks quiteelm- ilarlytothe adaptivelinear predictionmethod by Griffiths -both ofthem using linearpredict ion-error filteringasthe basistoest imat e theeinuscida' Irequenciea, except that in thelatterthe eigenstructureupdat ingmethodinvolves a signal pre-proeesaingwhich identifies thesignal and noiseeubapaces before theactual frequency est imationis performed.Theadvantageof having the pre-processing

22

isthatcert ainlevel of noisesuppression canbe achieved.Therefore,the eigee- struct ureupdat ing meth od will perform betterthanthe adaptivelinear prediction method inthecase of lowsignal-to-noiseratio (SNR).

Boththe eigenst ruct ureupdating met hod and the Hankelrank reduction method haveasignalpre-processingstep tosuppressthe effectofnoise in a differentfash- ion. TheSVD inthe lattermethoddecomposestheHankel matrixof ent ire data record into the signalandthe noise subspa.cesatonetime.That gives betternoise supprcssionthanthe way ofhavingmult ipleEVO'softhelocalizedcovariancerna- trices which areupdated atevery instantthrough[12].The methodologyseems to berobustif theprocess involvesonesingle SVO rather thanmanyEVO's. Also, it is mere computationallyadvantageousto have asingle SVD operationthanmulti- pie EVO's.Wit h regard to simplicity,the Hankelrankredu ctionmet hodinvolves less mathematicalsteps. Finally,the selection01appropriatememory fact or, a, inthe eigenstructureupdating methodimposes a trade-off between temporalres- olutionand noisesuppression[12}.Therefore,theHankelrank reductionmethod is selectedtobeusedinthe proposed cluttersuppressionscheme to estimatethe instan t aneousfrequencies of the ocean clutt ersignals from which thecluttercorn- ponents canbeidentifiedand suppressed.

23

Chapter 4

Frequency Estimation of Stationary Sine Waves

Based on the inherent time-varyi ng charac te risticofthe oceanclutterinIW radar.

theHankelrank reductionmethod whichis a lime-varying signalprocessing tech- nique,isutilizedinthe cluttereuppreaeionscheme proposedillthis thesisto track the clutter signal's frequenciesand thentosuppressthem. Before disclls!!ingthe time-varyingfrequency tracking technique ,itis wort hwhile to reviewtheback- ground of the frequencyestimatio nofthe stationary sinewaves sinceit forme the basisforthemethodto deal withthe non-sta tionar y (time-varying)prcccaa,In thischapter, the main focusisontheprocedu remethodology forest imat ingthe freq uenciesor the sine wavesbyparametricmod elling.Inolher word s,iti!lpO!l- aible tofita modeltotheprocess andthen to deter minethe parametersofthe modelCramwhich thesinusoidal frequencies can be obtained[25J. The methodis referredtoasthe autoregressive(AR)or linear predictionmodelling.In additio n to the common transfer-functionrepresent ati onapproach usedCorthe model-based spect ral estimationmethod, the slate-variable represenlationapproach isalso de- scribed here as it was usedintheHankel rankreduct ion method.It is round that both approachesyield thesameresult.The estimationorthesinusoidal frequencies in both noiselessand noisycaseswillbe discussed.

2'

4.1 Noisel e ss Case

A sampledsignal,1/(1-),composed ofMcomplex sinusoid, withno noisepresent isrepresented by the following equat ion,

"

Y(k)=?; c;expUw;k), k=1,2, ···,N (4.1) wherec;is the amplitude ;Wiisthe angula.r frequencyofthe itb complex sinusoid andNis the numberof data.samples. The angular frequenciesof thecom plex slnusoida areaeeumed tobeinvariantwith timeand they can be estim at ed by theautoregressive (AR)process (modelling) or linear pred ict ion methodt'J.which y(k}is givenbya linear combinat ion ofit!pastvalues andanadditivewhitenoise signal,w(k ),withzero mean .

£

,(k)=-~a;,(k- i)

+

w(k) (4.2) whereai,i=1,2 , ···.L,are knownas the ARcoefficientsandLis theorder of theAR process.Thatmeans thesignal,y(k ),ismodelled as theout put of an AR processwhose input isa whitenoisesource. Thea-dom ai ntrans fer functionof the AR process is

where

H(, )=A;,)

£ A(,)= l +~a;.-;

;=1

(4.3)

(4.4) The AR proceeeiscloselyrelated to thelinearprediction-errorfiltering when oper- ating onthe frequencyestima tionof thest atio nary sinewaves.Their relationship isdepicted inFig.4.1.The prediction-err orfilter is an all-zerofilterwitban im- pulse response of a finitedurationwhereaatheinversefilter in theAR modelisan all-pole filterwith anim pulse responseofaninfinite duration[261.Thezeros of the transferfunctionofthe predicricn-errcrfilt erare locatedat exactly the same

25

WHITE

NOISE~

^{l/ ' C%1}

~>

^YCk)

Cal

A(Z)

YCk ) - 1 L -

...J~>

^{PREDICTION ERROR}

Cbl

Figure 4.1:

C a'

an AR preeeee,(b) aliDurpredictioD~rrotFilla'

po!IitioDI(inside theunit circle onthe:plane)ASthepcleeofthetranaferIunc- tiou ofthe inverse filter of the ARmodel.This&Sauresthesta.bility of the fillen because intbe s domain(.t

=

a

+

jw),H(! )is consideredstab leifthe polesare locat ed inthe lert handsideofthe! planea.nd,for z

=

C',the lefthan.9planeis mapped int o theunit circle.H{..)isconsider ed marginallystableifthepoles a.re on theimaginary axis of the.splanewhere~=O.

Itwaa fintshowQ intheProoy'smethod described by Hilderbrand127)that in a ooiseleucondition,the &O.SUlarIrequeccieeofthe complex .inulOid.C&O.be obla.iocdfromthe a.ngularlocationsoftherootsofanMth orderpoIyoomiaJ A(z) on the unit circle00the zplane.Therootslying on the unitcircle are mapped ontotheimaginary axis of the3plane ("=jw) where the system ism&r~naJly sta ble. However , inthepracticalsit uatio n,Lcan be greaterthanor f!qualto M.Tuft,and Kumares4l1128,29]have demonst ratedthatunderthe inequality, M~L:S(N-M/ 2),thepolynomialA(.r)h...Mrccteon theunit circleonthe .rplue",itb their&DSUlulocation.s COJ'TeIlpondinstotheansu1ar frequencis of theMsinullOidsin the siy1al, and the(L-M)extraneow roobUf'uniformly

26

distributed &loog thecircumference imidetheunit circle.Thepropertythat the rootsofthe polynm ni&lA(z )on theunitcircleHzl

=

I)determine the~gula.r frequenciesofthe"ir.usoid..canbesh ownbythefollowiDS obse rvations.

Conside rasy"it emorpredictio n equation sin matri xform usedto determine thecoefficients,·ji.i::::1,2•...•L

[

9(L) .(L+1) .(L + 1) .(L)

•(N·-I) .(N ·-2)

.(1)

^.(2)

_. ] [G ^G, ' ] ['(

^.(L+2)

£+1 ) ]

. :::: - .

. . .

.(N - L)

G ,

.(N)

(4.')

}ja::::6} (4.6)

Rea.rrangeEq.(4.6)

V;

=

(blIY/ J (4.7)

and

a'::::!I.a" llJ,.. .•aLf (4.8)

Therelcre,

}ja'::::0 (4.9)

Yiisa(N-L)-by·(L + I )Toeplit z matrix formed fromaforwar dlinear prediction, i.e.y(k)=-Et':.1Giy(k-i ). i:::: 1.2. ·. ·, M, andithasarankorM.(Therank ofa matrixis definediI3thenumberolrowsorcolumnswhich nrclin early independent.) Toseethis.an M-by-Isinusoidal columnvectorisdefinedasfollows:

/;=[I,e-^j...'.e-¹iw;•• • •,e-^Lj...;jT, i=1,2," ',M (4.10)

Itisobserved thatany rowofYicanbewrittenas a linear com binat ion ortheAI independen tvectors in_f"~Thus,therank ofYiisMulongas Y;has at leastM ro ws.Thenull spaceolYihas a dimensionofL

+

^1-M.Sincea'lies in the null

27

spaceor

v ;

^thein ner product orIiand a'is zero{291.Tha t is,

and thisequation isrecognizedallthe transrerfuncti on of theprediction-error filter andevaluatedontheunit circle at%=eJ"",i

=

1.2•• . • •M.Then thepolynomial A{%}has roots.on the unitcircle,thatdetermi ne thesinu90idalIreqaenciee[28).

Not onlycan thesignal.lI{k).bepred icted from forward linearpredictio n.but itcan alsobepred ictedfrombackwardlinearpred iction.A sysremofpredict ion equationsfrombeckwsrd line arpredictionisobtainedtodeterminetile codlici enb, oi,i

=

1,2"",I, .asbelow:

[ ,"(2) ,'(3)

,'(N':L+I) ,'(3) ,'(4)

y'(N-L + 2) ...

y'(L+I)] [ _, ]

[ "(I)}

,'(1.+2) _, __ y'(2) •

. . _ . (.121

. . .

,'(N ) -L ,"(N- I.)

y.a ="

^(4.13)

RearrangeEq.(4.12)to

V;a ' =

0 (<.14)

where

V ;

~1b,IYoJ ^(4.15)

v;isa(N - L)·by·(L

+

¹⁾Han kel matrixinwhichtheNdatasam ples,y(k),arc complex conjugate dbecausey(k)is complexvalued.(tdenotescomplexconju- gate) .V;alsohasa rank ofMwhich can be seenfromthe previousobservation inYj.Sim ilarly,thepolynom ialA(z)hasroots ofeiw', i=1.2,·.· .M.

Transfer -functi onrepresen t at ionhallbeenwidelyused in mod el-based spcc.traJ estimationmethods.Thisapproachasused abovedemon strated thattheparam- eters or coefficientsof&IlARmodel couldbeut ilized todeter mi ne thesinusoidal

frequencies.The st&t.e vuiahlerepresentation howeveris AD alternativeapproach to analyzea linear system[30}.It willbeshown thatthis approach canalsobe used to describetheARmodel, andthenthe ARparametersare determinedto givethesinu50idalfrequencies.In general,a discrete-t ime system isrepresented byasetofstatevariableequations:

z(H I)~Az(k)

+

B, (k) V(k)~Cz(l)

+

D, (k)

('.16) ('.17) where.:r(k) isthe state vectordescribing thesystematthe Hh instantiy(l')isthe out putvector;v(k)is the input vector;A, B, CandDarethematrices determined fromthe constantsof thesystem[311.Considera secondorder AR process,

V(k )~-a,v(k-1) -a,.(k -2)

+

w(k ) (4.18) Definethe firststat evariable

.:r2(l' )

=

-oly(k - 1)-42y(k-2) Combine Eq.(4.18) andEq. (4.19)toget

FromEq.(4.19},increuingl' by 1 gives

Selecta seco-ic 'tate varia ble

Subst ituteEq.(4.22) intoEq.(4.21),itcan bewrittenIIlI

(4.19)

('.20)

('.21)

('.22)

z,(k)-a,v(k )

;tl(l')-01(%2(1-)

+

w(k))

%1(1')-01%2(1') -Glw(1') (4.23)

29

FromEq.(4.22), increasingkby 1 givee

z,(k+1) -a2y(k) -.,(r ,(k)

+

w{k»

(·" 2') Equation(4.23)andEq. (4.24) maybe combinedin matrixformas

FromEq.(4.20)andEq. (4.25),theAR processisnowrepresentedbythe following stale variableequations:

where

r(k+I)=Ar(k)+Bw(k)

,(k)=Cr( k)+Ow{k )

[0 -. , ] [-.,]

A= I _" .B = _', .C= IO.II.O =[lj

(1.2G) (1.27)

(•.28)

Thederiva.tionof asimple second order ARprocessintermsofstalevariable representationcanalsobeextendedtoa generalLorderARprocess wherethe matricesA,B,CAndD Arcas follows.

.. .

0

-" -"

0 -aL_1 -aL_1

0 -aL_2 -a',_2

A= .B=

-" -.,

-" ^-"

and

C=[0 00

...

0

I I·

^D=IIJ 30

(1.29)

('.30)

whereA,8,andCare the const a nt matriceswith the size ofL-by-L,L·by-land I-by-L,respectively.

Considerthe sampledsignal,y(k),in Eq. (4.1)again. Withoutthepresenceof noise,thesignal is exactlypredictableas alinear combinationofits past samples andthe predictionerroris zero. Consequently,y(k) can he adequately modelled as the output ofan ARprocessof theorderL

=

Mwithzero input,i.e.w(k)

=

0, orsim ply as theoutputofanoscillator[3D]

y(k)

~-t.';Y(k-i)

^(4.31)

In the state variablerepresentation , thesignal,y(k),isdescribedbythefoll owing setof equations:

x(k

+

1)~F'(k) y(k)~h'(k)

(4.32) (4.33) wherex(k)isanM-by-lstate vector.F and h aretheM·by·Mand l·by·Mstate matricesin the form ofAandCgiven inEq. (4.29) andEq, (4.30), respectively.

Itwillbeshownthat the eigenvaluesofFare exactly the same asthe roots ofthe MorderpolynomialA(z).(Theeigenvalues and theeigenvectorsofa matrixare discussed in AppendixB).Theanglesofthe eigenvalueswillgivethe frequencies ofthe sinusoids.IfFcan be estimated, thenthesinusoidal frequencieswillbe determi nedfromthe ang lesof theeigenvaluesofF.

Nowconsideran(N-L+1)·by· LHankelmat rixconstructeddirectlyfrom the sampledsignal,y(k),whereL>M and N

»

L.

[ ,(I) y(2) H

~

^{,(N-:L +I)}

y(2) y(3)

yiN-Lt 2)

31

y(L) ] ,(H I)

y(N)

(4.34)

Inter msoletate verlablee,Hca n bewritteninthefollowing form usingEq.(.1.:12) and Eq.(4.33).

H

=

hx(l) hx(2) hx(3)

hx(2) hx(3) hx(4)

"x(l,) hx(L+1) hr(L+2)

h(N-L+ l ) h:t(N -L+2) hx( l) hx(2) hFx(l) hFx(2) hF2x( l) hF2X(2)

Hcanbefactorizedas

hx(N) hx(L) hFx(L ) hF2x(L)

(01.35)

H = h hF

hF' [x(I ), x(2),x(3), .. .•x(L )

I

(-1.:161

Thematrix

a

is known asthe ohscrvability matrix andXis thestatevector matrix.Itis noticed that mat rix

a

^hasonlyMcolumns lindXhasonlyMrows.

Itmeans that therank ofHcannotbe greater thanM.Ifthere arcMdistinct sinusoidal frequencies,JJwill havea rankofMeventhoughthe sizeof1/ isgreater thanM-by·M .

Fcanbedeterminedfrom

a.

^If

a

is partitioned into twomatrices,8₁having rows from thefirst one to thesecond lastand81havingrows fromthe second one tothe lastone,then a relationsh ipbetweenF,61and 81willbe observed.That

(4.31)

32

where

h hF hF'

hF hF'

hF' (4.38)

Since Eq.(4.37)is overdete rmined,Fcanbesolvedbythe leastsquaremethodas

(4.39)

wherea~=(9[el)-ler. a[isthetranspose ofe1and(8 [ 61)- 1isthe inverse of(e [ e .). The eigenvaluesofFare determinedby(see Append ix8forthe comput a tion of the eigenvaluesof a matrix )

-A 0 0 -a M

1 -A 0 -aM_I

0 1 -A -a M _2

del =0 (4.40)

... 1-..\

-a,

o

¹ -al-"\

where A is ascalar parameter.The resultantequatio nof Eq,(4.40) is simplified in the following form.

(4.41)

Thisequation is calledthecharacteristic equation ofthe matrixF,and itsroots are theeigenvaluesof matrixF.Recallthe characteristicpolynomialin the transfer function oftheAiorder AR process,

M A(z)=1+~alz-;=0 A(z )

=

0can be rewrittenin the followingform:

33

(4.42)

(4.43)

Withrefe rencetoEq.(4.41)andEq.(4. 43).itill.hownthattheeigenvaluC!'Sof Fareexa.ctlythesameas therootsofthepoIynomi.JA(z ).Hence,theangul~

frequencies ofthecomple xsinusoidscanbedeterminedfrom thea.ng~ofthe eigenvalu esofFwhichareiathefonn of(eM.ei'-'>,...•ei..."')ontheunitcircle onthe z plane.

4.2 Noisy Case

It hasbeen showninthenoiseless casethat the Hankel matrixHexhi b itsalow rankproperly, and therankis equal tothe number ofcomplexeinusoidein the signal. Howev er, in the practical situat ion,thesignalis alwayscorruptedby noise.

Therefore,thelow rankprope rlyofHwill no longerhold.Infad,Hlendsto havea fullrank.

TheSingu larValue Decompoeition(SVD)baabeenrecogni zed asanumerical tool fordisplayingclose nesstolowrankora matrix(30) (seeApp endixB),andits struc t ureandnumericaldetailcanbeut ilizedtosuppres s noise as well.The SVD of anyrecta n gularmatrix, suchas the HankelmatrixHinEq. (4.34),takes the follow ingfo rrn,

11

=

USV

,

^T

~.,.;u; v! (4.44) whereSis a£· by-Ldiagonal matrix.Its diagonaleleme nts, kno wnasthesingular valuesofH,arearran gedasO"t~11,~...~O"M~O'M +t?:.••?:^f1[,~0;Ujand Piarethecorrespondi ngleft andright singularvectors which are tbeeig envecto rs or thematricesHHT andHT H, respectively. Simila rly,thesingular val uesare the square roo ts oftheeigenva luesof the ma t r icesHHT orHTH(seeAppendix Bfor thedefinition oftheSVDor a mat rix). Itcan beshownthat above a certain signal-to-noise(SNR) threshol d , a dat a matrixIIisdecomposed intotwovecto r eubepecee. One isthe signalsubspacespannedby the lert and theright singular vectorsassociated with theM largest singularvalues,andtheother isthe noise subspacespa nnedby the remaining(L - M)singula rvalues. ThusHcan be rearrangedas

H=( U,. U,) ( S' _o

₈0 ) (Yo

)T

2 Viz

35

(4.45)

where 51 isanM·by.Mdiagonal matrixcontaining theMprincipal singula rvalu('si V,andV,containthe correspondinglen andright singularvectors.52 hl\.'l the remain ing(L -M)smaller singularvalueseseociatedwiththe leftand theright singularvectors inV2and\-S.Ifthereisno noise,the(L-M)singularvalue will he zero. Thus the rank ofHisdeterminedbytheMprincipalsingularvalues.A signifi.:a nt break willbeobservedbeweenthesingularvalues,aMandCTM+I.when the SNR isabove a certain thresholdle vel.ThenHcan be app roximatedby the AIprincipalcomponentsas

RearrangeEq.(4.46)

H

z

Uls,vt

"

tr

^{r1; Ui}v[ ^(<.-t6)

(' .47)

ComparingEq. (4.47) and Eq.(4.36),theobservabilitymartix

e

canbe ident jflcd asUt S:n_{130].Usingthe}relat ionship betweenFand

e

^{derivedas in}Eq. (4,37), Fcan be determ inedandtheangular frequenciesof thecomplexsinusoidscall he estimatedfrom the angles of theeigenva luesofF.

TheSVD canbe used to decomposea Henkelmatrix (datamatrix) of1\signa l containinga finite numberofsinusoids plusnoiseintoasignal and a noisevector eubepecee providedthat the noise level is not very high.TheHa nkel matrixie thcn approximatedas detailedabove,bya red ucedorderHankelmat rixconforming to the signalsubspace defined by thefinite dominantsingularvalues.This lower rank Hankel matrixgivesestimate for the frequenciesof the sinusoidacontainedin the statemalrixF.

Aprocedure, based onthe SVD, was developedby Eckart a.nd Youngin1936 to findthe bestlower rankapproximation to a given matrix[32,331.First of all, let therank ofHbeL,andlety(M)bethe set or all(N -L+I)·by-Lmatrices

36

of rankM<L.Then for &IImatricesGiny{M),

UH -HI,;liB-Gil (4.<8)

whereif

=

U5 V^Tand

5

isobtained fromthe matri xSbyseltin~all exceptthe Atlarge:.tsin~ulA.rvalues tozero. ThematrixnormofEq.(4.48)isthe Frobe nius form,

IIH-Gil'=''''''(H-G)"(H

- GlJ

(4.<9)

where thetraceofama.trixisdefinedasthe sumor themat rix's diagonal elements andthe asterisk,., stands forcomplexconjugate transpose of a matri x.Thusif willbe thebestapproximatio n ofHifthecondit io nofEq. (4.48)is satisfied .

Thefrequenci es ofstationarysine waves can be estimat ed bytheparametric modelli n g.Thepar amet ri c modellingca.nberealizedbyboth thetransfer-function representa tion an d the state-variablerepresen tatio n.Although theestimat ionof thesinusoidalfreq uencies obtained fromthe model parametenarepresented by the tworepresent ati o nsin a differentway,i.e. the zeros

0'

thedenominatorpolyno mi&l ofthetransferfunction(or theARmodelandthe eigenvalu esorthesWema trix F,the res ultsafCthe same.Mor eover,theSVD can servetoreducetheda.taspace toa.dim ensionallysmalle rsi~na1space and the effec t ofnoise canbesubstantailly suppres sed.

37

Chapter 5

The Clutter S uppression Scheme

As men tioned inChapte r2, ocean du tterinHFradarcanbe viewedas two narrowba.nd frequency-mod ulatedsinuso id alsign als,withtheir centre Ircqc cn eiee givenbyEq.(2.2) .Based on thistime-varyingmodelofthe ocean clutt e rpresen ted in Kha n 's recen t ...rcrk 17),a clut tersupp ress ion schemeis developedtosdcctivdy suppressthecluttercomponentsfrom thereceivedradarsig n;tl. Theinstantancolls frequencies ofthecluttersigniUscanbetrackedbyatime-va ryingsignalproccs sing techniqu e,name ly Hankelrankreductionmethod,&ndtheclutte rsign~scanthe'll be iden t ified (romtheircentre frequencieswhicha1mOlltagree withtheavera,;c valuesoftheir instantaneous frequencies.The du t urIUPPTel!lion schemepropcw!d in this thesis involves twosteps.Thefirstoneisloestimatetheinsta.ntancolu frequen cies of theocea n duller3i&llals byapplyin&theHankel rankred uct ion method,andthe otheroneisto removethe3igna.lpowerAS!IOCiatedwit htheclutt er 3ignal3 fromthe timeseriesdata.50that the re3ultanldatasampleswillbeclut ter free. Thederiva tionoftheHankel rankreductionmethodand theproceduresto suppress thecluttersignalswillbedi3CUsscdinde tailinthischaplet.

38