Modulation Fm Sur Labview by Ameur1990

SimulationofaPhaseLockedLoo SimulationofaPhaseLockedLoopUsingLabVI pUsingLabVIEW EW.

T.J.Moir MasseyUniversityatAlbany InstituteofInformationandMathem InstituteofInformationandMathematicalSci aticalSciences ences Albany Auckland NewZealand Email: [email protected]

2 Abstract

A meth ethod for sim simulat ulatiing an anal analog ogue ue phas phasee-llocke ocked d loop (PLL (PLL) ) is show shownn. The The programminglanguagegivenisLabVIEW(trademarkNationalInstruments)butthe techn techniiques ques used used are qui quite gen general eral and and appl applyy equa equallly wel well to other other progr program amm ming languag languages esand and packages.LabVIEW packages.LabVIEW seems seemsanunl anunlik ikely ely candid candidate ateto to simu simulatea lateaPLL PLL as it is mo more re often often associ associated ated with with process process control control system systems s and and virtual virtual instrum instrumen entati tation on ratherthancommunicationsystems.HoweveritwillbeshownthatinfactLabVIEW givesapowerfulsolutionwhichisbotheasytoimplementandwithausergraphical interface interface comparabl comparable e to any mod modern ern programmi programming ng languag language. e. The simul simulation ation is full fully y interact interactivean iveanddemodul ddemodulatesordinaryFM atesordinaryFMli likearadioreceiver. kearadioreceiver. Keywords:

Phase-LockedLoop(PLL),Feedbacksystem Phase-LockedLoop(PLL),Feedb acksystem,LabVIEW ,LabVIEW

1.Introduction

ThePhase-LockedLoop(PLL)isoneofthemostcommonlyusedintegratedcircuits (ICs) (ICs) in use in mo moder dern n comm communic unicati ations ons system systems[1 s[1]. ]. Althou Although gh perha perhaps ps surpri surprisi sing ngly ly firstinventedasearlyas1932byBellescise[2]itnevergainpopularityuntiltheearly 1970 1970s s when when chea cheap p ICs ICs were were read readiily avai availlabl able. It qui quickl ckly foun ound appl appliicati cation on as a precisi precision on FMdem FM demodul odulator ator as a replacem replacement ent for Foster-Seely Foster-Seely discri discrimi minators. nators.Dig Digital ital commu communi nication cation system systems s quickl quickly yfol follow lowed ed and the PLL has found found applica application tionin in such area reas as mode odems, mobi obile com omm munication tionss, satel tellite receivers and tel television electronics.ThePLLisusedextensivelyinmodernelectronicsystemsbutitsdesignis often met met with with trepidation trepidation . This This is perhaps perhapsund understan erstandab dable lesi since nceto to fully fully understand understand the the operat operatiion of a PLL PLL requi requires res some some knowl knowled edge ge of comm communi unicati cation on syste system ms and and controlsystems,twosubjectswhicharetreatedinisolation.Forexampleseldomare PLLs PLLs covered covered in a taught taughtcon control trol course courseat at under undergra gradua duate te leve levelwhi lwhils lstt feedb feedback ack and and stabilityisonlybrieflymentionedwhencoveringPLLsinacommunicationcourse. The The purp purpos ose e of thi this pape paper r is to show show how a real realiisti stic PLL PLL can can be sim simulate ulated. d. The The sim simulat ulatiion uses uses Lab LabVIEW VIEW as it is easy easy to buil uild a grap graphhical cal interf terfac ace e whi which is interactiv interactive. e. Although Although such packages packages as MATLA MATLAB B and MATRIXx MATRIXx [3] could could equally equally wel well be used used the the resul result t wil will not not be as intera nteracti ctivve. For For exam exampl ple e in [3] [3] alth althou ough gh the the simu simulati lation onis is realis realisticit ticitisoft isofthe‘one-sh he‘one-shot’ ot’vari varietywherethe etywheretheprogram program mu must stbere-run bere-run toshowtheeffectofanychangesandcannotbemadetoruncontinuouslyinpseudo ‘real-tim ‘real-time’ e’li like kethe theapproac approachusedhere. husedhere.Thefin Thefinalgoalisa algoalisa single single PLL program which which canthenbeappliedtoawholerangeofpossibleapplicationssuchasdemodulationof FM, FM, Frequ Frequen ency cy Shif Shiftt Key Keying ing (FSK) (FSK) or to furthe further r investi nvestigate gate such such proble problem ms such such as multi-pathandadditivenoise(thresholding).

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2.TheBasicPLL

TheblockdiagramofagenericPLLisshowninFigure1below.

Figure1.BlockDiagramofGenericPLL

We consi consideran deran inputto inputto the PLL PLLwh whic ichis hisFM FM..Wh Whil ilstt stthe hereare reare other othertyp typesof esofin input put whichcanbeused,thisapproachgivestheopportunityofmodulatingvariousdifferent basebandsignalswhichcanbelaterusedtotestthePLL.Forexampleastepinputora sinu sinusoi soidal dal frequ frequen ency cy respons response e are comm commonly only used used for testing testing all all feedb feedback ack control control syste system ms. The The FM sign signal al whi which must ust be sim simulate ulated d is an FM modul odulated ated sin sine wave wave (al (althou though gh squa square re wav waves and and other other wav wavefor eform ms are are also also con conside sidere red) d) and and has the the analogueformf(t)where  f  ( t ) = cos[

 c

( t ) +

sin(

 m

t )]

Intheabove = ∆  /  m isdefinedtobetheusualFMmodulationindexwith c and res respecti ectivvely ely the the carr carriier and base aseband freq requenc encies in rad rad/s. The The depth of  m modulationisgivenby ∆ rad/s. InFigure1thePLLcomprisesaphasedetector(PD),avoltage-controlledoscillator (VCO)andafilter.ThePLLwilldemodulatetheFMandgiveandoutputwhichisthe ori origin ginal baseb aseban and d sign signal al.. The The PLL PLL woul would d normal ormallly oper operat ate e on the the interm termed ediiate ate frequency(IF)wavef frequency(IF)waveformofaradioreceiv ormofaradioreceiver. er. The The theory theory of the PLL is well well docum documented ented [1,4] [1,4] and and only only an outlin outline e will will be give given n here. ere. The The operat operatiion is easi easies est t to see see when when there there is no FM and and onl only a fixed xed carri carrier er frequ frequen ency cy is presen presentt (ie (ie ∆ω  = 0 ). When in lock ock the the outp output ut of the the VCO VCO wil will be a

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wave wavefform whi which is in phas phase-q e-qua uadra dratur ture e with with the the incom ncomiing wave wavefform. orm. The The phas phase e detector detector (PD) (PD) is a line linear ar multip ultipli lier er (for (for our exam exampl ple) e) and and give gives s an output output whic which h is approx approxim imatel atelyy zero with with an additi additive ve term at twice twicethe thecarr carrie ier r frequ frequen ency cy 2ω c . When When properly properly designe designed d (iethe (ie the bandwi bandwidth dthis ischose chosen napprop appropriatel riately) y)the the termat term at 2ω c willbe attenuatedbythefilter(apartfromsomeresidualleft-over)andthePLLoutputwillsit at zero. zero.Wh When en aba a baseb seban and dsi sign gnal al is presen presented(ie ted(ieFM FM) )the the VCO output outputwi will ll track the variationinphaseoftheincomingFMandthePLLoutput(theinputtotheVCO)will be the the rate rate of chan change ge of phas phase e (ie (ie instan stanta tanneous eous frequ requen ency cy). ). The The VCO VCO thus thus acts acts dyna dynam mical cally as an integ ntegrat rator or and and thi this is importan portant t when when exam examiining ning the the openopen-lloop frequencyresponse. The Thebe beauty auty ofthe PLL PLLiistha s thatt it can be anal analyysed in a line linear arfo form rm indep indepen enden dentt ofthe carrierfrequency.TheblockdiagramoftheclosedloopPLLisshowninFigure2.The VCOtransferfunctionH(s)isshownasapureintegratorandthephasedetectorasa summi summing ngju juncti nction on providi providing ngnega negativ tive e feedba feedback. ck. It remain remains s to find findthe thefi filter lterdy dynam namics ics F(s). in

+

Filter

PD

F(s)

DemodulatedSignal

 o

 K v  s

VCO H(s) Figure2.LinearPLL

Thefilterdynamicsarefoundbydrawingtheopen-loopBodeplotwhichshouldhave asimil asimilarformtotheoneshownin arformtotheoneshowninFigure3.(although Figure3.(althoughotherformsarepossibl otherformsarepossible) e)

5 dBGain -40dB/decade

f 1 0

-20dB/decade

 f 

Frequency(Hz)

f 2 -40dB/decade

Phase(degrees)

PhaseMargin Frequency(Hz) o

-180

Figure3.OpenLoopPLLBodePlot

This This typeof typeofPL PLLissome Lissometim timesknow esknownas nasa athi thirdorder rdorder typeII typeII PLLastherearetwo PLLastherearetwo integratorswithintheloop.ThefirstintegratoristheVCOandthesecondisanadded elec electro tronnic integra tegrator tor.. Sin Since two two integr tegrat ator ors s with with negati egativve feedb eedbac ack k resu resullts in an oscill oscillator, ator, apha a phase selea lead d(adv (advanc ance) e)stabi stabili lisation sation is needed. needed.Hen Hencet cetheoveral heoverallBode lBodepl plot ot has has the form form shown shown in Figur Figure e 3. This This particul particular ar desig design nis is prefe preferred rred asit as itha has s better better trackingabilitiesthanatypeIPLL.Thehigherthegainatlowfrequenciesresultsin goodtrackingandhencelowerror.(forthecontrolsystemdetailsseereference[5]) The The uni unity gain ain bandw andwiidth of the the PLL PLL shoul ould be chos chosen en high enou enough gh to trac trackk adequa adequatel telyy but but nottoo high highso so astole asto letttoomuc toomuch h 2  c throug through. h. Byexper Byexperiienceithas enceithas beenfoundthataunitygainbandwidthof 





givesgoodresults.



 f 

=

2  f c 10

6

3.LabVIEWSimulation

Thegraph The graphical ical userin user interface terface of LabVIEW LabVIEW together with with thebl the block ockdi diagram agram approach approach makesitagoodcandidateforsimulatingaPLL.Thesimulationisdividedintoseveral steps.Thegen steps.The generation eration ofFM, simu simulati latinga nga first firstorder orderli lineartime-i neartime-inv nvaria ariant ntsy system, stem, the VCO and and phase ase detec etecto tor r and and the the com composi osite des design. gn. The The spec speciificati cation on for the the simulationisasfollows: sampl samplin ing g frequ frequen ency cy 10kHz 10kHz,car ,carri rier er frequ frequen ency cy 2kHz, 2kHz, mo modul dulati ation on frequ frequen ency cy as a +/- percentageofthecarrierfrequencyuptosay+/-10%(+/-200Hz),unity-gaincrossover frequency400Hz,phase-m frequency400Hz,phase-marginaround55 arginaround55degrees. degrees. 3.1SimulationofFrequencyModulationandtheVCO

TheFMsignalgenerationiscoveredinoneofthelabVIEWexamplesandsoisoneof theeasiestofthestepstofoll theeasiestofthestepsto follow.Theblockdi ow.Theblockdiagrami agramisshowni sshowninFi nFigure4. gure4.

Figure4FMSimulationinLabVIEW

In common common with with thewhol the whole e PLL simul simulation ation theFMgenerator the FMgenerator (which (whichis is only onlya a slig slight ht modifi odifica cati tion on of a Lab LabVIEW VIEW exam exampl ple e progr program am) ) uses uses scal scalar ar quan quanti titi ties es rathe rather r than than arrays.Figure4showsasinewaveasthemodulatingsignalandasinewaveasthe carrier carrier signal signal.. Thecase statement statementenab enables les diff different erentki kinds nds ofbase of baseban band dsig signal nals snam namely ely,, square square,, trian triangl gle e and and sawtooth. sawtooth. The The vari various ous wavef waveform orm generato generators rs defaul defaultt to a vector vector outputandsotheymustbeconvertedtoscalarformbyindexingthearrayandtaking the the zero zeroth th value alue (the (the first rst sam sample) ple).. The The FM gen generat eratiion is sim similar to how FM is generatedusingtwosinusoidalgenerators.Theoutputofthebasebandsignalgenerator isfedintothe‘sweep’inputofthesecondwhichactsasthecarrierfrequency.Thesine generatorforthecarrieronlyacceptsscalarinputsforfrequencyandhencetheneed for for a scal scalar sim simulati ulation on.. Normal Normalis ised ed frequ frequen ency cy is used used through throughout out defin defined ed as actual actual frequenc frequency y(Hz)/Sa (Hz)/Samp mpli ling ngFrequ Frequency ency (Hz). Thesam The sampli plingfrequen ngfrequency cy of10kH of 10kHz z usedin thePLLsimulationwaschosenasitismorethantentimestheunitygainbandwidthof 400Hzandabouteighttimestheupperbreak-frequencyoftheopen-loopBodeplot. Thecontinuouslychangingamplitudeofthefirstsinewavegeneratorismultipliedby the the fixed xed carri carrier er freque requenc ncyy of 2kHz 2kHz/1 /10k 0kHz Hz and and thi this chan change ges s the the freque requenncy of the the secondoscillatorandproducesFM.Adcoffsetatthefirstoscillatoroutputisrequired sothatwhenthereisnobasebandsignal,unityismultipliedbythecarrierfrequencyto give give a contin continuou uous s wavef waveform orm.. The The ampl amplitu itude de of the first first oscil oscilla lator tor (baseb (baseban and d sign signal al) )

7

determin determines es the depth of mod modul ulation ation.. Ifthe If the baseban baseband d signal signal had ampl amplitud itude e unity unitythen then byoff-settingitsoutputbyunityandmultiplyingby2kHz/10kHzthesecondoscillator willsweepfrom0hzto4kHzwhichrepresents100%FMmodulation.Theamplitude ofthebasebandosci ofthebasebandoscill llator atorX100 X100 thereforerepresentspercen thereforerepresentspercentagedepthofmodul tagedepthofmodulation ation.. It is nom nomin inal ally ly setto set to 10% through throughout out whic which hrep represe resents nts a carrie carrier r frequ frequen ency cy whic which his is centredon2kHzandsweeps+/-200Hz. ItisworthmentioningtheVCOoperationinthissectionasitisnearlyidenticaltothe aboveFMgeneration.Figure5showstheLAbVIEWblockdiagramoftheVCOand thephasedetector.

Figure5VCOandPhaseDetector

The The VCO sim simulati ulation on is the FM generator generator with with noba no baseb seban and d input. input. As with with the FM generato generator r the VCO VCOin inpu putt has hasan an offs offset et ofunity ofunity sothatwh sothat when en the VCO inputiszero, inputiszero, unity multip tiplies the the VCO freeree-rrunning freq requency (2k (2kHz/1 z/10kHz) and gives a continuouswaveform.ThescalingoftheVCOis2kHz/voltandhencetheVCOgain is  K v = 12566 rad  /  s / volt .TheinputunitistakenasvoltstoconformtoarealVCO. Thi This gai gain is inheren herent t in the the VCO VCO and and must ust be accou account nted ed for when when cal calcul culatin ating the the overallgainoftheloop.ThephasedetectorisalinearmultiplierasshowninFigure5. ItstwoinputscomefromtheFMgeneratorandtheVCOoutput.TheVCOoutputisa sinewaveherebutcanbeeasilychangedtoasquarewaveasisthecaseinmostICs. Thephasedetectoroutputfeedsin Thephasedetectoro utputfeedsintot tothefi hefilterwhi lterwhichisdi chisdiscussednex scussednext. t. 3.2FilterSimulation

ExcludingthedynamicsoftheVCOwhichisoftheformofanintegrator,itremains tosimulatethefiltertransferfunctionwhichisoftheform 







F ( s ) =

(1 + sT 1 ) s (1 + sT 2 ) K 

forT1>T2.

8

Theaboveequationisa secondlineartim secondlineartime-in e-invari variant(LTI) ant(LTI) transferfuncti transferfunctionwhich onwhichcan can be spli split into into two first-ord first-order ertran transf sfer erfu func nction tions sin in cascad cascade. e.Lab LabVIEW VIEW in its basic basic form hasmany.viblockswhichareavailableforfilteringetcbutnonearedirectlyrelevant to implem plemen enti ting ng the above above equat equatiion. It was theref therefore ore deci decided ded to bui build a .vi .vi block block whichwouldimplementanyfirstorderLTIsystemandcascadethemtoconstructF(s). Itisamatterofchoiceastowhethertoimplementonesecond-orderLTIortwofirstorder order LTI syste system ms but but it was deci decide ded d to go for the sim simples plestt soluti solution. on. Consi Conside der r the generalfirst-ordersystemC(s)where       g (b0 + b1s )     C ( s) = (s + a) OnepossiblesignalflowgraphforC(s)isshowninFigure6.

Figure6.Signalflowgraphofgenericfir Figure6.Signalflowgraphofgenericfirst-ordersystem st-ordersystem

Where here y is the the syst system em outpu output, t, u is the the syst system em input put and and x is a state state vari variab ablle. For For LabV LabVIEW IEW to implem plemen entt this this an integra ntegrati tion on algori algorithm thm is requi required red.. There There are many any techniquesforintegrationbutperhapsthemostpopularmethodistouseTrapezoidal integr ntegrati ation. on. A Trapez Trapezoi oidal dal integra ntegrator tor is repres represen ented ted by the Bil Bilinear near transf transform orm.. In differenceequationformitbecomes 







 y k 

=  y k −1 +

T 

2

[uk  + uk  1 ] −

whereTisthesamplinginterval(0.1ms)andhastheLabVIEWblockdiagramshown inFigure7.

9

Figure7TrapezoidalIntegrationusingLabVIEW

Inthedia Inthe diagram gram delta deltatisthe tisthestepsizeT,and stepsizeT,andgai gainis0.5set nis0.5setextern externall ally.Whe y.Whencombi ncombined ned with with the flow flow graph graph a gene general ral first-ord rst-order er syste system m is constr construc ucted ted and and has has the form form of Figure8.

Figure8.Generalfirst-orderLTIsystem.

Intheabovediagramtheblockwiththeintegrationsymbolrepresentsthepreviously disc discus usse sed d Trap Trapez ezoi oida dall integra tegrator tor.. The The While loops oops in Figu Figure re 7 and and 8 hav have thei their r condi conditi tion onse sett to Fal False so that that they they only only iterate terate one tim time for for every every loopof oop of the main ain

10

extern external al Whi While loop. loop. When When imp mple lem menting enting feedb feedback ack in LabVIEW LabVIEW a sign signal al cannot cannot be connected connected directl directlyy back backas asin in abl a block ockdi diagra agram m.Ins . Insteadit teaditm mustbe stored in aregi a register ster andthepreviousvaluefedbackasshownwiththestatevariablexinFigure8.Thisis becauseadigital becauseadigitalsystem systemcannotrespondinstan cannotrespondinstantaneousl taneouslyastheremustbeat yastheremustbeatlea leastao staone ne steptimecompuationaldelayforinformationtopassfrominputtooutput.Theabove LabVI abVIEW EW prog progra ram m can can be con converte erted d into a sub .vi .vi and and used used as many any tim times as necessaryprovideditisdefinedasre-entrant.Itcanbeusedasanintegrator,phaseleadorasalow-passfilter. 3.3.CompositePLLSimulation

TheLTIblockusedintheprevioussectioncanbeusedtoconstructanintegratorand phase-le phase-leadcompensa adcompensator torforthePLL(iet forthePLL(ietheFil heFilter).It ter).Itisnecessa isnecessaryto rytocal calcul culatethe atethegain gain val values and and any any param paramete eters. rs. One One possi possibl ble e desi design gn for a band bandwi width dth of 400Hz 400Hz give gives s a phase-leadL(s)of 









 L( s ) = 10

( s + 794.2) ( s + 7942 )

sothatfortheLTIblocka=7942,g=10,b0=794.2andb1=1.Theintegratorhasagain whi which can can be found ound to be appr approx oxiimatel ately 2X10 2X106 but but thi this does does not not accou accounnt for any any existin existing g(hi (hiddenor ddenorim impli plicit)gain cit)gainsalready salready in theloop.These theloop. These hiddengain hiddengain terms consist consist oftheVCOgain(12566)andthePhasede oftheVCOgain(12566)andthePhasedetectorgain(0.5),atotalof tectorgain(0.5),atotalof6283.2.Div 6283.2.Dividi iding ng this this value value into theover the overal alllgai gain ngi gives ves a remai remaini ning nggai gain nof of318 318.141 .141and and itis it is this this gain gain whichmustbeusedontheintegratorie318.14/s.ThisisshowninFigure9below.

Figure9.PartofthePLLshowingthefilterdynamics .

11

Thegainadjustparameterisnominallysettounitybutcanbevariedtoseetheeffect ofvaryingtheoverallgainoftheloop.ThecompletePLLFMdemodulatorisshowin Figure10below. The The trans transiient ent respon response se of the loop is found ound by FM modul odulating ating a squa square re wave. wave. An extern external al first-ord first-order erfi filte lter rsetat setat a900H a 900Hz z cut-offfrequ cut-offfrequenc encywas ywascons constructe tructed dusi usingthe ngthe LTIblockandisanintegralpartofthePLLblock.Thecut-offfrequencycanbevaried fromthefrontpanel.TheresultingtransientresponseisshowninFigure11belowfor a10Hzbaseband a10Hzbaseband signa signal.l. Adepthofmodulati Adepthofmodulationof onof+/-200Hz +/-200Hz was usedo usedonthe2kHz nthe2kHz carrier.TheFMmodulationindexforthisexampleistherefore = ∆  /   m =20. Iftheexternalfil Iftheexternalfilterisset terissetto toahigh ahighvalue(4kHz)sothatitisin value(4kHz)sothatitisineff effectiv ectiveandthegain eandthegainof of theloopisincreasedsignificantlytheeffectoftwicethefrequencyofthecarrierfeedthroughcanbeillustratedinFigure12.

Figure10MainsimulationloopforPLLandFMgenerator.

12

Figure11TransientresponseofPLLtoa10Hzsquarewave .

Figure12Illustratesfeed-throughtermforhighbandwidth andnoexternalfiltering.

13

Itisimportanttonotethatintheabovesimulationsthereisnohardlimiterpresentas wouldbeina wouldbeinareal realradioreceive radioreceiver.SincetheFMhasa r.SincetheFMhasaconstantamp constantampli litudeofunitythere tudeofunitythere is no need. eed. Howe Howeve ver, r, if addi additi tive ve noi noise or co-ch co-chan anne nell inter nterfferen erence ce is adde added d to the the simulationitisthennecessarytoaddeitherahardlimiterorsomeformofAutomatic Gai Gain Control Control (AGC) (AGC).. Thi This is beca becaus use e the phas phase e detec detector tor is a linear near multi ultipl pliier and and hence hence any anycha change nge in ampl amplitu itude deof ofthe theFM FM signa signallwi will ll resultina resultina change change ofloop-gain ofloop-gain.. Thiscanleadtoinstabilityasthegaincanbeeithertoolowortoohigh.(seefigure3) 4.Conclusionsandfurtherwork

A sim simulat ulatiion of a PLL PLL usi using Lab LabVIEW VIEW has been een desc descri ribbed in som some dept depthh. The The sim simulati ulation on diff differs ers from fromprev previou ious s studie studies s in thatit that itiis full fullyy interac interactiv tive e and not a ‘one‘oneshot’ shot’ type type sim simulati ulation. on.Unl Unlik ike e a real PLL all allof ofthe thepara parame meters ters can bevar be varie ied din in realrealtime time to see theef the effe fect. ct. For exam exampl ple e thegai the gain nis is easil easilyy increa increased sed or decreas decreased edan and dthe the externalfi externalfiltercut-offcanbe ltercut-offcanbevari varied.At ed.At present presentforsim forsimpli plicity citytthePLLisdesi hePLLisdesignedfora gnedfora givenfixedcarrierfrequencyandbandwidthbutthattoocanbemadeinteractivewith thedesign equationsenteredas equationsenteredas equationsinLabVIEW equationsinLabVIEWandcontinuousl andcontinuouslyupdated.The yupdated.The PLLbehavesinan PLLbehavesinanide identic nticalfash alfashionto ionto anICPLLandassuchisid anICPLL andassuchisideal ealfor forin investi vestigatin gating g suchpropertiesasco-channelinterferenceinFM,thresholding,multi-pathandsoon.

5.References.

[1]Specialissueonphase-lockedloops.,IEEETrans.onCommunications,vol.COM20,No.10,Oct.1982. [2]H.deBellescise,Laréceptionsynchrone,OndeÉlectrique,vol.11,1932. [3] [3] T.J T.J.Moi .Moir, r, Sim Simula ulation tion of a Pha Phase-Loc -Lockked Loop oop usi using Matri trixx. Ele Electron troniic EngineeringJune1995,pp45-49 [4]F.M.Gardner,Phase-lockLoopTechniques,NewYorkWiley1979. [5] [5] T.J.M T.J.Moi oir, r, Digi Digital tal contro controll syst system em compe compennsati sation on usi using the the span span rati ratio, o, Jour Journa nall A,vol31,1,1990,pp65-69