文本1周期信號

PartI

PeriodicSignals

Time-descriptions

Thefactthatthegreatmajorityoffunctionswhichmayusefullybeconsideredassignalsarefunctionsoftimelendsjustificationtothetreatmentofsignaltheoryintermsoftimeandoffrequency.AperiodicsignalwillthereforebeconsideredtobeonewhichrepeatsitselfexactlyeveryTseconds,whereTiscalledtheperiodofthesignalwaveform;thetheoreticaltreatmentofperiodicwaveformsassumesthatthiactrepetitionitendedthroughoutalltime,bothpastandfuture.Inpractice,ofcourse,signalsdonotrepeatthemselvesindefiniy.Nevertheless,awaveformsuchastheoutputvoltageofamainsrectifierpriortosmoothingdoesrepeatitselfverymanytimes,anditsysisasastrictlyperiodicsignalyieldsvaluableresults.Inothercases,suchastheelectrocardiogram,thewaveformisquasi-periodicandmayusefullybetreatedastrulyperiodicforsomepurpose.Itisworthnotingthatatrulyrepetitivesignalisofverylittleinterestinacommunicationchannel,senofurtherinformationisconveyedafterthefirstcycleofthewaveformhasbeenreceived.Oneofthemainreasonsfordiscussingperiodicsignalsisthlearunderstandingoftheirysisisagreathelpwhendealingwithaperiodicandrandomones.

Acompletetime- descriptionofsuchasignalinvolvesspecifyingitsvaluepreciselyateveryinstantoftime.Insomecasesthismaybedoneverysimplyusingmathematicalnotation.Fortunay,itisinmanycasesusefultodescribeonlycertainaspectsofasignalwaveform,ortorepresentitbyamathematicalformulawhichisonlyapproximate.Thefollowingaspectsmightberelevparticularcases:

theaveragevalueofthesignal,

thepeakvaluereachedbythesignal,

theproportionofthetotaltimespentbetweenvalueaandb,

theperiodofthesignal.

Ifitisdesiredtoapproximatethewaveformbyamathematicalexpression,suchtechniquesasapolynomialexpansion,aTaylorseries,oraFourierseriesmaybeused.Apolynomialofordernhavingtheform

ftaatat2at3 （1-1）

0 1 2 3

maybeusedtofittheactualcurveat(n+1)arbitrarypoints.Theaccuracyoffitwillgenerallyimproveasthenumberofpolynomialterms reases.Itshouldalsobenotedthattheerrorbetween

thetruesignalwaveformandthepolynomialwillnormally everylargeawayfromtheregionofthefittedpoints,andthatthepolynomialitselfcannotbeperiodic.Whereasapolynomialapproximationfitstheactualwaveformatanumberofarbitrarypoints,thealternativeTaylorseriesapproximationprovidesagoodfittoasmoothcontinuouswaveforminthevicinityofoneselectedpoint.ThecoefficientsoftheTaylorseriesarechosentomaketheseriesanditsderivativesagreewiththeactualwaveformatthispoint.Thenumberoftermsintheseriesdeterminestowhatorderofderivativethisagreementwillextend,andhencetheaccuracywithwhichseriesandactual

waveformagreeintheregionofthepointchosen.ThegeneralformoftheTaylorseriesfor

approximatingafunction

fata

dfa

dt

ta2

2!

d2fa

dt2

tan

n!

dnfa

dtn

ft

ftintheregionofthepointtaisgivenby

（1-2）

Generallyspeaking,thefittotheactualwaveformisgoodintheregionofthepointchosen,butrapidlydeterioratestoeitherside.ThepolynomialandTaylorseriesdescriptionsofasignalwaveformarethereforeonlytobe mendedwhenoneisconcernedtoachieveaccuracyoveralimitedregionofthewaveform.Theaccuracyusuallydecreasesrapidlyawayfromthisregion,althoughitmaybeimprovedbyludingadditionalterms(solongastlieswithintheregionofconvergenceoftheseries).Theapproximationsprovidedbysuchmethodsareneverperiodicinformandcannotthereforebeconsideredidealforthedescriptionofrepetitivesignals.

BycontrasttheFourierseriesapproximationiswellsuitedtotherepresentationofasignalwaveformoveranextendedinterval.Whenthesignalisperiodic,theaccuracyoftheFourierseriesdescriptionismaintainedforalltime,sethesignalisrepresentedasthesumofanumberofsinusoidalfunctionswhicharethemselvesperiodic.BeforeexaminingindetailtheFourierseriesmethodofrepresentingasignal,thebackgroundtowhatisknownasthe‘frequency- ’approachwillbeintroduced.

Frequency-descriptions

Thebasicconceptionoffrequency- ysisisthatawaveformofanycomplexitymaybeconsideredasthesumofanumberofsinusoidalwaveformsofsuitableamplitude,periodicity,and

relativephase.Acontinuoussinusoidalfunctionsintisthoughtofasa‘singlefrequency’

waveoffrequencyradians/second,andthefrequency- descriptionofasignalinvolvesitsbreakdownintoanumberofsuchbasicfunctions.ThisisthemethodofFourier ysis.

Thereareanumberofreasonswhysignalrepresentationintermsofasetofcomponentsinusoidalwavesoccupiessuchacentralroleinsignalysis.Thesuitabilityofasetofperiodicfunctionsforapproximatingasignalwaveformoveranextendedintervalhasalreadybeenmentioned,anditwillbeshownlaterthattheuseofsuchtechniquescausestheerrorbetweentheactualsignalanditsapproximationtobeminimizedinacertainimportanse.Afurtherreason

whysinusoidalfunctionsaresoimportsignalysisisthattheyoccurwidelyinthephysicalworldandareverysusceptibletomathematicaltreatment;alargeandextremelyimportantclassofelectricalandmechanicalsystems,knownas‘linearsystems’,respondssinusoidallywhendrivenbyasinusoidaldisturbingfunctionofanyfrequency.Allthesemanifestationsofsinusoidalfunctioninthephysicalworldsuggestthatsignalysisinsinusoidaltermswillsimplifytheproblemofrelatingasignaltounderlyingphysicalcauses,ortothephysicalpropertiesofasystemordevicethroughwhichithaspassed.Finally,sinusoidalfunctionsformasetofwhatarecalled‘orthogonalfunction’,theratherspecialpropertiesandadvantageofwhichwillnowbediscussed.

Orthogonalfunctions

Vectorsandsignals

Adiscussionoforthogonalfunctionsandoftheirvalueforthedescriptionofsignalsmaybeconvenientlyintroducedbyconsideringtheogybetweensignalsandvectors.Avectorisspecifiedbothbyitsmagnitudeanddirection,familiarexamplesbeingandvelocity.Suppose

wehavetwovectorsV1andV2;geometrically,wedefhecomponentofvectorV1along

vectorV2byconstructingtheperpendicularfrom ofV1

V1C12V2Ve

ontoV2.Wethenhave

（1-3）

wherevectorVeistheerrorintheapproximation.Clearly,thiserrorvectorisofminimumlengthwhenitisdrawnperpendiculartothedirectionofV2.ThuswesaythatthecomponentofvectorV1alongvectorV2isgivenbyC12V2,whereC12ischosensuchastomaketheerrorvectorassmallaspossible.Afamiliarcaseofanorthogonalvectorsystemistheuseofthreemutually

perpendicularaxeso-ordinategeometry.

Thesebasicideasaboutthecomparisonofvectorsmaybeextendedtosignals.Supposewewishtoapproximateasignalf1tbyanothersignalorfunctionf2toveracertainintervalt1tt2;inotherwords

f1tC12f2t fort1tt2

WewishtochooseC12toachievethebestapproximation.Ifwedefheerrorfunction

fetf1tC12f2t （1-4）itmightappearatfirstsightthatweshouldchooseC12soastominimizetheaveragevalueoffetoverthechoseninterval.Thedisadvantageofsuchanerrorcriterionisthatlargepositiveand

negativeerrorsoccurringatdifferentinstantswouldtendtocanceleachotherout.Thisdifficultyisavoidedifwechoosetominimizetheaveragesquared-error,ratherthantheerroritself(thisisequivalenttominimizingthesquarerootofthemean-squarederror,or’rm.s’error).Denotingthe

e

averageoff2tby,wehave

1 t2f2tdt 1 t2ftCft2dt

（1-5）

t2t1t1 e

t2t1t1

1 122

DifferentiatingwithrespecttoC12andputtingtheresultingexpressionequaltozerogivesthevalueofC12forwhichisaminimum.Thus

d 1 t2 ftCft2dt0

122 1

1

dC ttt

1 122

Expandingthebracketandchangingtheorderofintegrationanddifferentiatinggives

C t2ftftdt t2f2tdt

12t 1 2 t 2

1 1

（1-6）

Signaldescriptionbysetsoforthogonalfunctions

Supposethatwehaveapproximatedasignal f1toveracertainintervalbythefunction f2tsothatthemeansquareerrorisminimized,butthatwenowwishtoimprovetheapproximation.Itwillbedemonstratedthataveryattractiveapproachistoexpressthesignalintermsofasetof

functions f2t,f3t,f4t,etc.,whicharemutuallyorthogonal.Supposetheinitial

approximationis

f1tC12f2t （1-7）

andthattheerrorisfurtherreducedbyputting

f1tC12f2tC13f3t （1-8）

where f2tandf3tareorthogonalovertheintervalofinterest.NowthatwehaveorporatedtheadditionaltermC13f3t,itisinterestingtofindwhatthenewvalueof

C12mustbeinorderthatthemeansquareerrorisagainminimized.Wenowhave

fetf1tC12f2tC13f3t （1-9）

andthemeansquareerrorintheintervalt1tt2

1

1 t2

istherefore

2

t2t1

tf1tC12f2tC13f3t.dt （1-10）

DifferentiatingpartiallywithrespecttoC12tofindthevalueofC12forwhichthemeansquareerrorisagainminimized,andchangingtheorderofdifferentiationandintegration,wehaveagain

C t2ftftdt t2f2tdt

12t 1 2 t 2

1 1

（1-11）

Inotherwords,thedecisiontoimprovetheapproximationby orporatinganadditionaltermin

doesnotrequireustomodifythecoefficient,providedthatf3tisorthogonaltof2tinthechosentimeinterval1.Bypreciselysimilarargumentswecouldshowthatthevalueof C13would

beunchangedifthesignalweretobeapproximatedbyf3talone.

Thisimportantresultmaybeextendedtocovertherepresentationofasignalintermsofawholesetoforthogonalfunctions.Thevalueofanycoefficientdoesnotdependuponhowmanyfunctionsfromthecompletesetareusedintheapproximation,andisthusunalteredwhenfurthertermsareluded.Theuseofasetoforthogonalfunctionsforsignaldescriptionisogoustotheuseofthreemutuallyperpendicular(thatis,orthogonal)axesforthedescriptionofavectorinthree-dimensionalspace,andgivesrisetothenotionofa‘signalspace’.Accuratesignalrepresentationwilloftenrequiretheuseofmanymorethanthreeorthogonalfunctions,sothatwe

mustthinkofasignalwithinsomeintervalmultidimensionalspace.

t1tt2asbeingrepresentedbyapointina

Tosummarize,thereareanumberofsetsoforthogonalfunctionsavailablesuchastheso-calledLegendrepolynomialsandWalshfunctionsfortheapproximatedescriptionofsignalwaveform,ofwhichthesinusoidalsetisthemostwidelyused.Setsinvolvingpolynomialsintarenotbytheirverynatureperiodic,butmaysensiblybeusedtodescribeonecycle(orless)ofaperiodicwaveform;outsidethechoseninterval,errorsbetweenthetruesignalanditsapproximationwillnormallyreaserapidly.Adescriptionofonecycleofaperiodicsignalintermsofsinusoidialfunctionswill,however,beequallyvalidforalltimebecauseoftheperiodicnatureofeverymemberoftheorthogonal.

TheFourierseries

ThebasisoftheFourierseriesisth omplexperiodicwaveformmaybe ysedintoanumberofharmonicallyrelatedsinusoidalwaveswhichconstituteanorthogonalset.Ifwehaveaperiodic

signalftwithaperiodequaltoT,thenftmayberepresentedbytheseries

ftA0Ancosn1tBnsinn1t （1-12）

n1 n1

where12T.Thusftisconsideredtobemadeupbytheadditionofasteadylevel(A0)toanumberofsinusoidalandcosinusoidalwavesofdifferentfrequencies.Thelowestofthese

frequenciesis1(radianspersecond)andiscalledthe‘fundamental’;wavesofthisfrequencyhaveaperiodequaltothatofthesignal.Frequency21iscalledthe‘secondharmonic’,31isthe‘thirdharmonic’,andsoon.Certainrestrictions,knownastheDirichletconditions,mustbe

ceduponftfortheaboveseriestobevalid.Theintegralftdtoveracompleteperiodmustbefinite,andmaynothavemorethanafinitenumberofdiscontinuitiesinanyfinite

interval.Fortunay,theseconditionsdonotexcludeanysignalwaveformofpracticalinterest.

Evaluationofthecoefficients

1

WenowturntothequestionofevaluatingthecoefficientsA0,AnandBn.Usingtheminimumsquareerrorcriteriondescribedinforegoingtext,andwritingforthesakeofconvenience,wehave

A0

2

fxdx,

A1

fxcosnx.dx,

B1

fxsinnx.dx（1-13）

n n

Althoughinthemajorityofcasesitisconvenientfortheintervalofintegrationtobesymmetricalabouttheorigin,anyintervalequalinlengthtooneperiodofthesignalwaveformmaybechosen.

Manywaveformofpracticalinterestareeitherevenoroddfunctionsoftime.If ftiseventhenbydefinitionftft,whereasifitisoddftft.Ifftisevenandwemultiplyitbytheoddfunctionsinn1ttheresultisalsoodd.ThustheintegrandforeveryBnisodd.Nowwhenanoddfunctionisintegratedoveranintervalsymmetricalaboutt0,theresultisalwayszero.HencealltheBcoefficientsarezeroandweareleftwithaseriescontaining

onlycosines.Bysimilararguments,ifftisoddtheAcoefficientsmustbezeroandweareleftwithasineseries.Itisindeedintuitivelyclearthatanevenfunctioncanonlybebuiltupfroma

numberofotherfunctionswhicharethemselveseven,andviceversa.

WehavealreadyseenhowtheFourierseriesissimplifiedinthecaseofanevenoroddfunction,bylosingeitheritssineoritscoserms.Adifferenttypeofsimplificationoccursinthecaseofawaveformpossessingwhatisknowas‘half-wavesymmetry’.Inmathematicalterms,half-wavesymmetryexistswhen

ftftT2 （1-14）

InotherwordsanytwovaluesofthewaveformseparatedbyT2willbeequalinmagnitudeandoppositeinsign.Generalizing,onlyoddharmonicsexhibithalf-wavesymmetry,andthereforeawaveformofanycomplexitywhichhassuchsymmetrycannotcontainevenharmoniccomponents.Conversely,awaveformknowntocontainanysecond,,orotherharmoniccomponentscannotdisyhalf-wavesymmetry.

Usually,wehavealwaysintegratedoveracompletecycletoderivethecoefficients.Howeverin

thecaseofanoddorevenfunctionitissufficient,andoftensimpler,tointegrateoveronlyonehalfofthecycleandtomultiplytheresultby2.Furthermoreifthewaveisnotonlyevenoroddbutalsodisyshalf-wavesymmetry,itisenoughtointegrateoveronequarterofacycleandmultiplyby4.Thesecloserlimitsareadequateinsuchcasesthefunctionthatisbeingintegratedisrepetitive,repeatinicewithinoneperiodwhenthefunctioniseitherevenorodd,andfourtimeswithinoneperiodwhenitalsoexhibitshalf-wavesymmetry.

Choiceoftimeorigin,andwaveformpower

TheamountofworkinvolvedalculatingtheFourierseriescoefficientsforaparticularwaveformshapeisreducedifthewaveformiseitherevenorodd,andthismayoftenbearrangedbyajudiciouschoiceoftimeorigin(thatis,shiftoftimeorigin)2.ThisshifthasthereforemerelyhadtheeffectofconvertingaFourierseriescontainingonlysinetermsintoonecontainingonlycosineterms;theamplitudeofacomponentatanyonefrequencyis,aswewouldexpect,unaltered.Foracomplicatedwaveformwhichisneitherevennorodd,itmustbeexpectedtoludebothsineandcosermsinitsFourierseries.

Asthetimeoriginofawaveformisshifted,thevarioussineandcosinecoefficientsofitsFourierserieswillchange,butthesumofthesquaresofanytwocoefficients AnandBnwillremainconstant,whi eansthattheaveragepowerofthewaveform,aconceptfamiliartoelectrical

engineers,isunaltered.

Theaboveideaslea turallyto ternativetrigonometricformfortheFourierseries.Ifthetwofundamentalcomponentsofawaveformare

A1cos1tandB1sin1t

theirsummaybeexpressedin ternativeformusingtrigonometricidentities

AcostBsint A2B2costtan1B1

1 1 1 1 1 1 1

A2B2sinttan1B1

A1

（1-15）

1 1 1

A1

Thusthesineandcosinecomponentsataparticularfrequencyareexpressedasasinglecosineorsinewavetogetherwithaphaseshift.IfthisprocedureisappliedtoallharmoniccomponentsoftheFourierseries,wegetthealtiveforms

ftA0Cncosn1tnor

N1

ftA0Cnsinn1tn

N1

（1-16）

where

A2B2

Cn ,ntan1Bn

An,n

tan1AnBn

（1-17）

Finally,wenotethats ethemeanpowerrepresentedbyanycomponentwaveis

n n n

0

A2B22C22 （1-18）

andthepowerrepresentedbythetermequalto

A0issimplyA2,thetotalaverag