外文资料翻译信号与系统.docx
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外文资料翻译信号与系统
外文资料
SignalsandSystem
Signalsarescalar-valuedfunctionsofoneormoreindependentvariables.Oftenforconvenience,whenthesignalsareone-dimensional,theindependentvariableisreferredtoas“time”Theindependentvariablemaybecontinuesordiscrete.Signalsthatarecontinuousinbothamplitudeandtime(oftenreferredtoascontinuous-timeoranalogsignals)arethemostcommonlyencounteredinsignalprocessingcontexts.Discrete-timesignalsaretypicallyassociatedwithsamplingofcontinuous-timesignals.Inadigitalimplementationofsignalprocessingsystem,quantizationofsignalamplitudeisalsorequired.AlthoughnotpreciselyCorrectineverycontext,discrete-timesignalprocessingisoftenreferredtoasdigitalsignalprocessing.
Discrete-timesignals,alsoreferredtoassequences,aredenotedbyfunctionswhoseargumentsareintegers.Forexample,x(n)representsasequencethatisdefinedforintegervaluesofnandundefinedfornon-integervalueofn.Thenotationx(n)referstothediscretetimefunctionxortothevalueoffunctionxataspecificvalueofn.Thedistinctionbetweenthesetwowillbeobviousfromthecontest.
Somesequencesandclassesofsequencesplayaparticularlyimportantroleindiscrete-timesignalprocessing.Thesearesummarizedbelow.
Theunitsamplesequence,denotedbyδ(n)=1,n=0,δ(n)=0,otherwise
(1)
Thesequenceδ(n)playarolesimilartoanimpulsefunctioninanaloganalysis.
Theunitstepsequence,denotedbyu(n),isdefinedas
U(n)=1,n≧0u(n)=0,otherwise
(2)
Exponentialsequencesoftheform
X(n)=
(3)
Playaroleindiscretetimesignalprocessingsimilartotheroleplayedbyexponentialfunctionsincontinuoustimesignalprocessing.Specifically,theyareeigenfunctionsofdiscretetimelinearsystemandforthatreasonformthebasisfortransformanalysistechniques.When︳α︳=1,x(n)takestheform
x(n)=A
(4)
Becausethevariablenisaninteger,complexexponentialsequencesseparatedbyintegermultiplesof2πinω(frequency)areidenticalsequences,I.e:
(5)
Thisfactformsthecoreofmanyoftheimportantdifferencesbetweentherepresentationofdiscretetimesignalsandsystems.
Ageneralsinusoidalsequencecanbeexpressedas
x(n)=Acos(
n+Φ)(6)
whereAistheamplitude,
thefrequency,andΦthephase.
Incontrastwithcontinuoustimesinusoids,adiscretetimesinusoidalsignalisnotnecessarilyperiodicandifitistheperiodicandifitis,theperiodis2π/ω0isaninteger.
Inbothcontinuoustimeanddiscretetime,theimportanceofsinusoidalsignalsliesinthefactsthatabroadclassofsignalsandthattheresponseoflineartimeinvariantsystemstoasinusoidalsignalissinusoidalwiththesamefrequencyandwithachangeinonlytheamplitudeandphase.
Systems:
Ingeneral,asystemmapsaninputsignalx(n)toanoutputsignaly(n)throughasystemtransformationT{.}.Thedefinitionofasystemisverybroad.withoutsomerestrictions,thecharacterizationofasystemrequiresacompleteinput-outputrelationshipknowingtheoutputofasystemtoacertainsetofinputsdosenotallowustodeterminetheoutputofasystemtoothersetsofinputs.Twotypesofrestrictionsthatgreatlysimplifythecharacterizationandanalysisofasystemarelinearityandtimeinvariance,alternativelyreferredasshiftinvariance.Fortunately,manysystemcanoftenbeapproximatedbyalinearandtimeinvariantsystem.Thelinearityofasystemisdefinedthroughtheprincipleofsuperposition:
T{ax1(n)+bx2(n)}=ay1(n)+by2(n)(7)
WhereT{x1(n)}=y1(n),T{x2(n)}=y2(n),andaandbareanyscalarconstants.
TimeinvarianceofasystemisdefinedasTimeinvariance
T{x(n-n0)}=y(n-n0)(8)
Wherey(n)=T{x(n)}and
isaintegerlinearityandtimeinvarianceareindependentproperties,i.e,asystemmayhaveonebutnottheotherproperty,bothorneither.
Foralinearandtimeinvariant(LTI)system,thesystemresponsey(n)isgivenby
y(n)=
(9)
wherex(n)istheinputandh(n)istheresponseofthesystemwhentheinputisδ(n).Eq(9)istheconvolutionsum.
Aswithcontinuoustimeconvolution,theconvolutionoperatorinEq(9)iscommutativeandassociativeanddistributesoveraddition:
Commutative:
x(n)*y(n)=y(n)*x(n)(10)
Associative:
[x(n)*y(n)]*w(n)=x(n)*[y(n)*w(n)](11)
Distributive:
x(n)*[y(n)+w(n)]=x(n)*y(n)+x(n)*w(n)(12)
Incontinuoustimesystems,convolutionisprimarilyananalyticaltool.Fordiscretetimesystem,theconvolutionsum.InadditiontobeingimportantintheanalysisofLTIsystems,namelythoseforwhichtheimpulseresponseifoffinitelength(FIRsystems).
Twoadditionalsystempropertiesthatarereferredtofrequentlyarethepropertiesofstabilityandcausality.Asystemisconsideredstableintheboundedinput-bounderoutput(BIBO)senseifandonlyifaboundedinputalwaysleadstoaboundedoutput.AnecessaryandsufficientconditionforanLTIsystemtobestableisthatunitsampleresponseh(n)beabsolutelysummable
ForanLTIsystem,
Stability
(13)
BecauseofEq.(13),anabsolutelysummablesequenceisoftenreferredtoasastablesequence.
Asystemisreferredtoascausalifandonlyif,foreachvalueofn,sayn,y(n)doesnotdependonvaluesoftheinputfornh(n)=0forn<0(14)
BecauseofEq.14.asequencethatiszeroforn<0isoftenreferredtoasacausalsequence.
1.Frequency-domainrepresentationofsignals
Inthissection,wesummarizetherepresentationofsequencesaslinearcombinationsofcomplexexponentials,firstforperiodicsequenceusingthediscrete-timeFourierseries,nextforstablesequencesusingthediscrete-timeFouriertransform,thenthroughageneralizationofdiscrete-timeFouriertransform,namely,thez-transform,andfinallyforfinite-extentsequenceusingthediscreteFouriertransform.Insection1.3.3.wereviewtheuseoftheserepresentationincharacteringLITsystems.
Discrete-timeFourierseries
Anyperiodicsequencex(n)withperiodNcanberepresentedthroughthediscretetimeseries(DFS)pairinEqs.(15)and(16)
Synthesisequation:
=
(15)
Analysisequation:
=
(16)
Thesynthesisequationexpressestheperiodicsequenceasalinearcombinationofharmonicallyrelatedcomplexexponentials.ThechoiceofinterpretingtheDFScoefficientsX(k)eitheraszerooutsidetherange0≦k≦(N-1)orasperiodicallyacceptedconvention,however,tointerpretX(k)asperiodictomaintainadualitybetweentheanalysisandsynthesisequations.
2.DiscreteTimeFourierTransform
Anystablesequencex(n)(i.e.onethatisabsolutelysummable)canberepresentedasalinearcombinationofcomplexexponentials.Foraperiodicstablesequences,thesynthesisequationtakestheformofEq.(17),andtheanalysisequationtakestheformofEq.(18)
synthesisequation:
x(n)=
(17)
analysisequation:
X(ω)=
(18)
TorelatethediscretetimeFourierTransformandthediscretetimeFourierTransformseries,considerastablesequencex(n)andtheperiodicsignalx1(n)formedbytimealiasingx(n),i.e
(19)
ThentheDFScoefficientsofx1(n)areproportionaltosamplesspacedby2π/NoftheFourierTransformx(n).Specifically,
X1(k0=1/NX(ω)
(20)
Amongotherthings,thisimpliesthattheDFScoefficientsofaperiodicsignalareproportionaltothediscreteFourierTransformofoneperiod.
3.ZTransform
AgeneralizationoftheFourierTransform,theztransform,permitstherepresentationofabroaderclassofsignalsasalinearcombinationofcomplexexponentials,forwhichthemagnitudesmayormaynotbeunity.
TheZTransformanalysisandsynthesisequationsareasfollows:
synthesisequations:
x(n)=
(21)
analysisequations:
X(z)=
(22)
FromEqs.(18)and(22),X(ω)isrelatetoX(z)byX(ω)=X(z)z=
I.e,forastablesequence,theFourierTransformX(ω)istheZTransformevaluatedonthecontour|z|=1,referredtoastheunitcircle.
Eq.(22)convergeonlyforsomevalueofzandnotothers,TherangeofvaluesofzforwhichX(z)converges,i.e,theregionofconvergence(ROC),correspondstothevaluesofzforwhichx(n)z-nisabsolutelysummable.
Wesummarizethepropertiesofthez-transformbutalsooftheROC.Forexample,thetwosequences
and
Havez-transformsthatareidenticalalgebraicallyandthatdifferonlyintheROC.
ThesynthesisequationasexpressedinEq.(21)isacontourintegralwiththecontourencirclingtheoriginandcontainedwithintheregionofconvergence.Whilethisequationprovidesaformalmeansforobtainingx(n)fromX(z),itsevaluationrequirescontourintegration.Suchanintegertediousandusuallyunnecessary.WhenX(z)isarationalfunctionofz,amoretypicallyapproachistoexpandX(z)usingapartialfractionofequation.Theinversez-transformoftheindividualsimplertermscanusuallythenberecognizedbyinspection.
ThereareanumberofimportantpropertiesoftheROCthat,togetherwithpropertiesofthetimedomainsequence,permitimplicitspecificationoftheROC.Thispropertiesaresummarizedasfollows:
Propotiey1.TheROCisaconnectedregion.
Propotiey2.Forarationalz-transform,theROCdoesnotcontainanypolesandisboundedbypoles.
Propotiey3.Ifx(n)isarightsidedsequenceandifthecircle│z│=r0isintheROC,thenallfinitevaluesofzforwhich0<│z│Propotiey4.Ifx(n)isaleftsidedsequenceandifthecircle│z│=r0isintheROC,thenallvaluesofzforwhich0<│z│Propotiey5.Ifx(n)isastableandcasualsequencewitharationalz-tra