外文资料翻译信号与系统Word文档格式.docx

外文资料翻译信号与系统Word文档格式.docx

《外文资料翻译信号与系统Word文档格式.docx》由会员分享，可在线阅读，更多相关《外文资料翻译信号与系统Word文档格式.docx（18页珍藏版）》请在冰豆网上搜索。

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）doesnotdependonvaluesoftheinputforn<

n0.AnecessaryandsufficientconditionforanLTIsystemtobecausalisthatitsunitsampleresponseh（n）bezeroforn<

0.ForanLTIsystem.Causality：

h（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:

（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│<

r0willbeintheROC.

Propotiey4.Ifx（n）isaleftsidedsequenceandifthecircle│z│=r0isintheROC,thenallvaluesofzforwhich0<

Propotiey5.Ifx（n）isastableandcasualsequencewitharationalz