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.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)senseifandonlyifaboun
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