最新Chapter-9-The-Laplace-Transform教学讲义PPT课件.ppt

《最新Chapter-9-The-Laplace-Transform教学讲义PPT课件.ppt》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、Chapter-9-The-Laplace-TransformContentsTheLaplaceTransform(BilateralandUnilateral)andtheInverseLTTheRegionofConvergenceforrationalLTPropertiesofLTAnalysisLTIsystemusingLTCTFouriertransformenablesustodoalotofthings,e.g.—AnalyzingfrequencyresponseofLTIsystem

2、s—SolvingLCCDE’s—Sampling—Modulation……Whydoweneedyetanothertransform?MotivationfortheLaplaceTransformOneviewofLTisasanextensionoftheFTtoallowanalysisofbroaderclassofsignalsandsystems.TherelationshipbetweenFTandLTabsoluteintegrabilityneededExample:etu(t)The

3、relationshipbetweenFTandLTFTisaspecialcaseofLTifX(s)canbeevaluatedats=jω.Example9.1Whena>0ThentheLaplacetransformSinceandThatis,Forexample,fora=0,x(t)istheunitstep(x(t)=u(t))withLaplacetransformExample9.2ThenorrequireRe{s+a}<0,orRe{s}<-aThatis,Acriticaliss

4、ueindealingwithLaplacetransformisconvergence:——X(s)generallyexistsonlyforsomevaluesofs,locatedinwhatiscalledtheregionofconvergence(ROC),forwhichtheintegralinLTtransformconvergesROC={sothat}DependsonlyonσnotonωConclusionsTheLThasconvergenceproblemsastheFT.—

5、—TheLTmayconvergeforsomevaluesofRe{s}andnotforothers.InspecifyingtheLTofasignal,bothX(s)andROCarerequired.IftheROCofLTincludejω-axis,wecangetReRes-planes-planeImIm-a-aROCofEx.9.1ROCofEx.9.2GraphicalVisualizationoftheROCAconvenientwaytodisplaytheROCisshowna

6、sthefollowingfigure:Examples:orInthesameway,weobtainNotethat,LTalsohasashortage.Anarbitraryperiodicsignal,suchashasnoLTexisted.BecauseitsROCisnotexisted.Example9.3Example9.4RationalLaplaceTransformsMany(butbynomeansall)Laplacetransformsofinteresttousarerat

7、ionalfunctionsofs(theprecedingexamples)N(s),D(s)—polynomialsinsAnyx(t)consistingofalinearcombinationofcomplexexponentialsfort>0orfort<0hasarationalLaplacetransform.ZerosandPolesRootsofD(s)=polesofX(s);RootsofN(s)=zerosofX(s).TherepresentationofX(s)throughi

8、tspolesandzerosinthes-planeisreferredtoasthepole-zeroplotofX(s)Example9.3:ImRe-11-2s-planeNotation:×——poleO——zeroExample9.5Thepole-zeroplotisQ:Doesx(t)haveFT?ImRe-112s-plane0Property1:TheROCofX(s)consistsofst