9(2) 信号与系统 周建华 光电学院教学课件

9 7AnalysisandcharacterizationofLTIsystemsusingtheLaplacetransform 9TheLaplaceTransform 9TheLaplaceTransform 9 7 1Causality TheROCassociatedwiththesystemfunctionforacausalsystemisaright halfplane Forasystemwitharationalsystemfunction causalityofsystemisequivalenttotheROCbeingtheright halfplanetotherightoftheright mostpole Example9 179 18FindtheROCofthefollowingsystemsanditscausalproperty a b 9TheLaplaceTransform Example9 19Determinewhetherornotthefollowingsystemiscausal 9TheLaplaceTransform Conclusion 1 Thesystemiscausal theROCistotherightoftherightmostpole Theconverseisnotingeneraltrue unlessthesystemfunctionisrational 9TheLaplaceTransform 2 Asystemisanticausalifitsimpulseh t 0fort 0 Thesystemisanticausal theROCistotheleftoftheleftmostpole Theconverseisnotingeneraltrue unlessthesystemfunctionisrational 9TheLaplaceTransform 9 7 2Stability AnLTIsystemisstableifandonlyiftheROCofitssystemfunctionH s includestheentirej axis i e Re s 0 2 AncausalsystemwithrationalsystemfunctionH s isstableifandonlyifallofthepolesofH s lieintheleft halfofthes plane i e allofthepoleshavenegativerealparts 9TheLaplaceTransform Example9 209 21Determinewhetherornotthefollowingsystemsarestable a b 9TheLaplaceTransform 9 7 3LTIsystemscharacterizedbylinearconstant coefficientdifferentialequations linearconstant coefficientdifferentialequations 9TheLaplaceTransform Systemfunction 9TheLaplaceTransform Zerosofthesystemfunction polesofthesystemfunction Accordingtoitspoles wecandetermineitsstableandcausalproperties 9TheLaplaceTransform a todetermineh t b Accordingtox t andinitialrestcondition todeterminey t Example9 23Determineh t 9TheLaplaceTransform Example9 25SupposeanLTIsystemwithinputandoutput Whetherornotthesystemisstableorcausal Determinethedifferentialequationofthesystem 9TheLaplaceTransform 9 8systemfunctionalgebraandblockdiagramrepresentations 9TheLaplaceTransform 9 8 1systemfunctionsforinterconnectionsofLTIsystems a Parallelinterconnection 9TheLaplaceTransform h1 t H1 s h2 t H2 s x t y t b Seriesinterconnection 9TheLaplaceTransform h1 t H1 s h2 t H2 s x t y t a Feedbackinterconnection 9TheLaplaceTransform 9TheLaplaceTransform Example FindthesystemfunctionH s ofthefollowingsystem H1 s H2 s H1 s H3 s x t y t 9 8 2blockdiagramrepresentationsforcausalLTIsystemsdescribedbydifferentialequationsandrationalsystemfunctions Anadder Multiplicationbyacoefficient Anintegrator 9TheLaplaceTransform Thebasicoperations 9TheLaplaceTransform Blockdiagram 1 directform 直接型 2 cascadeform 级联型 3 parallelform 并联型 9TheLaplaceTransform 补充 根据简化的梅森公式 可由系统函数画出直接型方框图 1 名词定义闭合环路 是一条顺向的闭合回路 前向路径 从系统输入点到系统输出点 不包含任何环路的信号流通路径 1 Directformoftheblockdiagram H1 s H2 s H1 s H3 s x t y t 9TheLaplaceTransform 2 简化的梅森公式 第i条闭合环路的传输值 所有闭合环路的传输值之和 第k条前向路径的传输值 所有前向路径的传输值之和 9TheLaplaceTransform ThebasicexpressionofH s 9TheLaplaceTransform 一般情况将H s 写成以下形式 9TheLaplaceTransform Example9 28 Solution 9TheLaplaceTransform Example9 29 Solution 3 x t y t 2 9TheLaplaceTransform Example9 30 Solution 9TheLaplaceTransform 9TheLaplaceTransform Example9 31 Solution 3 x t y t 2 6 4 2 9TheLaplaceTransform 9TheLaplaceTransform Example 4 x t y t 7 6 5 9TheLaplaceTransform 也可由方框图 根据梅森公式 写出系统函数的表达式 2 Cascadeformoftheblockdiagram Hi s isfirst orderorsecond ordersystem Ifsisrealpole Hi s isfirst ordersystem Ifsisconjugationimaginarypole Hi s issecond ordersystem 9TheLaplaceTransform 即 把H s 因式分解为一阶或二阶子系统级联 相乘 的形式 9TheLaplaceTransform Example9 30 Solution first ordersystem first ordersystem 9TheLaplaceTransform 1 x t y t 2 x t y t 1 x t 2 y t 9TheLaplaceTransform Example Solution first ordersystem second ordersystem 9TheLaplaceTransform 1 x t y t 2 x t 2 y t 9TheLaplaceTransform 1 x t 2 2 y t 3 Parallelformoftheblockdiagram Hi s isfirst orderorsecond ordersystem 9TheLaplaceTransform 即 把H s 部分分式展开为一阶或二阶子系统并联 相加 的形式 9TheLaplaceTransform Example9 30 Solution first ordersystem first ordersystem 9TheLaplaceTransform 1 x t y t 2 9TheLaplaceTransform Example Solution 9TheLaplaceTransform first ordersystem second ordersystem 9TheLaplaceTransform 1 x t y t 2 x t 2 y t 9TheLaplaceTransform 1 y t 2 x t 2 y t 9 9theunilateralLaplacetransform 9TheLaplaceTransform 如果拉普拉斯变换的积分下限变为 我们把这种变换称为单边拉普拉斯变换 Definition theunilateralLaplacetransform 9TheLaplaceTransform 9TheLaplaceTransform Note Ifx t 0fort 0 itsunilateralLaplacetransformisidenticaltoitsbilateralLaplacetransform2 TheROCofunilateralLaplacetransformmustbearight halfplane 3 Ifx t 0fort 0 itsinverseunilateralLaplacetransformisidenticaltoitsinversebilateralLaplacetransform Example9 329 339 34DeterminetheunilateralLaplacetransformforthefollowingsignals a b c 9TheLaplaceTransform Example9 359 36DeterminetheinverseunilateralLaplacetransformforthefollowingsignals a b 9TheLaplaceTransform Table9 3 9 9 2propertiesoftheunilateralLaplacetransform 9TheLaplaceTransform note MostpropertiesoftheunilateralLaplacetransformissametothoseofthebilateralLaplacetransform thesignificantdifferencesbetweenthemarethepropertiesofdifferentiationandintegrationinthetimedomain 6 Differentiationinthetimedomain 9TheLaplaceTransform 9TheLaplaceTransform 9TheLaplaceTransform Example wehaveknown DeterminetheunilateralLaplacetransformofx t 9TheLaplaceTransform Solution Fromtheabovefigure wecanknow Method1 accordingtoproperty 9TheLaplaceTransform Method2 accordingtothedefinition 8 Integrationinthetimedomain 9TheLaplaceTransform 9 Theinitial andfinal valuetheorems Ifx t 0fort 0andx t containsnoimpulsesorhigher ordersingularitiesatt 0 then 9TheLaplaceTransform 9 9 3solvingdifferentialequationsusingtheunilateralLaplacetransform UsethedifferentiationintimedomainpropertyoftheunilateralLaplacetransformtosolvelinearconstant coefficientdifferentialequationswithnonzeroinitialconditions 9TheLaplaceTransform Example9 38Asystemcharacterizedbythedifferentialequation Withinitialconditions Consider 9TheLaplaceTransform 9TheLaplaceTransform Solution ApplyingtheunilateralLaplacetransformtobothsides 9TheLaplaceTransform Ifa 2 3 5 then Zero inputresponse 零输入响应 Zero stateresponse 零状态响应 Example9 38Asystemcharacterizedbythedifferentialequation Withinitialrestconditions Consider 9TheLaplaceTransform 9TheLaplaceT