9(1) 信号与系统 周建华 光电学院教学课件

Inchapter4andinchapter5 weconsiders j andz ej intheFouriertransform Now we llconsiders j intheLaplacetransform Andz rej intheZ transform 本章要求 正确理解拉普拉斯变换及其定义式 熟练掌握由信号时域特点判断其拉氏变换收敛范围的定性分析方法 包括 有限持续期信号 左边信号 右边信号以及双边信号 牢记常用典型信号的拉氏变换 正确理解拉氏变换的基本性质 特别注意单边拉氏变换和双边拉氏变化的区别 必须弄清楚初值定理和终值定理的使用条件 熟练掌握从基本变换对出发 灵活运用拉氏变换的性质求解信号拉氏变换的基本方法 包括正变换和反变换 掌握采用部分分式展开法求解拉氏反变换的方法 熟练掌握运用拉氏变换分析LTI系统的方法 熟练掌握由系统函数H s 判断系统因果性和稳定性的定性分析方法 对给定系统的数学模型或模拟框图或信号流图会求其H s 和h t 了解由H s 的零极点分布确定系统频响的几何方法 能熟练运用Mason公式化简信号流图 能应用拉氏变换分析具体电路 9TheLaplaceTransform 9 TheLaplaceTransform 9 1TheLaplaceTransform LTI 9TheLaplaceTransform 1 Definition TheLaplacetransformofthesignalx t is 9TheLaplaceTransform OperatorL x t denotestheLaplacetransformofx t Thetransformrelationshipbetweenx t andX s 9TheLaplaceTransform a Especially whens j aboveequationbecomesTheFouriertransformofsignalx t therelationshipbetweentheFouriertransformandtheLaplacetransform 9TheLaplaceTransform b Ontheotherhand 9TheLaplaceTransform Example9 1ComputetheLaplacetransformofthefollowingsignals a b 9TheLaplaceTransform Solution a b 2 RegionofConvergence ROC ROC RangeofsforX s toconvergeRepresentation A InequalityB RegioninS plane 9TheLaplaceTransform 9TheLaplaceTransform x t 的拉普拉斯变换相当于x t e t的傅利叶变换 在满足x t e t绝对可积条件下的 的取值范围称为拉普拉斯变换的收敛域 9TheLaplaceTransform ExampleforROC 9TheLaplaceTransform Ins plane thehorizontalaxisisRe s or axis TheverticalaxisisIm s orj axis 9TheLaplaceTransform Example9 2 ComputetheLaplacetransformofthefollowingsignal 9TheLaplaceTransform Solution 9TheLaplaceTransform ROC 9TheLaplaceTransform and havesameLaplacetransformrepresentation buttheirROCisdifferent L L Note forasignalx t wemustgiveouttherepresentationofLaplacetransformanditsROC Example9 39 4DeterminetheLaplacetransformofthefollowingsignals a b 9TheLaplaceTransform 9TheLaplaceTransform a solution 9TheLaplaceTransform b solution 9TheLaplaceTransform 9TheLaplaceTransform 3 Thepole zeroplotofX s X s canberepresentedtheratiooftwopolynomials thenumeratorpolynomial thedenominatorpolynomial 9TheLaplaceTransform Definition ThezerosofX s therootsofthenumeratorpolynomialN s iscalledthezerosofX s ThepolesofX s therootsofthedenominatorpolynomialD s iscalledthepolesofX s 9TheLaplaceTransform 9TheLaplaceTransform TherepresentationofX s throughitspolesandzerosinthes planeisreferredtothepole zeroplotofX s Definition Inthes plane use X toindicatethepolesofX s anduse O toindicatethezerosofX s 9TheLaplaceTransform Ontheotherhand IfM N s X s X s have M N polesatinfinity IfM N s X s 0 X s have N M zerosatinfinity 9TheLaplaceTransform Example 9TheLaplaceTransform Example 2 1 Re s Im s s plane Note ThealgebraicformofX s doesnotbyitselfidentifytheROCfortheLaplacetransform Thatis acompletespecification towithinascalefactor aLaplacetransformconsistsofthepole zeroplotofthetransform togetherwithitsROC 9TheLaplaceTransform 9TheLaplaceTransform Example9 5DeterminetheLaplacetransformofthefollowingsignal 9TheLaplaceTransform Solution 9TheLaplaceTransform 112 Re s Im s s plane 9 2TheRegionofConvergenceforLaplaceTransform Property1 TheROCofX s consistsofstripsparalleltoj axisinthes plane Property2 ForrationalLaplacetransform theROCdoesnotcontainanypoles Property3 Ifx t isoffinitedurationandisabsolutelyintegrable thentheROCistheentires plane 9TheLaplaceTransform 9TheLaplaceTransform Example9 6ComputetheLaplacetransformofx t Solution Property4 Ifx t isrightsided andifthelineRe s 0isintheROC thenallvaluesofsforwhichRe s 0willalsointheROC 9TheLaplaceTransform Property5 Ifx t isleftsided andifthelineRe s 0isintheROC thenallvaluesofsforwhichRe s 0willalsointheROC 9TheLaplaceTransform Property6 Ifx t istwosided andifthelineRe s 0isintheROC thentheROCwillconsistofastripinthes planethatincludesthelineRe s 0 9TheLaplaceTransform 9TheLaplaceTransform Example9 7DeterminetheLaplacetransformofx t 9TheLaplaceTransform Property7 IftheLaplacetransformX s ofx t isrational thenitsROCisboundedbypolesorextendstoinfinity Inaddition nopolesofX s arecontainedintheROC Property8 IftheLaplacetransformX s ofx t isrational thenifx t isrightsided theROCistheregioninthes planetotherightoftherightmostpole Ifx t isleftsided theROCistheregioninthes planetotheleftoftheleftmostpole 9TheLaplaceTransform 9TheLaplaceTransform Example9 8 DeterminetheinverseLaplacetransformofX s 9 3TheInverseLaplaceTransform So 9TheLaplaceTransform Appendix PartialFractionExpansion 补充 部分分式展开法求拉氏反变换 Considerafractionpolynomial DiscussthreecasesofD s 0 fordistinctrealroots conjugationcomplexroots andsameroots 9TheLaplaceTransform Case1 Distinctrealroots thus 9TheLaplaceTransform CalculateA1 Multiplytwosidesby s p1 Lets p1 so Generally 9TheLaplaceTransform 9TheLaplaceTransform Example9 99 109 11DeterminetheinverseLaplacetransformofX s a b c 9TheLaplaceTransform Case2 Conjugationcomplexroots 9TheLaplaceTransform p1isacomplexrootofD s thus 9TheLaplaceTransform CalculateA1 9TheLaplaceTransform 9TheLaplaceTransform Othercomputationissametocase 1 distinctrealroots 9TheLaplaceTransform Example DeterminetheinverseLaplacetransform 9TheLaplaceTransform Solution Case3 Sameroot 9TheLaplaceTransform 9TheLaplaceTransform thus 9TheLaplaceTransform Forfirstorderpoles Multiplytwosidesby s p1 r Forr orderpoles 9TheLaplaceTransform 9TheLaplaceTransform So 9TheLaplaceTransform using Wecanobtainx t 9TheLaplaceTransform Example WehaveknownthesystemfunctionofaLTIsystem Determineitsunitstepresponse 9TheLaplaceTransform Solution 9TheLaplaceTransform Inabovethreecases weemphasizen m thatis intherepresentationofX s Ifn m howcanwedo Suchasthefollowingexample 9TheLaplaceTransform Example Determinex t ofX s 9TheLaplaceTransform Solution 如果m n 则必须将有理分式化为多项式加真分式的形式 然后进行部分分式展开 9TheLaplaceTransform 9TheLaplaceTransform Example Determinex t ofX s 9TheLaplaceTransform Solution Then wecanobtainx t 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot Foracertainvalueofs1 then Re 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 零点向量 零点到s1的向量 极点向量 极点到s1的向量 X s1 的模等于各零点向量模的乘积除以各极点向量摸的乘积再乘以系数M X s1 的相位等于各零点向量相位的和减去各极点向量相位的和 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 即 傅立叶变换X j 1 的模等于各零点向量模的乘积除以各极点向量模再乘以系数M X j 1 的相位等于各零点向量相位的和减去各极点向量相位的和 If 零点向量 零点到虚轴上某点j 1的向量 极点向量 极点到虚轴上某点j 1的向量 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 9 4 1first ordersystems 一阶系统 Thedifferentialequationforafirst ordersystemisoftenexpressedinthdform Thecoefficient isthetimeconstantofthesystem SuchasRCcircuit 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot TheBodeplotofthefrequencyresponse 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 讨论 1 增加 H j 的模单调下降 H j 的相位从0下降到 2 2 当 1 时 H j 的模下降了3dB 而此时H j 的相位是 4 所以把 1 称为3dB点或者转折频率点 称为系统的时间常数 一阶连续系统 表现为低通滤波器特性 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 9 4 2second ordersystems Thelinearconstant coefficientdifferentialequationforasecond ordersystemis SuchasRLCcircuit 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 讨论 1 1 两个极点都在实轴上 如图9 19a 由0变到 则 H j 单调下降 而H j 的相位由0变到 近似为一个低通滤波器特性 2 0 1 两个极点是共轭复数 h t 有振荡 此时频率响应如图9 20 此时对某个频率点的较窄的频率范围内 有一个陡峭的尖峰 越小 尖峰越大 这可以用来对一些正弦信号进行选频性放大 所以 此时可以做一个选频放大器 或者说近似为带通滤波器特性 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot Re 0 Im 9 4 2all passsystems 全通系统 全通系统 对于所有 频率响应的模都是一个常数 极点与零点关于虚轴对称 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot 9 4GeometricEvaluationoftheFourierTransformformthePole ZeroPlot Exercise 10 9 9 5PropertiesoftheLaplaceTransform 1 Linearity 9TheLaplaceTransform Example9 13 FindtheROCforx t Wehaveknown 9TheLaplaceTransform 极点 1 抵消了 收敛域扩大了 Solution 2 Timeshifting 9TheLaplaceTransform 9TheLaplaceTransform Example DeterminetheLaplacetransformoftheimpulsetrains 9TheLaplaceTransform Solution 说明 上式中每一个冲激的收敛域均为Re s 但无穷多个冲激线性组合后 总的收敛域变为Re s 0 即收敛域缩小了 这是由于线性组合过程中产生了新的极点s 0 因此性质 1 线性组合 中关于收敛域的概念只适合于有限项的线性组合 9TheLaplaceTransform 3 Shiftinginthes domain 4 Timescaling Figure9 24 b c error 9TheLaplaceTransform Specially 5 Conjugation 6 Convolutionproperty 9TheLaplaceTransform 9TheLaplaceTransform Example 若有极点被抵消的情况 则收敛域会扩大 9TheLaplaceTransform Example wehaveknownthesystemfunctionofacausalandLTIsystem Determinetheoutputy1 t andy2 t Anditsinputsignalis 9TheLaplaceTransform Solution Becausethesystemiscausal itsROCisRe s 2 9TheLaplaceTransform Discussion Asfortheaboveexample iftheinputisx t e 4t H 4 isnotconvergent So y t doesnotexist 9TheLaplaceTransform 的拉普拉斯变换不存在 证明如下 说明 9TheLaplaceTransform 所