信号与系统英文版教学课件：ch6 The Laplace Transform.ppt

Ch6. The Laplace Transform1n With Laplace transform, we expand the application in which Fourier analysis can be used.n The Laplace transform provides us with a representation for signals as linear combinations of complex exponentials of the form with s=σ + jωn The Laplace transform (拉拉普普拉拉斯斯变变换换) is a generalization of the continuous-time Fourier transform. INTRODUCTIONCh6. The Laplace Transform21. The Laplace TransformLet s = σ+ jω, and using X(s) to denote this integral, we obtainFor some signals which have not Fourier transforms, if we preprocess them by multiplying with a real exponential signal , then they may have Fourier transforms.The Laplace transform of x(t) 1) Development of The Laplace TransformWe will denote the transform relationship between x(t) and X(s) asCh6. The Laplace Transform3ØThe Laplace transform is an extension of the Fourier transform; the Fourier transform is a special case of the Laplace transform when σ= 0.generallyThat is, the laplace transform of x(t) can be interpreted as the Fourier transform of x(t) after multiplication by a real exponential signal. The real exponential may be decaying or growing in time, depending on whether is positive or negative.ØIn specifying the Laplace transform of a signal, both the algebraic expression and the range of values of s for which this expression is valid are required.ØThe range of values of s for which the integral in X(s) converges is referred to as the region of convergence(ROC).Ch6. The Laplace Transform4Example 6.1 Consider the signalFor convergence, we require that Re{s + α} > 0, or Re{s} > –α , Thus,region of convergence (ROC ) (收敛域)Ch6. The Laplace Transform5Example 6.2 Consider the signalFor convergence, we require that Re{s + α} < 0, or Re{s} < –α , Thus,Ch6. The Laplace Transform6–αReIms-plane–αReIms-planeROC for Example 6.1ROC for Example 6.2 Ch6. The Laplace Transform7Example 6.3 Consider the signalUsing Euler’s relation, we can writeThus,Ch6. The Laplace Transform8Consequently,Some useful LT pairs:Ch6. The Laplace Transform92) the pole-zero plot (极零图极零图) Generally, the Laplace transform is rational, i.e., it is a ratio of polynomials in the complex variable s: ØThe roots of N(s) are referred to as the zeros (零点零点) of X(s) ; and the roots of D(s) are referred to as the poles (极点极点) of X(s).ØThe representation of X(s) through its poles and zeros in the s-plane is referred to as the pole-zero plot (极零图极零图) of X(s). ØMarking the locations of the roots of N(s) and D(s) in the s-plane (s平面平面) and indicating the ROC provides a convenient pictorial way of describing the Laplace transform. ØX(s) will be rational whenever x(t) is a linear combination of real or compllex exponentials.Ch6. The Laplace Transform10ØExcept for a scale factor, a complete specification of a rational Laplace transform consists of the pole-zero plot of the transform, together with its ROC. Example 6.4 Consider the signal–1 1 2ReIms-planePole-zero plot and ROC Ch6. The Laplace Transform11Øin general, if the order of the denominator exceeds the order of the numerator by k, X(s) will have k zeros at infinity. Similarly, if the order of the numerator exceeds the order of the denominator by k, X(s) will have k poles at infinity.3) the pole and zero plot at infinityCh6. The Laplace Transform122. The Region of Convergence For Laplace Transform1) Property 1The ROC of X(s) consists of stripes parallel to the jω-axis in the s-plane.2) Property 2 For rational Laplace transforms, the ROC does not contain any poles.3) Property 3 If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-plane. Ch6. The Laplace Transform13Example 6.5LetThe pole at s = -α is removableIn this example , since x(t) is of finite length, it follows from property 3 that the ROC is the entire s-plane. But in the form of above equation , X(s) would appear to have a pole at s=-a ,which ,from property 3 ,would be inconsistent with an ROC that consists of the entire s-plane.In fact, s=-a are both pole and zero of X(s).Ch6. The Laplace Transform144) Property 4 If x(t) is right sided, and if the line Re{s} = σ0 is in the ROC, then all values of s for which Re{s} > σ0 will also be in the ROC; and the ROC of a right-sided signal is a right-half plane. σRReIm5)Property 5 If x(t) is left sided, and if the line Re{s} = σ0 is in the ROC, then all values of s for which Re{s} < σ0 will also be in the ROC; and the ROC of a left-sided signal is a left-half plane.σLReImCh6. The Laplace Transform156) Property 6 If x(t) is two sided, and if the line Re{s} = σ0 is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line Re{s} = σ0.σLσRReIm7) Property 7If the Laplace transform X(s) of x(t) is rational, then its ROC is bounded by poles or extends to infinity. In addition, no poles of X(s) are contained in the ROC.8) Property 8 If the Laplace transform X(s) of x(t) is rational, then if x(t) is right sided, the ROC is the region in the s-plane to the right of the rightmost pole. If x(t) is left sided, the ROC is the region in the s-plane to the left of the leftmost pole.Ch6. The Laplace Transform16Example 6.6LetReIms-planeROC corresponding to a right-sided signal ROC corresponding to a left-sided signal ROC corresponding to a two-sided signal There are three possible ROCs that can be associated with this algebraic expression, corresponding to three distinct signals.Ch6. The Laplace Transform173. The Inverse Laplace TransformMultiplying both sides by , we obtain Changing the variable of this integration from ω to s and using the fact that σ is constant, so that ds = jdω. Thus, the basic inverse Laplace transform equation is:ØThe inverse Laplace transform equation states that x(t) can be represented as a weighted integral of complex exponentials. ØThe formal evaluation of the integral for a general X(s) requires the use of contour integration (围线积分) in the complex plane. ØFor the class of rational transforms, the inverse Laplace transform can be determined by using the technique of partial-fraction expansion. Ch6. The Laplace Transform拉普拉斯变换与傅里叶变换的关系拉普拉斯变换与傅里叶变换的关系1 1）付氏变换与拉氏变换的形式相似，基本差别：）付氏变换与拉氏变换的形式相似，基本差别： 付氏变换时域与变换域变量皆为实数（付氏变换时域与变换域变量皆为实数（ ）） 拉氏变换时域变量为实数，变换域变量为复数（拉氏变换时域变量为实数，变换域变量为复数（ ））2 2）物理意义）物理意义 傅傅氏：氏：将将 分解成许多形式为分解成许多形式为 的指数项之和，的指数项之和，每一对正、负每一对正、负 组成一个余弦振荡，振幅为组成一个余弦振荡，振幅为 拉氏：拉氏：将将 分解成许多形式为分解成许多形式为 的指数项之和，每的指数项之和，每一对正、负一对正、负 组成一个变幅的余弦振荡组成一个变幅的余弦振荡, ,振幅为振幅为 3 3）傅立叶变换是双边）傅立叶变换是双边拉普拉斯变换中拉普拉斯变换中 的一种特殊情况，因的一种特殊情况，因此，求两者反变换的此，求两者反变换的积分路径不同。

积分路径不同 18Ch6. The Laplace Transform拉普拉斯反变换的求法：拉普拉斯反变换的求法：( (一）部分分式展开法一）部分分式展开法 19Ch6. The Laplace Transform（二）（二） 围线积分法（留数法）围线积分法（留数法） 拉氏反变换：拉氏反变换： 留数定理：留数定理： 上式左边的积分是在上式左边的积分是在s平面内沿一不通过被积函数极点的封闭曲线平面内沿一不通过被积函数极点的封闭曲线C进行的，右边则是在此围线进行的，右边则是在此围线C中被积函数各极点上留数之和中被积函数各极点上留数之和 为应用留数定理，在求拉氏反变换的积分线（为应用留数定理，在求拉氏反变换的积分线（ ）上应补足一条积分线以构成一个封闭曲线上应补足一条积分线以构成一个封闭曲线 当然要求必须有当然要求必须有 或或或或上式在满足以下两个条件上式在满足以下两个条件（约当引理）时成立（约当引理）时成立 ①① 时，时， 一致地趋近于零；一致地趋近于零； ②② 因子因子 的指数的指数st的实部应小于的实部应小于 ，， 即即 20Ch6. The Laplace Transform一般条件一般条件①①都能满足（都能满足（ 除外），除外）， 当当 或或条件条件②②满足满足 即即 积分沿左半圆弧进行；积分沿左半圆弧进行； 积分沿右半圆积分沿右半圆弧进行。

因此弧进行因此＝围线中被积函数＝围线中被积函数 所有极点的留数之和所有极点的留数之和 留数的求取：留数的求取： [ 的极点即为的极点即为 的极点的极点] 规则规则ⅠⅠ： 若若 为一阶极点，为一阶极点，则则（（ⅠⅠ）） 规则规则ⅡⅡ：： 若若 为为p阶极点，阶极点， 则则（（ⅡⅡ）） 21Ch6. The Laplace Transform22Example 6.7LetPerforming the partial-fraction expansion, we obtain -2ReImReIm-1Ch6. The Laplace Transform23Example 6.8LetCompute the x(t) with contour integration method.X(s) has two first-order poles: and a second-order pole:From the Residue Theorem, Ch6. The Laplace Transform24Ch6. The Laplace TransformIf ROC isThen If ROC isIf ROC is25Ch6. The Laplace TransformExample 6.9LetX(s) has a pair of conjugate poles:FromSo26Ch6. The Laplace Transform274. Geometric Evaluation of The Fourier Transform From The Pole-Zero PlotA general rational Laplace transform has the form:and it can be factored into the form: where βi, αj are zeros and poles of X(s), respectively. Im s1 αRes-planeComplex plane representation of the vectors s1, a, and s1–a representing the complex numbers s1, a and s1 – a respectively. Ch6. The Laplace Transform28Let’s take an example to show how to evaluate the Fourier transform from the pole-zero plot:Given -2 -1ReImω s-planeGeometrically, from Figure, we can write|X(jω)| is the reciprocal of the product of the lengths of the two pole vectors(极点矢量); arg X(jω) is the negative of the sum of the angles of the two vectors. zero vectors(零点矢量)Ch6. The Laplace Transform295. Properties of The Laplace Transform1) Linearity2) Time ShiftingIfand thenNote: ROC is at least the intersection of R1 and R2, which could be empty, also can be larger than the intersection. IfthenCh6. The Laplace Transform303) Shifting in the s-Domain Ifthen4) Time Scaling IfthenConsequence: if x(t) is real and if X(s) has a pole or zero at s = s0 , then X(s) also has a pole or zero at the complex conjugate point s = s0*. 5) Conjugation When x(t) is real: Consequence:Ch6. The Laplace Transform316) Convolution Property Ifand then7) Differentiation in the Time Domain Ifthen8) Differentiation in the s-Domain Ch6. The Laplace Transform329) Integration in the Time Domain 10) The Initial- and Final-Value Theorems(初初值值和和终终值值定理定理) Initial-value theorem :Final -value theorem :Conditions : x(t)=0 for t<0 and that x(t) contains no impulses or higher order singularities at the origin.Ch6. The Laplace Transform33Example 6.10 Consider the signalWe knowAnd from the time shifting property,So thatHere, the pole at s = 0 is removableCh6. The Laplace TransformExample 6.11 Determine the Laplace transform of Sawtooth 0 T t Ex(t) = + +E0 T t0 T t 0 T tSolution A: 34Ch6. The Laplace TransformSolution B: 0 T t 0 t 0 T t = *35Ch6. The Laplace Transform36Example 6.12 Determine the Laplace transform ofSinceFrom the differentiation in the s-domain property,In fact, by repeated application of this property, we obtainCh6. The Laplace Transform37Example 6.13 Use the initial-value theorem to determine the initial-value ofIn the Time Domain:Ch6. The Laplace Transform386. Analysis and Characterization of LTI Systems Using The Laplace Transform 1) System functionWe know, in the time domain, the input and the output of an LTI system are related through Convolution by the impulse response of the system. Thus y(t) = h(t) *x(t) supposeFrom Convolution Property Y(s) = H(s) X(s) system function(transfer function)For ,H(s) is the frequency response of the LTI system.Ch6. The Laplace Transform39ØThe ROC associated with the system function for a causal system is a right-half plane.§An ROC to the right of the rightmost pole does not guarantee that a system is causal. §For a system with a rational system function, causality of the system is equivalent to the ROC being the right-half plane to the right of the rightmost pole.2) Relating Causality to the System function For a causal LTI system, the impulse response is zero for t<0 and thus is right sided.ØA system is anticausal if its impulse response h(t)=0,for t>0. The ROC associated with the system function for a anticausal system is a left-half plane.Ch6. The Laplace Transform40Example 6.14 Consider a system with impulse response Since h(t) = 0 for t < 0, this system is causal. The system function: It is rational and the ROC is to the right of the rightmost pole, consistent with our statement. Ch6. The Laplace Transform41Example 6.15 Consider the system functionFor this system, the ROC is to the right of the rightmost pole. The impulse response associated with the systemis nonzero for –1 < t < 0. Hence, the system is not causal. Since the system function is irrational. Ch6. The Laplace Transform42ØA causal system with rational system function H(s) is stable if and only if all of the poles of H(s) lie in the left-half of the s-plane ―i.e., all of the poles have negative real parts. ØAn LTI system is stable if and only if the ROC of its system function H(s) includes the jω-axis [i.e., Re{s} = 0]. stableh(t) absolutely integrableh(t) has FTROC of h(t)’s LT contains jω axis3) Relating Stability to the System function The stability of an LTI system is equivalent to its impulse response being absolutely integrable, in which case the Fourier transform of the impulse response converges. sinceCh6. The Laplace Transform43Example 6.16 Consider an LTI system with system function H(s) have two poles: s1=-1,s2=2If the system is know to be causal, the ROC will be ,thusThe system is unstable.If the system is know to be stable, the ROC is ,thusIf the ROC of H(s) is ,thenThe system is anticausal and unstable.Ch6. The Laplace Transform44For an LTI system which is described by a linear constant-coefficient differential equation of the formIts system function (transfer function) is:Thus, the system function for a system specified by a differential equation is always rational. 4) LTI System Characterized by Linear Constant-Coefficient Differential Equations Ch6. The Laplace Transform45Example 6.17 Given the following information about an LTI system: 1. The system is causal. 2. The system function is rational and has only two poles, at s = –2 and s = 4. 3. If x(t) = 1, then y(t) = 0. 4. The value of the impulse response at is 4.Determine the system function of the system.From fact 2, we writeCh6. The Laplace Transform46From fact 3, p(s) must have a root at s = 0 and thus is of the form p(s) = sq(s) From fact 4 and 1, The highest powers in s in both the denominator and the numerator are identical, that is, q(s) must be a constant. We let q(s) = k. It’s easy to find that k = 4. So thatCh6. The Laplace Transform477. System Function Algebra and Block Diagram Representations The use of the Laplace transform allows us to replace time-domain operations such as differentiation, convolution, time shifting, and so on, with algebraic operations. In this section we take a look at another important use of system function algebra in analyzing interconnections of LTI systems and synthesizing systems as interconnections of elementary system building blocks. Ch6. The Laplace Transform48ØThe parallel interconnection of two systems:h1(t)H1(s)x(t)y(t)=x(t)*h1(t)+x(t)*h2(t)h2(t)H2(s) y(s)=x(s) H1(s)+x(s)H2(s)x(s)h(t)=h1(t)+h2(t)H(s)=Y(s)/X(s)= H1(s)+H2(s)h1(t)H1(s)x(t)h2(t)H2(s)x(s)y(t)Y(s)ØThe series interconnection of two systems:h(t)=h1(t)*h2(t)H(s)=Y(s)/X(s)= H1(s) H2(s)Ch6. The Laplace Transform49ØThe feedback interconnection of two systems:h1(t)H1(s)x(t)h2(t)H2(s)x(s)y(t)Y(s) e(t)z(t)+-E(s)Z(s)Y(s)= H1(s)E(s)E(s)=X(s) - Z(s)Z(s)= H2(s)Y(s) Ch6. The Laplace Transform50Example 6.18 Consider the causal LTI system with system function This system can also be described by the differential equationy(t) x(t)1/s–3Block diagram representation of the causal LTI system Ø1/s is the system function of a system with impulse response u(t), i.e., it is the system function of an integrator. Ch6. The Laplace Transform51Example 6.19 Consider a causal LTI system with system function Here z(t) is the output of the first subsystem, y(t) is the output of the overall system.z(t) y(t) x(t)1/s–32Ch6. The Laplace Transform52Example 6. 20 Consider a second-order LTI system with system function z(t)x(t)1/s–3–21/sCh6. The Laplace Transform53y(t)x(t)1/s–32-21/s-64Direct-form (直接型) representation for the system in Example 6.20 Ch6. The Laplace Transform54x(t)1/s–2–12y(t)–11/s3Cascade-form (串联型) representation for the system in Example 6.20 Ch6. The Laplace Transform55y(t)x(t)–11/s–81/s–2 6 2Parallel-form (并联型) representation for the system in Example 6.20 Ch6. The Laplace Transform56Ch6. The Laplace Transform例例 二阶连续反馈系统的仿真：二阶连续反馈系统的仿真：开开环传递函数函数在在MATLAB中建立仿真模型如下：中建立仿真模型如下：阶跃响应的仿真结果：阶跃响应的仿真结果：57Ch6. The Laplace Transform反馈系统的转移函数：反馈系统的转移函数：58Ch6. The Laplace Transform59Ch6. The Laplace Transform608. Signal Flow Graph Analysis (信号流图分析信号流图分析)1) Signal Flow Graph Representation1 1。

由方程作流图由方程作流图 例例作图规则：作图规则： （（1）首先把方程式写成因果关系式：果＝）首先把方程式写成因果关系式：果＝f（（因）因）；； 如选如选 为果：为果： （（2 2）方程式中的各个变量用）方程式中的各个变量用“○○”表示，称作结点；表示，称作结点； ○ ○ (3)(3)变量之间的因果关系用线段来表示，称作支路变量之间的因果关系用线段来表示，称作支路 其特点：其特点： ⅰⅰ）有向，因）有向，因 果（支路的方向表示信号流动的方向）果（支路的方向表示信号流动的方向） ⅱⅱ）支路旁边标上因变量的系数（传输值））支路旁边标上因变量的系数（传输值） a ⅲⅲ）每一个结点的变量等于流入它的变量与相应支路传输值的乘积的代）每一个结点的变量等于流入它的变量与相应支路传输值的乘积的代数和 如如 1 sY(s) 1/s Y(s) X(s) -aCh6. The Laplace Transform61例例 求求 的流图的流图 1） （（1）选）选 为果：为果： （（2）选）选 为果：为果： 【注意】各方程的【注意】各方程的果变量不能相同果变量不能相同 2 2）用结点表示变量（结点还兼有加法器的作用））用结点表示变量（结点还兼有加法器的作用） 3 3）用支路表示因果关系并标注传输值）用支路表示因果关系并标注传输值 -a/b -c/b -d/f -e/f 若若由此可画流图：由此可画流图：ab+1cdef+1一个方程组的流图不是唯一的一个方程组的流图不是唯一的 ,但其解答是唯一的但其解答是唯一的 !Ch6. The Laplace Transform62例例 求一阶系统的流图求一阶系统的流图 ——时域时域 模型模型 ——复域复域 模型模型 、、 、、 ——复量复量 作流图：结点作流图：结点3个个—— 、、 、、 1/s 1-a0 1若只有若只有 、、 两个复量：两个复量： 则流图为：则流图为： H(s) 其中其中 流图和框图都用于描述系统方程，但流图更简洁，使用更方便。

流图和框图都用于描述系统方程，但流图更简洁，使用更方便 Ch6. The Laplace Transform632 2由框图作流图由框图作流图 规则：规则： （（1）用变量表示结点，）用变量表示结点， （（2）方框用支路代替，且有向支路旁边记上传输值）方框用支路代替，且有向支路旁边记上传输值 （（3）加法器用）加法器用“○○”表示叫做和点表示叫做和点 如由（如由（1 1）式可得一阶系统的框图：）式可得一阶系统的框图： 则由上述规则很容易得出其流图则由上述规则很容易得出其流图3. 由电路图作流图由电路图作流图 规则：规则： （（1）选回路电流及节点电压为信号变量，找出从）选回路电流及节点电压为信号变量，找出从输入到输出的流程输入到输出的流程 （（2）列方程组）列方程组 (3) 作流图作流图 Ch6. The Laplace Transform64Example 6. 21 Consider again the system in example 6.20. Use signal flow graph to represent it. y(t)x(t)1/s–32-21/s-64y(t)x(t)1/s–32-21/s-6411Ch6. The Laplace Transform65 支路传输值支路传输值：支路因果变量间的转移函数：支路因果变量间的转移函数入支路入支路：流向结点的支路：流向结点的支路 出之路出之路：流出结点的支路：流出结点的支路源结点源结点(Source nodes) ：只有输出，通常表示输入信号，如：只有输出，通常表示输入信号，如x[n]汇结点汇结点(Sink nodes) :只有输入，通常表示输出信号，如只有输入，通常表示输出信号，如――→ y[n]和点和点：多输入，单输出：多输入，单输出 分点分点：多输出单输入：多输出单输入闭环（环闭环（环Loop ））：顺向闭合路径，如：顺向闭合路径，如w1[n] → c → w2[n] →a → w1[n]自环自环(self-loop)：仅包含有一条支路的闭环（只与一个结点相接触）：仅包含有一条支路的闭环（只与一个结点相接触）开路径开路径：从某一结点连续经一些顺向（有向线段的方向一致）路径至另一结点：从某一结点连续经一些顺向（有向线段的方向一致）路径至另一结点前前向向路路径径(forward path)：：从从源源结结点点至至汇汇结结点点的的开开路路径径（（不不包包含含有有任任何何环环路路的的信号流通路径），信号流通路径），Ch6. The Laplace Transform662) Analysis mothods of Signal Flow Graph1 1。

简化求解简化求解 化简的目的：化简的目的： 将流图逐步简化，最终在激励与输出间仅有一将流图逐步简化，最终在激励与输出间仅有一条支路，从而直接得出输出与激励的关系条支路，从而直接得出输出与激励的关系 简化规则：简化规则： （（1 1））支路串联支路串联（顺向的支路串联可合并成一条支路，（顺向的支路串联可合并成一条支路，并消去中间结点）并消去中间结点） H1 H2 H3 X1 X2 X3 X4 X1 X4 H1H2H3 （（2 2））支路并联支路并联（若干支路并联可用一等效支路代替）（若干支路并联可用一等效支路代替） X1 H2 X2 H1H3 X1 X2 H1+H2+H3 Ch6. The Laplace Transform67(3)(3)结点的消除结点的消除 X1 X0 a1X2 a3 a2 X3 X1 a1X2 a3 a2 X3 a1X1 X0 a1X2 a3 a2 X3 X1 a3X2 a3 a2 X3 a1规则：各条路径的传输值等于流入规则：各条路径的传输值等于流入X0和流出和流出X0的传输的传输值的乘积。

值的乘积 X1 X3 X2 X4 a1 a3 a2 a4 a2 a3 a1 a4X1 X3 X2 X4 a1 a3 a2 a4 X0 Ch6. The Laplace Transform68(4)(4)自环的消除自环的消除 X1 X2 X0 规则：消除自环后，该结点所有入支路的传输值应规则：消除自环后，该结点所有入支路的传输值应俱除以俱除以（（1－－t））因子，而出支路的传输值不变。

因子，而出支路的传输值不变 X1 X3 X2 X0 t a1 a3 a2 X1 X3 X2 X0 a1 a2 t X1 X2 X0 Ch6. The Laplace Transform692 2 MasonMason’’s s Formula ( Formula (梅森规则梅森规则):):In a signal flow graph, the transfer value (transfer function) between any source node and sink node or mixed node can be determined by the following equation:Hereis the graph determinant(特征行列式特征行列式).Ch6. The Laplace Transform70is the gain of each loop. is the multiplication of the gains of two loops which have no shared nodes and branches . is the graph determinant of the left graph after removing the k-th forward parth. is the gain of the k-th forward parth between the source node and the sink node.Ch6. The Laplace Transform71Example 6. 22 Compute the transfer functions between nodes A and B, and nodes A and C in the following signal flow graph.X(s)Y(s)1/s2A31/s1511–10C–141/sW(s)BCh6. The Laplace Transform72–11/sA to B:After removing G1 , the left graph is:Thus,X(s)Y(s)1/s2A31/s1511–10C–141/sW(s)BCh6. The Laplace Transform73A to C:After removing G1 , the left graph is:Thus,Y(s)1/s15–10–141/sX(s)Y(s)1/s2A31/s1511–10C–141/sW(s)BCh6. The Laplace Transform749. The Unilateral Laplace Transform bilateral Laplace transform: (双边拉普拉斯变换双边拉普拉斯变换)unilateral Laplace transform:(单边单边拉普拉斯变换拉普拉斯变换) ØThe lower limit of integration, , signifies that we include in the interval of integration any impulses or higher order singularity functions concentrated at t = 0. (奇异函数)ØThe bilateral transform depends on the entire signal from t = –∞ to t = +∞, whereas the unilateral transform depends only on the signal from to ∞. ØThe bilateral transform and the unilateral transform of a causal signal are identical.1) The Unilateral Laplace Transform and Inverse TransformCh6. The Laplace Transform75Example 6.23Consider the signalThe bilateral transform X(s) for this example can be obtained from Example 6.1 and the time-shifting property:By contrast, the unilateral transform isØThe evaluation of the inverse unilateral Laplace transforms is also the same as for bilateral transforms, with the constraint that the ROC for a unilateral transform must always be a right-half plane.ØThe ROC for the unilateral transform is always a right-half plane. Ch6. The Laplace Transform76 In fact, we could recognize as the bilateral transform of x(t)u(t). Since and Thus,Example 6.24 Consider the unilateral Laplace transform Determine x(t).For the unilateral transform, the ROC must be the right-half plane to the right of the rightmost pole of .In this case, the ROC consists of all points s with Re{s} > –1. Ch6. The Laplace Transform77 Thus,üunilateral Laplace transform provide us with information about signals only for Example 6.25 Consider the unilateral Laplace transform Taking inverse transforms of each term results inCh6. The Laplace Transform782) Properties of the Unilateral Laplace Transform üTime scaling:üConvolution: assuming that x1(t) and x2(t) are identically zero for t < 0. üDifferentiation in the time domain :Ch6. The Laplace Transform79 Proof of this property for first-derivative of x(t): Similarly, the unilateral Laplace transform of second-derivative of x(t) can be obtained by repeating using the property:Ch6. The Laplace Transform803) Solving Differential Equations Using the Unilateral Laplace TransformExample 6.26 Consider the system characterized by the differential equation with initial conditionsLet x(t) = αu(t). Determine the output y(t).Applying the unilateral transform to both sides of the differential equation, we obtain Ch6. The Laplace Transform81or equivalently,Thus, we obtainzero-state response zero-input response ØThe unilateral Laplace transform is of considerable value in analyzing causal systems which are specified by linear constant-coefficient differential equations with nonzero initial conditions (i.e., systems that are not initially at rest).Ch6. The Laplace Transform824) Representation of Circuits in s-domainThe relations between I and V in the time domain for R,L,C are:Respectively.Apply unilateral Laplace transform to each equation to obtain For a circuit, if we obtain the representation for the basic elements in the circuit in the s-domain, then we also obtain the circuit in the s-domain. Ch6. The Laplace Transform83ØAn inductor with inductance L and initial current may be taken as an inductor with inductance L and zero initial current cascaded with a impulse source voltage with area . ØA capacitor with capacitance C and initial voltage may be taken as a capacitor with capacitance C and zero initial voltage cascaded with a step source voltage with step . IR(s)VR (s)R+-IL(s)VL (s)sL+-–+IC(s)VC (s)+-+–Representation of the three basic elements in the s-domain with initial state being equalized as a source voltageCh6. The Laplace Transform84IR(s)VR (s)R+-VL (s)sL+-VC (s)+-Representation of the three basic elements in the s-domain with initial state being equalized as a source currentØAnother expression of the relation between current and voltage of three basic elements in the s-domain can induce another model:Ch6. The Laplace Transform85First draw the s-domain model of the given circuit and the initial conditions are equalized as source voltages at the same time. Example 6.27 Determine the current i1(t) in the following circuit. Given the initialConditions and inputR1=0.2ΩL=0.5Hx(t) + －C=1FR2=1Ω– +i1(t)i2(t)R1sLX(s) + －R2+ –I1(s)I2(s) － +Ch6. The Laplace Transform86Using KVL to analyze the circuit in the s-domain to obtain following equations :Or equivalently,Thus,Consequently,R1sLX(s) + －R2+ –I1(s)I2(s) － +Ch6. The Laplace Transform87 SUMMARY 2.The properties of the ROC of the Laplace transform and the relationship between the ROC and the poles;1. The bilateral and unilateral Laplace transform;3. The inverse Laplace transform;4. The properties of the bilateral and unilateral Laplace transforms (note the similarities and the differences);5. The pole-zero plot; 6. Laplace transform provides a very useful analytical tool in the analysis and study of continuous-time LTI systems;7. The concepts of the zero-state response and the zero-input response. Ch6. The Laplace Transform作业：9.21(b)(d)(c)(j)9.22(a)(c)(e)(g)9.26 9.27 9.289.31 9.329.34 9.35 9.38 9.4088。