信号与系统课件第9章拉普拉斯变换讲解

CHAPTER 9 THE LAPLACE TRANSFORM 6.0 INTRODUCTION,With Laplace transform, we expand the application in which Fourier analysis can be used.,The Laplace transform provides us with a representation for signals as linear combinations of complex exponentials of the form with s= + j,The Laplace transform (拉普拉斯变换) is a generalization of the continuous-time Fourier transform.,以傅里叶变换为基础的频域分析方法的优点在于：它给出的结果有着清楚的物理意义 ，但也有不足之处。 傅里叶变换只能处理符合狄利赫勒条件的信号，而实际中会遇到许多信号，例如(t)、t(t)、sint(t)等，它们并不满足绝对可积条件，虽然通过求极限的方法可以求得它们的傅里叶变换，但其变换式中常常含有冲激函数，使分析计算较为麻烦。 而有些信号（如单边指数信号 ）是不满足绝对可积条件的，因而其对信号的分析受到限制； 另外傅氏变换分析法只能求取零状态响应。,因此，有必要寻求更有效而简便的方法，人们将傅里叶变换推广为拉普拉斯变换（LT: Laplace Transform）。,优点： 求解比较简单，特别是对系统的微分方程进行变换时，初始条件被自动计入，因此应用更为普遍。 缺点： 物理概念不如傅氏变换那样清楚。,拉普拉斯变换（简称拉氏变换）可看作一种广义的傅氏变换，将频域扩展为复频域，简化了信号的变换式，扩大了信号的变换范围，为分析系统响应提供了统一和规范化的方法。,本章内容及学习方法,本章首先由傅氏变换引出拉氏变换，然后对拉氏正变换、拉氏反变换及拉氏变换的性质进行讨论。 本章重点在于，以拉氏变换为工具对系统进行复频域分析。 最后介绍系统函数以及H(s)零极点概念，并根据他们的分布研究系统特性，分析频率响应，还要简略介绍系统稳定性问题。 注意与傅氏变换的对比，便于理解与记忆。,9.1 THE LAPLACE TRANSFORM,Let s = + j, and using X(s) to denote this integral, we obtain,For 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 is an extension of the Fourier transform; the Fourier transform is a special case of the Laplace transform when = 0.,从傅氏变换到拉氏变换:,傅氏变换信号不满足绝对可积条件的原因：,则,1拉普拉斯正变换,2拉氏逆变换,3拉氏变换对,积分下限用 目的是把 时出现的冲激包含进去，这样，利用拉氏变换求解微分方程时，可以直接引用已知的初始状态 ，但反变换的积分限并不改变。,称为单边拉氏变换,4. 拉氏变换的收敛域,通常把使 满足绝对可积条件的 值的范围称为拉氏变换的收敛域。,收敛域：使F(s)存在的s的区域称为收敛域。 记为：ROC(region of convergence) 实际上就是拉氏变换存在的条件；,Example 9.1 Consider the signal,For convergence, we require that Res + 0, or Res ,Thus,Example 9.2 Consider the signal,For convergence, we require that Res + 0, or Res ,Thus,ROC for Example 9.1,ROC for Example 9.2,Example 9.3 Consider the signal,Using Eulers relation, we can write,Thus,Consequently,Some useful LT pairs:,Marking the locations of the roots of N(s) and D(s) in the s-plane (s平面) and indicating the ROC provides a convenient pictorial way of describing the Laplace transform.,Generally, the Laplace transform is rational, i.e., it is a ratio of polynomials in the complex variable s:,The roots of N(s) are referred to as the zeros (零点) of X(s) ; and the roots of D(s) are referred to as the poles (极点) of X(s).,The representation of X(s) through its poles and zeros in the s-plane is referred to as the pole-zero plot (极零图) of X(s).,Except 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.,About the infinity (无穷远点): in 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.,Example 9.4 Consider the signal,Pole-zero plot and ROC,9.2 THE REGION OF CONVERGENCE FOR LAPLACE TRANSFORM,Property 1: The ROC of X(s) consists of stripes parallel to the j-axis in the s-plane.,Property 2: For rational Laplace transforms, the ROC does not contain any poles.,Property 3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s-plane.,Example 9.5 Let,The pole at s = - is removable,Property 4: If x(t) is right sided, and if the line Res = 0 is in the ROC, then all values of s for which Res 0 will also be in the ROC; and the ROC of a right-sided signal is a right-half plane.,Property 5: If x(t) is left sided, and if the line Res = 0 is in the ROC, then all values of s for which Res 0 will also be in the ROC; and the ROC of a left-sided signal is a left-half plane.,Property 6: If x(t) is two sided, and if the line Res = 0 is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line Res = 0.,Property 7: If 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.,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.,Example 9.6 Let,9.3 THE INVERSE LAPLACE TRANSFORM,Multiplying both sides by , we obtain,Changing the variable of this integration from to s and using the fact that is constant, so that ds = jd.,Thus, the basic inverse Laplace transform equation is:,The inverse Laplace transform equation states that x(t) can be represented as a weighted integral of complex exponentials.,The formal evaluation of the integral for a general X(s) requires the use of contour integration (围线积分) in the complex plane.,For the class of rational transforms, the inverse Laplace transform can be determined by using the technique of partial-fraction expansion.,Example 9.7 Let,Performing the partial-fraction expansion, we obtain,Example 9.8 Let,Compute the x(t) with contour integration method.,X(s) has two first-order poles: and a second-order pole:,From the Residue Theorem,9.4 GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT,A general rational Laplace transform has the form:,Complex plane representation of the vectors s1, a, and s1a representing the complex numbers s1, a and s1 a respectively.,Lets take an example to show how to evaluate the Fourier transform from the pole-zero plot:,Given,Geometrically, from Figure, we can write,|X(j)| is the reciprocal of the product of the lengths of the two pole vectors(极点矢量); arg X(j) is the negative of the sum of the angles of the two vectors.,zero vectors(零点矢量),9.5 PROPERTIES OF THE LAPLACE TRANSFORM,9.5.1 Linearity,then,Note: ROC is at least the intersection of R1 and R2, which could be empty, also can be larger than the intersection.,考虑信号x(t)=x1(t)-x2(t)的拉氏变换,9.5.2 Time Shifting,If,then,9.5.3 Shifting in the s-Domain,If,then,9.5.4 Time Scaling,If,then,Consequence:,Consequence: 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*.,6.5.5 Conjugation,When x(t) is real:,9.5.6 Convolution Property,then,9.5.7 Differentiation in the Time Domain,If,then,9.5. 8 Differentiation in the s-Domain,9.5.9 Integration in the Time Domain,9.5.10 The Initial- and Final-Value Theorems(初值和终值定理),Initial-value theorem :,Final -value theorem :,若t0,x(t)=o且t=0不包括任何冲激或高阶奇异函数，则：,初值定理,初值定理条件: 必须存在。当F(s)为有理分式时必须为真分式，时域中意味着 f (t) 在t=0处不包含冲激及其导数；当F(s)部位真分式时，必须先用长除法将F(s)分成一个多项式与一个真分式F0(s)之和，然后再从F0(s)求初值。,例.求初值,由初值定理：,应用初值定理先求真分式,终值定理,条件： 存在，相当于在复频域中 的极点都在S平面的左半平面和原点仅有单极点。,例.给定 ，试求 的终值。,解:因为F(s)的极点为s=0、-1和-2，满足终值定理的条件,求终值首先判断极点位置。,Example 9.9 Consider the signal,We know,And from the time shifting property,So that,Example 9.10 Determine the Laplace transform of,Since,From the differentiation in the s-domain property,In fact, by repeated application of this property, we obtain,Example 9.11 Use the initial-value theorem to determine the initial-value of,9.6 ANALYSIS AND CHARACTERIZATION OF LTI SYSTEMS USING THE LAPLACE TRANSFORM,The Laplace transforms of the input and the output of an LTI system are related through multiplication by the Laplace transform of the impulse response of the system.,Y(s) = H(s) X(s),The 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.,Example 9.12 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.,Example 9.13 Consider the system function,For this system, the ROC is to the right of the rightmost pole.,Since the system function is irrational.,An LTI system is stable if and only if the ROC of its system function H(s) includes the j-axis i.e., Res = 0.,ROC of h(t)s LT contains j axis,Determine h(t) for each of the following case: 1.The system is stable. 2.The system is causal. 3.The system is neither stable nor causal.,Example If,1. The system is stable.,ROC: (-1,2),2. The system is causal.,ROC:,ROC:,3. The system is neither stable nor causal,A 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.,H(s)有极点位于j 轴上，且只能是一阶极点,a).系统稳定：,H(s)全部极点位于s平面的左半平面,b).系统临界稳定：,c).系统不稳定：,H(s)有位于右半平面的极点 ，或虚轴上有重极点,例：判断系统的稳定性,解：H(s)的特征方程为 s3-s2+2=0,特征根为 s1=-1；s23=1j,s23=1j在S右半平面（有正实部），系统不稳定,解：H(s)的特征方程为 s3+5s2+9s+5=0,特征根为 s1=-1；s23=-2j,所有极点（特征根）在S左半平面（有负实部），系统稳定,(s+1)(s+2)2+1=0,For an LTI system which is described by a linear constant-coefficient differential equation of the form,Its system function (transfer function) is:,Thus, the system function for a system specified by a differential equation is always rational.,Consider a continuous-time LTI system for which the input x(t) and output y(t) are related by the differential equation: Let X(s) and Y(s) denote Laplace transforms of x(t) and y(t), respectively, and let H(s) denote the Laplace transform of h(t), the system impluse response.,Example 9.14,Determine h(t) for each of the following case: 1.The system is stable. 2.The system is causal. 3.The system is neither stable nor causal.,Example 9.14 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,From 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.,Its easy to find that k = 4. So that,9.7 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.,Example 9.15 Consider the causal LTI system with system function,This system can also be described by the differential equation,1/s is the system function of a system with impulse response u(t), i.e., it is the system function of an integrator.,Example 9.16 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.,Example 9. 17 Consider a second-order LTI system with system function,Direct-form (直接型) representation for the system in Example 9.17,Cascade-form (串联型) representation for the system in Example 9.17,y(t),x(t),Parallel-form (并联型) representation for the system in Example 9.17,9.8 Signal Flow Graph Representation (信号流图),Formally, a signal flow graph is a network of directed branches that connect at nodes. Associated with each node is a variable or node value.,Input, output, node value, direction, branch (j, k),Two special types of nodes:,Source nodes(源结点): nodes that have no entering branches, which are used to represent the injection of external inputs or signal sources into a graph. Sink nodes(汇结点): nodes that have only entering branches, which are used to extract outputs from a graph.,Loop(环),self-loop(自环),forward path (前向通路),mixed node(混合结点),Example 9. 18 Consider again the system in example 6.17. Use signal flow graph to represent it.,Masons Formula (梅森规则):,Here is the graph determinant(特征行列式).,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:,is 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.,Example 9. 19 Compute the transfer functions between nodes A and B, and nodes A and C in the following signal flow graph.,A to B:,After removing G1 , the left graph is:,Thus,A to C:,After removing G1 , the left graph is:,Thus,9.9 THE UNILATERAL LAPLACE TRANSFORM,bilateral Laplace transform: (双边拉普拉斯变换),unilateral Laplace transform: (单边拉普拉斯变换),The 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),The bilateral transform depends on the entire signal from t = to t = +, whereas the unilateral transform depends only on the signal from to .,The bilateral transform and the unilateral transform of a causal signal are identical.,The ROC for the unilateral transform is always a right-half plane.,Example 9.20 Consider the signal,The bilateral transform X(s) for this example can be obtained from Example 9.1 and the time-shifting property:,By contrast, the unilateral transform is,The 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.,In fact, we could recognize as the bilateral transform of x(t)u(t).,Since,and,Thus,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 Res 1.,Thus,unilateral Laplace transform provide us with information about signals only for,Example 9.22 Consider the unilateral Laplace transform,Taking inverse transforms of each term results in,9.9.1 Properties of the Unilateral Laplace Transform,Time scaling:,Convolution: assuming that x1(t) and x2(t) are identically zero for t 0.,Differentiation in the time domain :,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:,9.9.2 Solving Differential Equations Using the Unilateral Laplace Transform,Applying the unilateral transform to both sides of the differential equation, we obtain,or equivalently,Thus, we obtain,The 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).,9.9.3 Representation of Circuits in s-domain,Apply unilate