电子科大信号系统英文版课件chapter9 laplacetransform-b

,9 The Laplace Transform,9. The Laplace Transform,9.1 The Laplace Transform,(1) Definition,(2) Region of Convergence ( ROC ),ROC: Range of for X(s) to convergeRepresentation: A. Inequality B. Region in S-plane,9 The Laplace Transform,Example for ROC,9 The Laplace Transform,(3) Relationship between Fourier and Laplace transform,Example 9.1 9.2 9.3 9.5,9 The Laplace Transform,9.2 The Region of Convergence for Laplace Transform,Property1: The ROC of X(s) consists of strips parallel to j-axis in the s-plane.Property2: For rational Laplace transform, the ROC does not contain any poles.Property3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s- plane,9 The Laplace Transform,Property4: If x(t) is right sided, and if the line Res=0 is in the ROC, then all values of s for which Res0 will also in the ROC.,9 The Laplace Transform,Property5: If x(t) is left sided, and if the line Res=0 is in the ROC, then all values of s for which Res0 will also in the ROC.,9 The Laplace Transform,Property6: 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 .,9 The Laplace Transform,9 The Laplace Transform,Property7: 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.Property8: 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 9.7 9.8,9 The Laplace Transform,Appendix Partial Fraction Expansion,Consider a fraction polynomial:,Discuss two cases of D(s)=0, for distinct root and same root.,9 The Laplace Transform,(1) Distinct root:,thus,9 The Laplace Transform,Calculate A1 : Multiply two sides by (s-1):,Let s=1, so,Generally,9 The Laplace Transform,(2) Same root:,thus,For first order poles:,9 The Laplace Transform,Multiply two sides by (s-1)r :,For r-order poles:,So,9 The Laplace Transform,9.3 The Inverse Laplace Transform,So,9 The Laplace Transform,The calculation for inverse Laplace transform:(1) Integration of complex function by equation.(2) Compute by Fraction expansion. General form of X(s):,Important transform pair:,Example 9.9 9.10 9.11,9 The Laplace Transform,9.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot,General form of X(s):,Relation between Fourier and Laplace transform:,9 The Laplace Transform,or,9 The Laplace Transform,9.4.1 First-order System,Pole-zero plot:,System function of first-order system:,9 The Laplace Transform,9 The Laplace Transform,9.4.2 All-Pass System,Pole-zero plot:,System function :,9 The Laplace Transform,Frequency response :,9 The Laplace Transform,9.5 Properties of the Laplace Transform,9.5.1 Linearity of the Laplace Transform,Example 9.13,9 The Laplace Transform,9.5.2 Time Shifting,9 The Laplace Transform,9.5.3 Shifting in the s-Domain,9 The Laplace Transform,9 The Laplace Transform,9.5.4 Time Scaling,Especially,9 The Laplace Transform,9.5.5 Conjugation,When x(t) is real, X(s)=X*(s*),9.5.6 Convolution Property,9 The Laplace Transform,9.5.7 Differentiation in the Time Domain,9 The Laplace Transform,9.5.8 Differentiation in the s-Domain,Example 9.14 9.15,9 The Laplace Transform,9.5.9 Integration in the Time-Domain,Under the specific constrains that x(t)=0 for t0 contains no impulses or highter order singularities at the origin,Initial-value theorem:Final-value theorem:,9 The Laplace Transform,9.5.10 The Initial- and Final-Value Theorems,Example 9.16,Page 691: Table 9.1,9 The Laplace Transform,9.5.11 Table of Properties,Page 692: Table 9.2,9.6 Some Laplace Transform Pairs,9 The Laplace Transform,9.7 Analysis and Characterization of LTI Systems Using the Laplace Transform,System output: Y(s)=H(s)X(s),LTI system,x(t),y(t),H(s) - System function ( Transfer/transition function ),9 The Laplace Transform,9.7.1 Causality,(1) The ROC associated with the system function for a causal system is a right-half plane.(2) 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.,Causal LTI system: h(t)=0 for t0 .,Example 9.17 9.18 9.19,9 The Laplace Transform,9.7.2 Stability,(1) An LTI system is stable if and only if the ROC of its system function H(s) includes the j-axis.(2) A causal system with rational system function H(s) is stable if 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.,Example 9.20 9.21,9 The Laplace Transform,9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential Equations,Differential equation:,Example 9.24,so,9 The Laplace Transform,9.7.4 Examples Relating System Behavior to the System Function,Example 9.25 9.26,9 The Laplace Transform,9.8 System Function Algebra and Block Diagram Representation,9.8.1 System Function for Interconnections of LTI Systems,9 The Laplace Transform,(1) Parallel interconnection,For overall system: h(t)=h1(t)+h2(t) and H(s)=H1(s)+H2(s),9 The Laplace Transform,(2) Series interconnection,For overall system: h(t)=h1(t)*h2(t) and H(s)=H1(s)H2(s),9 The Laplace Transform,(3) Feedback interconnection,For overall system:,9 The Laplace Transform,9.8.2 Block Diagram Representations for Causal LTI Systems Described by Differential Equations and Rational System Functions,Basic elements: (1) Integrator (2) Amplifier (3) Adder,9 The Laplace Transform,Block Diagram construction: (1) Direct form (2) Parallel form : H(s) = H1(s) + H2(s) (3) Series form : H(s) = H1(s) H2(s),Exam