Chapter 9 The Laplace Transform

§9 §4 Complex Frequency Domain Analysis of LTI System Chapter 9The Laplace Transform • Representation of Aperiodic Signals • LTI System’s Response to Aperiodic Signals • LTI System Description and Performance • LTI System’s Full Response

Problems: 2, 5, 7, 8, 9, 13, 21(a b i j), 22(a b c), 28, 31, 32, 33, 35, 34

9.0 Introduction F Key : • Condition Fourier Transform A. Fourier Transform and Its Limitation are satisfied by some kind of signals • Problems Signals not satisfy the condition

Basic signal: represents Basic signal: represents B. Cause × Condition not satisfied C. Solution

9.1 The Laplace Transform Pairs When We have 反正 2 1 1 2 反正 9.1.1 The Transform Pairs

A. Inverse Transform Integral line for Inverse Transform 1 1 t 9.1.2 Understanding of Laplace Transform

if 2 2 and Similarity of Similarity of and B. Transform or

<example> <9.1> Region of Convergence <9.2> Condition： L <9.1> 9.1.3 Convergence Region of Laplace Transform and

L If convergence region contains axis F 9.1.4. Relation between Laplace and Fourier Transform

generally 9.2 The Region of Convergence for Laplace Transform 9.2.1 Region of Convergence A. Understanding ROC by Convergence Condition “In this area， is defined” Condition：

Convergence Region 1 is defined in this area Integral line for Inverse Transform B. Understanding ROCby Inverse Transform

<9.1> 1 0 Convergence Condition for Fourier Transform 1 F 0 C. Example of Convergence Region < Proof >

Convergence Region Convergence Region ROC: Contains axis Integral line Integral line Not allowed doesn't exist Doesn’t Contains axis ROC: As proved in above :

F <9.2> < Proof >

ROC: Contains axis Integral line Integral line Not allowed doesn't exist Doesn’t Contains axis ROC: As proved in above : ROC ROC

<9.1> <9.2> Integral line Integral line D. The Importance of Convergence Region <9.1+9.2>

Pole ： Root of the numerator Convergence Region L （可证） Zero: Root of the numerator <9.4> Convergence Region L （可证） 9.2.2 Zeros and Poles of Laplace Transform <9.3>

L （可证） Rational Laplace Transform ROC <9.5>

1. ROC: ROC ROC: bounded by poles Example <9.3> At poles, Doesn't exist ROC 1 Signal 1 ROC 2 Signal 2 ROC 3 Signal 3 9.2.3. General Rule for Convergence Region 2. Rational Laplace Transform

L <Example> L L and ROC 3. Finite Duration Signals ROC：entire s-plane

Right sided signal Signal: right sided ：Right Plane 4. Right Plane

Signal: left sided ：Left Plane 5. Left sided signal Left Plane

left Signal: two-sided 6.

①③②④ ② ① ③ ④ ROC ③ ④ ① ② × Laplace Transform √ Laplace Transform <9.7> 由<9.1>、<9.2>： ROC ？

1. Strips; 2. Rational L-Transform: 1 2 4 3 7. Rational L-Transform ROC: bounded by poles left sided signal two sided signal two sided signal right sided signal ① ~ ④Corresponding to different signals

rightmost pole rightmost pole <Example> Signal: right sided 8. Rational L-Transform

leftmost pole leftmost pole Signal: left sided 9. Rational L-Transform

zero pole 9.4 Geometric Evaluation of Fourier Transform from the Pole-Zero Plot 9.4.1 Geometric Evaluation of Laplace Transform from the Pole-Zero Plot A. The method 零点相位极点相位

2. Example

as above, if , let 9.4.2. Geometric Evaluation of Fourier Transform from the Pole-Zero Plot A. The method

B. Example

9.5.1. Linearity ② 零极相消，则会扩大 9.5 Properties of The Laplace Transform 收敛域： ① 一般情况，为交叠区实用判别法：由合成零极点图/信号特征确定

右右 <Example> 右

9.5.2. Time Shifting <proof>

9.5.4. Time Scaling 9.5.3. Shifting in S-Domain

缩伸 <Proof> <Example : >

R <9.1> <9.2> -R * Special case for a = -1 <Example> 原点反折

9.5.5. Convolution Property Usage of the Property:

<Example>Special case

9.5.6. Differential in the Time-Domain <Example> Special case

9.5.7. Differential in the S-Domain <Example>

9.5.8. Integration in the Time-Domain

②contains noor orexist ① Where Where ② 9.5.9. The Initial- and Final-value Theorems If ①( causal signal ) Then

From the theorem, we have <Example> 验证Laplace变换的正确性 In<9.4>

A. Contour Integral 围线积分、留数定理 (复杂） B. Partial-Fraction Expansion Integral line in ROC 部分分式展开 (简单） (大多数情况) < < 9.3 The Inverse Laplace Transform 9.3.1 General Calculation Method

m 阶多项式 n 阶多项式 Partial-Fraction Expansion 多项式 + 9.3.2 Partial-Fraction Expansion for Rational Function ( see Appendix ) A. Fundamental Idea

Basic Transform Pairs for Partial-Fraction Expansion

B. Division

① Where C. Partial-Fraction Extension

<Example>

Chapter 9 The Laplace Transform

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