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Signals and Systems Lecture 24
The Laplace Transform ROC of Laplace Transform Inverse Laplace Transform
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Chapter 9 The Laplace Transform
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Chapter 9 The Laplace Transform
Defining ——Laplace Transform 1. The relationship
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Chapter 9 The Laplace Transform
2. Region of Convergence(收敛域) Dirichlet Condition 1 : ROC:对给定的 ,使其拉氏变换存在的 σ对应的 S平面上的区域。
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Chapter 9 The Laplace Transform
Example 9.1 pole-zero plot 零极点图 Example 9.2 pole-zero plot
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Chapter 9 The Laplace Transform
Particularly, The Fourier transform of does not exist.
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Chapter 9 The Laplace Transform
Example 9.3
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Chapter 9 The Laplace Transform
Example 9.3
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Chapter 9 The Laplace Transform
Example 9.4 entire S plane pole-zero plot does not exist.
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Chapter 9 The Laplace Transform
Zeros: Poles: 1. The direction of signals ROC of X(s) 2. The position of poles §9.2 The Properties of ROC Property 1: The ROC of X(s) consists of strips parallel to the jω-axis in the s-plane ——Depends only on σ
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Chapter 9 The Laplace Transform
Property 2: For rational Laplace transforms, the ROC does not contain any poles. Property 3: If is of finite duration and is absolutely integrable, then the ROC is the entire s-plane. ① when
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Chapter 9 The Laplace Transform
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Chapter 9 The Laplace Transform
Example 零极点 抵消 pole-zero plot pole: zero: zeros:
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Chapter 9 The Laplace Transform
Property 4: If is right sided, finite
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Chapter 9 The Laplace Transform
Property 5: If is left sided, Property 6: If is two sided, ROC:
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Chapter 9 The Laplace Transform
Example 9.7 If b>0, If b≤0, has no Laplace transform.
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Chapter 9 The Laplace Transform
Property 7: If the Laplace transform of is rational, ① is right sided, ② is left sided, Example 9.8 left sided right sided two sided
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Chapter 9 The Laplace Transform
Basic Laplace Pairs Poles ROC none
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Chapter 9 The Laplace Transform
1. The direction of signals 2. The position of poles ROC of X(s)
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Chapter 9 The Laplace Transform
1. 2. 3.
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Chapter 9 The Laplace Transform
§9.3 The Inverse Laplace Transform defining Example 9.9 Determine the inverse Laplace transform for all possible ROC.
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Summary The Laplace Transform and its RoC Inverse Laplace Transform
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Problem Set P (a) (b) P (a) (b)
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