Chapter-2-Laplace-Transform-Student

Chapter-2-Laplace-Transform-Student - ELEC 215: Tim Woo...

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Unformatted text preview: ELEC 215: Tim Woo Spring 2009/10 Chapter 2 - 1 Chapter 2: Laplace Transform Spring 2009/10 Lecture: Tim Woo ELEC 215: Tim Woo Spring 2009/10 Chapter 2 - 2 Laplace Transform Where we are Differential equations State-space model CTFT Hardware Implementation System Characteristics System Responses Closed-loop Systems Continuous-time z-Transform Difference equations State-space model DTFT Hardware Implementation System Characteristics System Responses Closed-loop Systems Discrete-time Mapping Done in 211 To be covered In progress Done Will be covered if available Open-loop Systems Open-loop Systems ELEC 215: Tim Woo Spring 2009/10 Chapter 2 - 3 Expected Outcome • In this chapter, you will be able to – Calculate the (Bilateral) Laplace transform of time-domain signals – Summarize the properties of Laplace transform – Analyze the characteristics of signals and system from the information of s-plane without evaluating the inverse of Laplace transform. – Calculate the inverse Laplace transform associates with different Regions of Convergence. – Compare the conditions between Bilateral Laplace Transform and Unilateral Laplace Transform – Summarize the properties of Unilateral Laplace Transform ELEC 215: Tim Woo Spring 2009/10 Chapter 2 - 4 Outline • Section 9.0 Introduction • Section 9.1 The Laplace Transform • Section 9.2 The Region of Convergence for Laplace Transforms • Section 9.5 Properties of the Laplace Transform • Section 9.6 Some Laplace Transform Pairs • Section 9.3 The Inverse Laplace Transform • Section 9.9 The Unilateral Laplace Transform ELEC 215: Tim Woo Spring 2009/10 Chapter 2 - 5 9.0 Introduction • Recall that Fourier transform is useful because a signal can be represented in terms of complex exponentials of the form through Fourier transform. • Many nice properties of the Fourier transform hold for the general complex exponentials • The generalization of the Fourier transform to the case of general complex exponentials leads to the Laplace Transform . t j e ω ω σ j s e st + = where st e ( ) ∫ ∞ ∞ − − = dt e t x j X t j ω ω ) ( ( ) ∫ ∞ ∞ − − = dt e t x s X st ) ( Continuous-Time Fourier Transform Laplace Transform ELEC 215: Tim Woo Spring 2009/10 Chapter 2 - 6 9.0 Introduction • The Laplace transform is also more broadly applicable. – Fourier Transform • The response of stable LTI system (convergence problem) – Says impulse response, step response, frequency response – Laplace Transform • The investigation of the stability or instability of systems • The response of both stable and unstable LTI system – Says impulse response, step response, frequency response • Analysis of feedback system • Since we’ve already studied Continuous-Time Fourier Transform, we will discuss this topic at a rather quick pace, focusing only on new concepts . ELEC 215: Tim Woo Spring 2009/10 Chapter 2 - 7 9.1 The Laplace Transform • The Laplace Transform of a signal x ( t ) is given by...
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This note was uploaded on 03/27/2011 for the course ELEC 215 taught by Professor Prof.kamtimwo during the Spring '11 term at HKUST.

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Chapter-2-Laplace-Transform-Student - ELEC 215: Tim Woo...

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