Chapter-4-Analysis-and-characterization-of-LTI-systems-Student

# Chapter-4-Analysis-and-characterization-of-LTI-systems-Student

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ELEC 215: Tim Woo Spring 2009/10 Chapter 4 - 1 Chapter 4: Analysis and characterization of LTI systems Spring 2009/10 Lecture: Tim Woo

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ELEC 215: Tim Woo Spring 2009/10 Chapter 4 - 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 4 - 3 Expected Outcome • In this chapter, you will be able to – Identify several properties of LTI systems such as causality and stability from the knowledge of the pole locations and the ROC of s - and z -plane – Apply a suitable transform to obtain the system function for an LTI systems characterized by a linear constant-coefficient differential or difference equation – Analysis the behavior of the system function by using a suitable transform.

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ELEC 215: Tim Woo Spring 2009/10 Chapter 4 - 4 Outline • Section 9.7 Analysis and Characterization of LTI systems using the Laplace Transform • Section 10.7 Analysis and Characterization of LTI systems using the z-transforms
ELEC 215: Tim Woo Spring 2009/10 Chapter 4 - 5 Introduction One of the applications of the Laplace transform (or z-transform) is in the analysis and characterization of LTI systems. The convolution property states the transforms of the input and output of an LTI system are related through multiplication by the transform of the impulse response of the system. Thus, As discussed in Section 3.2 in ELEC 211, if we input an complex exponential signal of the form (or ) into an LTI system, then the output is n j n e z ω = () ) ( ) ( s X s H s Y = ) ( ) ( z X z H z Y = Continuous-Time System response Discrete-Time system response t j st e e = −∞ = = k k z k h z H ] [ ) ( n z z H n y ) ( ] [ = Continuous-Time System response Discrete-Time system response = τ d h s H ) ( ) ( st e s H t y ) ( ) ( = where where

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ELEC 215: Tim Woo Spring 2009/10 Chapter 4 - 6 Introduction • If the ROC of H ( s ) ( or H ( z ) ) contains the imaginary axis (or unit circle), then for ( or ) , H ( s ) (or H ( z ) ) is the frequency response of the continuous-time (or discrete-time) LTI system. • Many properties of LTI systems can be closely associated with the characteristics of the system function in the s -plane (or z -plane). ω j s = j e z =
ELEC 215: Tim Woo Spring 2009/10 Chapter 4 - 7 Introduction • In this chapter, we first discuss the analysis of causality and stability of a continuous-time LTI system. In addition, the design of stable and causal LTI system will also be addressed. • In the second part, we use the same approach for discrete time LTI system as we develop z-transform, which is the discrete-time counterpart of the Laplace transform.

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Chapter-4-Analysis-and-characterization-of-LTI-systems-Student

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