Chapter-3-z-Transform-Student

# Chapter-3-z-Transform-Student - Chapter 3 z-Transform...

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ELEC 215: Tim Woo Spring 2009/10 Chapter 3 - 1 Chapter 3: z-Transform Spring 2009/10 Lecture: Tim Woo

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ELEC 215: Tim Woo Spring 2009/10 Chapter 3 - 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 3 - 3 Expected Outcome • In this chapter, you will be able to – Calculate the (Bilateral) z-transform of time-domain signals – Summarize the properties of z-transform – Analyze the characteristics of signals and system from the information of z-plane without evaluating the inverse of z- transform. – Calculate the inverse z-transform associates with different Regions of Convergence. – Compare the conditions between Bilateral z-transform and Unilateral z-transform – Summarize the properties of Unilateral z-transform

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ELEC 215: Tim Woo Spring 2009/10 Chapter 3 - 4 Outline • Section 10.0 Introduction • Section 10.1 The z-transform • Section 10.2 The Region of Convergence for z-transform • Section 10.5 Properties of the z-transform • Section 10.6 Some z-transform Pairs • Section 10.3 The Inverse z-transform • Section 10.9 The Unilateral z-transform
ELEC 215: Tim Woo Spring 2009/10 Chapter 3 - 5 10.0 Introduction • In this chapter, we use the same approach for discrete time as we develop z-transform, which is the discrete- time counterpart of the Laplace transform. • The motivations for and properties of the z-transform closely parallel those of the Laplace transform. • However, we will encounter some important distinctions between the z-transform and the Laplace transform that arise from the fundamental differences between continuous-time and discrete-time signals and systems.

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ELEC 215: Tim Woo Spring 2009/10 Chapter 3 - 6 10.1 The z-Transform Definition : The z-transform of a signal x [ n ] is given by Note that if z = e j ω (i.e. | z | = 1 ), we have the discrete-time Fourier transform (DTFT). In general, z is a continuous complex variable and we can write it as z = r e j , where r = | z | > 0 is the magnitude and ω is the angle of z which is just the DTFT of x [ n ] r -n . } ] [ { ) ] [ ( ) ]( [ ) ( ) ( n n n j n n n j j r n x F e r n x e r n x e r X z X −∞ = −∞ = = = = = ) ( ] [ z X n x Z −∞ = = n n z n x z X ] [ ) ( )} ( { ] [ ]} [ { ) ( 1 z X Z n x and n x Z z X = = The z-transform evaluated on the unit circle (along ω varies from 0 to 2 π ) corresponds to the DTFT of x [ n ].
ELEC 215: Tim Woo Spring 2009/10 Chapter 3 - 7 10.1 The z-Transform •S i n c e z can be any point on the z-plane, generally there exists some z which makes or X ( z ) does not converge.

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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-3-z-Transform-Student - Chapter 3 z-Transform...

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