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Unformatted text preview: 4 z-Transform 4.1 Introduction 4 . 2 z-T r a n s f o r m 4.3 The Z-Plane and the Unit Circle 4.4 Properties of the z-transform 4.5 Transfer Function, Poles and Zeroes 4.6 Physical Interpretation of Poles and Zeroes 4.7 The Inverse z-transform -transform, like the Laplace transform, is an indispensable mathematical tool for the design, analysis and monitoring of systems. The z-transform is the discrete-time counter-part of the Laplace transform and a generalisation of the Fourier transform of a sampled signal. Like Laplace transform the z-transform allows insight into the transient behaviour, the steady state behaviour, and the stability of discrete-time systems. A working knowledge of the z-transform is essential to the study of digital filters and systems. This chapter begins with the definition of the Laplace transform and the derivation of the z-transform from the Laplace transform of a discrete-time signal. A useful aspect of the Laplace and the z-transforms are the representation of a system in terms of the locations of the poles and the zeros of the system transfer function in a complex plane. In this chapter we derive the so-called z-plane, and its associated unit circle, from sampling the s-plane of the Laplace transform. We study the description of a system in terms the system transfer function. The roots of the transfer function, the so-called poles and zeros of transfer function, provide useful insight into the behaviour of a system. Several examples illustrating the physical significance of poles and zeros and their effect on the impulse and frequency response of a system are considered. x ( m – k ) x ( m ) z – k Z Z Z Z Z Z Z Z Z Z Z Z Sec. 4.1 Introduction 2 4.1 Introduction The Laplace transform and its discrete-time counterpart the z-transform are essential mathematical tools for system design and analysis, and for monitoring the stability of a system. A working knowledge of the z-transform is essential to the study of discrete-time filters and systems. It is through the use of these transforms that we formulate a closed-form mathematical description of a system in the frequency domain, design the system, and then analyse the stability, the transient response and the steady state characteristics of the system. A mathematical description of the input-output relation of a system can be formulated either in the time domain or in the frequency domain. Time-domain and frequency domain representation methods offer alternative insights into a system, and depending on the application it may be more convenient to use one method in preference to the other. Time domain system analysis methods are based on differential equations which describe the system output as a weighted combination of the differentials (i.e. the rates of change) of the system input and output signals. Frequency domain methods, mainly the Laplace transform, the Fourier transform, and the z-transform, describe a system in terms of its response to the individual frequency constituents of the input signal. In terms of its response to the individual frequency constituents of the input signal....
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This note was uploaded on 10/16/2011 for the course MECHATRONI 111 taught by Professor Jung during the Spring '11 term at Hanyang University.
- Spring '11