18_Z_Transforms

18_Z_Transforms - Chapter 18 The z-Transform and the...

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685 Chapter 18 The z -Transform and the Discrete-Time Fourier Transform Chapter Outline 18.1 THE TWO-SIDED z -TRANSFORM. ..................................................... 686 18.1.1 Definition. ................................................................................................................................... 686 18.1.2 Region of Convergence . ...................................................................................................... 688 18.1.3 MATLAB Experiments. ....................................................................................................... 692 18.2 PROPERTIES OF THE TWO-SIDED z -TRANSFORM. ......................... 693 18.2.1 Introduction. ............................................................................................................................... 693 18.2.2 Basic Properties. ..................................................................................................................... 693 18.2.3 Inverse z -Transforms. ............................................................................................................. 695 18.2.4 MATLAB Experiments. 699 18.3 THE ONE-SIDED z -TRANSFORM. ...................................................... 700 18.3.1 Introduction. 700 18.3.2 Definition of the One-Sided z -Transform. .................................................................... 700 18.3.3 Properties of the One-Sided z -Transform. 701 18.3.4 Solution of Difference Equations in Advance Form . ............................................. 702 18.4 DISCRETE-TIME FOURIER TRANSFORM. ....................................... 703 18.4.1 Introduction. 703 18.4.2 Definition. 703 18.4.3 Properties of the DTFT . 705 18.4.4 The Generalized DTFT. 707 18.4.5 Relationship to the z -Transform. ...................................................................................... 710 18.5 CHAPTER SUMMARY. ....................................................................... 711 18.6 HOMEWORK FOR CHAPTER 18. 716 The analysis and design of continuous signals and systems depends heavily on the Fourier and Laplace transforms. We would like to have similar tools available for the analysis and design of discrete signals and systems. It is the purpose of this chapter to develop two transforms that play the role of the Laplace and Fourier transforms for discrete-time signals and systems. The first important transform is the two-sided z -transform. A number of basic properties of the z -transform are discussed. The one-sided z -transform is also presented. It is shown how to solve difference equations using the one-sided z -transform. There is a striking resemblance between the z -transform and the Laplace transform. The concepts that underlie these two mathematical objects are the same, and there are parallels between many of their properties. The presentation
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686 Chapter 18 The z -Transform and the Discrete-Time Fourier Transform of the z -transforms is made with this observation in mind; it parallels the presentation of the Laplace transform in Chapter 9. The second important transform is the discrete-time Fourier transform (DTFT). This transform is defined in this chapter and a number of its important properties are introduced. The DTFT plays the role of the Fourier transform for discrete-time signals and systems. This transform allows us to define the amplitude and phase spectrum and the energy or power spectrum of a discrete-time signal (Chapter 20). This transform also ultimately leads to a method for computing the Fourier transform from the samples of the signal (Chapter 20). The motivation of this chapter is to present compactly some mathematical tools we will need for discrete-time signal and system analysis. The discussion of discrete-time signals and systems begins in earnest in the next chapter. Summary of Sections Section 18.1: We introduce the two-sided z -transform. Section 18.2: We develop the basic properties of the z -transform. Section 18.3: We discuss the one-sided z -transform and its application to the solution of difference equations.
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This note was uploaded on 09/10/2009 for the course ECE 60367 taught by Professor Meehan during the Spring '09 term at Virginia Tech.

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18_Z_Transforms - Chapter 18 The z-Transform and the...

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