This preview shows pages 1–3. Sign up to view the full content.
685
Chapter 18
The
z
Transform and the DiscreteTime
Fourier Transform
Chapter Outline
18.1 THE TWOSIDED
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 TWOSIDED
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 ONESIDED
z
TRANSFORM.
......................................................
700
18.3.1 Introduction.
700
18.3.2 Definition of the OneSided
z
Transform.
....................................................................
700
18.3.3 Properties of the OneSided
z
Transform.
701
18.3.4 Solution of Difference Equations in Advance Form .
.............................................
702
18.4 DISCRETETIME 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 discretetime signals and systems.
The first important transform is the twosided
z
transform.
A number of basic
properties of the
z
transform are discussed.
The onesided
z
transform is also
presented.
It is shown how to solve difference equations using the onesided
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 686
Chapter 18
The
z
Transform and the DiscreteTime 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 discretetime 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 discretetime
signals and systems.
This transform allows us to define the amplitude and phase
spectrum and the energy or power spectrum of a discretetime 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 discretetime signal and system analysis.
The discussion of
discretetime signals and systems begins in earnest in the next chapter.
Summary of Sections
Section 18.1:
We introduce the twosided
z
transform.
Section 18.2:
We develop the basic properties of the
z
transform.
Section 18.3:
We discuss the onesided
z
transform and its application to the
solution of difference equations.
This is the end of the preview. Sign up
to
access the rest of the document.
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.
 Spring '09
 MEEHAN

Click to edit the document details