Page 176 - Schaum's Outline of Theory and Problems of Signals and Systems

Chapter 4

The z-Transform and Discrete-Time

LTI Systems

4.1 INTRODUCTION
In Chap. 3 we introduced the Laplace transform. In this chapter we present the
z-transform, which is the discrete-time counterpart of the Laplace transform. The z-trans-
form is introduced to represent discrete-time signals (or sequences) in the z-domain (z is a
complex variable), and the concept of the system function for a discrete-time LTI system
will be described. The Laplace transform converts integrodifferential equations into
algebraic equations. In a similar manner, the z-transform converts difference equations
into algebraic equations, thereby simplifying the analysis of discrete-time systems.
The properties of the z-transform closely parallel those of the Laplace transform.
However, we will see some important distinctions between the z-transform and the
Laplace transform.

4.2 THE Z-TRANSFORM
In Sec. 2.8 we saw that for a discrete-time LTI system with impulse response h[n], the
output y[n] of the system to the complex exponential input of the form z" is

where

A. Definition:

The function H(z) in Eq. (4.2) is referred to as the z-transform of h[n]. For a general
discrete-time signal x[n], the z-transform X(z) is defined as
m
X(Z) x[n]z-" (4.3)
=
n= -OD
The variable z is generally complex-valued and is expressed in polar form as

where r is the magnitude of z and R is the angle of z. The z-transform defined in Eq.
(4.3) is often called the bilateral (or two-sided) z-transform in contrast to the unilateral (or