Page 177 - Schaum's Outline of Theory and Problems of Signals and Systems

166 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

one-sided) z-transform, which is defined as

Clearly the bilateral and unilateral z-transforms are equivalent only if x[n] = 0 for n < 0.
The unilateral z-transform is discussed in Sec. 4.8. We will omit the word "bilateral"
except where it is needed to avoid ambiguity.
As in the case of the Laplace transform, Eq. (4.3) is sometimes considered an operator
that transforms a sequence x[n] into a function X(z), symbolically represented by

The x[n] and X(z) are said to form a z-transform pair denoted as

B. The Region of Convergence:
As in the case of the Laplace transform, the range of values of the complex variable z
for which the z-transform converges is called the region of convergence. To illustrate the
z-transform and the associated ROC let us consider some examples.

EXAMPLE 4.1. Consider the sequence
x[n ] = a"u[n] a real

Then by Eq. (4.3) the z-transform of x[n] is

For the convergence of X(z) we require that

Thus, the ROC is the range of values of z for which laz -'I < 1 or, equivalently, lzl > lal. Then

Alternatively, by multiplying the numerator and denominator of Eq. (4.9) by z, we may write X(z) as
z
X(z) = - Izl > la1
z-a

Both forms of X(z) in Eqs. (4.9) and (4.10) are useful depending upon the application.
From Eq. (4.10) we see that X(z) is a rational function of z. Consequently, just as with
rational Laplace transforms, it can be characterized by its zeros (the roots of the numerator
polynomial) and its poles (the roots of the denominator polynomial). From Eq. (4.10) we see
that there is one zero at z = 0 and one pole at z =a. The ROC and the pole-zero plot for
this example are shown in Fig. 4-1. In z-transform applications, the complex plane is
commonly referred to as the z-plane.