Mathematical Foundations - Control of Color Imaging Systems: Analysis and Design - page 112

Im( z )

ROC of X ( z )

a <| z |< a -1

a

x x

O Re( z )

a -1

FIGURE 3.4 Pole - zero plot and ROC of Example 3.12.

S OLUTION

1 X

1

1 X

1

1

x(n)z n (a) n z n a n z n (az) n (az 1 ) n

X(z) ¼ ¼ þ ¼ þ

n¼1 n¼1 n¼ 0 n¼1 n¼ 0

This can be simpli ed as

1 1 1 1

(az) n (az 1 ) n (az) n (az 1 ) n X(z) ¼ þ ¼ 1 þ þ

n¼ 1 n¼ 0 n¼ 0 n¼ 0

1 1

¼ 1 þ

az þ

(3 : 47)

1 1 az 1

or

z(z þ a a 1 )
(z a)(z a 1 )

X(z) ¼

For convergence of X(z), both sums in Equation 3.47 must converge. This requires
that jaz 1 jzj < a 1 and jazj < 1 and j < 1, or equivalently, jzj > a. Therefore, the

ROC of X(z)isa < jzj < a 1 . The pole - zero plot as well as ROC of X(z) is shown in

Figure 3.4.

The z-transforms of some elementary functions and their ROCs are listed in
Table 3.3.

3.5.1 P ROPERTIES OF Z -T RANSFORM
Most common properties of z-transform are listed below:

(a) Linearity : z-Transform is a linear transform, that is, if

Z

x 1 ( n ) ! X 1 ( z ) , ROC ¼ R 1 ( 3 : 48 )