The z-Transform - Digital Signal Processing: Fundamentals and Applications - page 138

performing inverse z-transform (which we shall study later), we are restricted to the causal sequence.
Now let us study the following typical examples.

EXAMPLE 5.1
Given the sequence

xðnÞ¼uðnÞ

find the z-transform of xðnÞ.
Solution:
From the definition of Equation (5.1) , the z-transform is given by

z 1 n z 1 z 1 2

N N

uðnÞz n X ðzÞ¼ ¼ ¼ 1 þ þ þ/

n ¼0 n ¼0

This is an infinite geometric series that converges to

z
z 1

X ðzÞ¼

1
1 r when jr j<1.

with a condition jz 1 j<1. Note that for an infinite geometric series, we have 1 þ r þ r 2 þ/ ¼

The region of convergence for all values ofz is given as jzj >1.

EXAMPLE 5.2
Consider the exponential sequence

xðnÞ¼a n uðnÞ

and find the z-transform of the sequence xðnÞ.
Solution:
From the definition of the z-transform in Equation (5.1) , it follows that

az 1 n az 1 az 1 2

X ðzÞ¼ N
n ¼0

¼ N
n ¼0

a n uðnÞz n

¼ 1 þ þ þ/

Since this is a geometric series that will converge for jaz 1 j<1, it is further expressed as

z
z a ;

X ðzÞ¼ for jzj > jaj

The z-transforms for common sequences are summarized in Table 5.1 . Example 5.3 illustrates how
to find the z-transform using Table 5.1 .