The z-Transform - Digital Signal Processing: Fundamentals and Applications - page 140

n

c. xðnÞ¼ð0:5Þ uðnÞ

n

d. xðnÞ¼ð0:5Þ sinð0:25pnÞuðnÞ

e. xðnÞ¼e 0:1n cosð0:25pnÞuðnÞ

Solution:
a. From Line 3 in Table 5.1 , we get

10z
z 1

X ðzÞ¼Zð10uðnÞÞ ¼

b. Line 9 in Table 5.1 leads to

X ðzÞ ¼ 10Zðsinð0:2pnÞuðnÞÞ

10sinð0:25pÞz
z 2 2zcos 0:25p þ 1 ¼

7:07z
z 2 1:414z þ 1

¼

c. From Line 6 in Table 5.1 , we obtain
X ðzÞ¼Z ð0:5Þ

n u n ¼

z
z 0:5

d. From Line 11 in Table 5.1 , it follows that

X ðzÞ¼Z ð0:5Þ sin 0:25pn u n ¼

0:5 sinð0:25pÞz
z 2 2 0:5 cos 0:25p z þ 0:5 2

n

0:3536z
z 2 1:4142z þ 0:25

¼

e. From Line 14 in Table 5.1 , it follows that

z z e 0:1 cos 0:25p
z 2 2e 0:1 cos 0:25p z þ e 0:2

X ðzÞ¼Z e 0:1n cos 0:25pn uðnÞ ¼

zðz 0:6397Þ
z 2 1:2794z þ 0:8187

¼

5.2 PROPERTIES OF THE Z-TRANSFORM
In this section, we study some important properties of the z-transform. These properties are widely
used in deriving the z-transfer functions of difference equations and solving the system output
responses of linear digital systems with constant system coefficients, which will be discussed in the
next chapter.
Linearity: The z-transform is a linear transformation, which implies
Zðax 1 ðnÞþbx 2 ðnÞÞ ¼ aZðx 1 ðnÞÞ þ bZðx 2 ðnÞÞ (5.2)

where x 1 ðnÞ and x 2 ðnÞ denote the sampled sequences, while a and b are the arbitrary constants.