The z-Transform - Digital Signal Processing: Fundamentals and Applications - page 158

5.7. Given two sequences

x 1 ðnÞ¼ 2 dðnÞþ 5 dðn 2 Þ and x 2 ðnÞ¼ 4 dðn 4 Þ

a. determine the z-transform of convolution of the two sequences using the convolution
property of z-transform

XðzÞ¼X 1 ðzÞX 2 ðzÞ

b. determine the convolution by the inverse z-transform
x n ¼ Z 1 ðX 1 ðzÞX 2 ðzÞÞ

from the result in (a).
5.8. Using Table 5.1 and z-transform properties, find the inverse z-transform for each of the
following functions:

7 z
z þ 1

3 z
z 0 : 5

a. XðzÞ¼ 5

b. XðzÞ¼ 3 z

8 z
ðz 0 : 8 Þ þ

2 z
ðz 0 : 8 Þ

ðz 0 : 5 Þ þ

2

3 z

c. XðzÞ¼

2

z þ 1 : 414 z þ 1

þ z 3
z 0 : 75
5.9. Using the partial fraction expansion method, find the inverse of the following z-transforms:

5 z 5
z 1

z 2
ðz 1 Þ

2 þ z 10

d. XðzÞ¼

1

a. XðzÞ¼

0 : 3 z 0 : 24
b. XðzÞ¼ z
ðz 0 : 2 Þðz þ 0 : 4 Þ
c. XðzÞ¼ z
ðz þ 0 : 2 Þðz

z

2

2

z þ 0 : 5 Þ

zðz þ 0 : 5 Þ
ðz 0 : 1 Þ

d. XðzÞ¼

2

ðz 0 : 6 Þ
5.10. A system is described by the difference equation

yðnÞþ 0 : 5 yðn 1 Þ¼ 2 ð 0 : 8 Þ n uðnÞ

Determine the solution when the initial condition is yð 1 Þ¼ 2.

5.11. Using the partial fraction expansion method, find the inverse of the following z-transforms:

1

a. XðzÞ¼

z

2

þ 0 : 2 z þ 0 : 2