Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 84

3.24. Determine the stability for each of the following linear systems.
a. yðnÞ¼ P k ¼ 0 ð 1 : 5 Þ k xðn kÞ
b. yðnÞ¼ P k ¼ 0 ð 0 : 5 Þ k xðn kÞ
3.25. Given the sequence

8
<

2 ; k ¼ 0 ; 1 ; 2

hðkÞ¼ 1 ; k ¼ 3 ; 4

:

0 elsewhere;

where k is the time index or sample number,
a. sketch the sequence hðkÞ and the reverse sequence hðkÞ ;
b. sketch the shifted sequences hðk þ 2 Þ and hðk 3 Þ .
3.26. Given the sequence

8
<

1 k ¼ 0 ; 1

2 k ¼ 2 ; 3

hðkÞ¼

:

2 k ¼ 4

0 elsewhere

where k is the time index or sample number,
a. sketch the sequence hðkÞ and the reverse sequence hðkÞ ;
b. sketch the shifted sequences hðk þ 1 Þ and hðk 2 Þ .
3.27. Using the sequence definitions

8
<

8
<
:

2 ; k ¼ 0 ; 1 ; 2 2 ; k ¼ 0

hðkÞ¼ 1 ; k ¼ 3 ; 4 and xðkÞ¼ 1 ; k ¼ 1 ; 2

:

0 elsewhere 0 elsewhere

evaluate the digital convolution

N

yðnÞ¼ xðkÞhðn kÞ

k ¼ N

a. using the graphical method;

b. using the table method;

c. applying the convolution formula directly.

3.28. Using the sequence definitions

8
<

8
<

2 ; k ¼ 0 ; 1 ; 2 2 ; k ¼ 0

xðkÞ¼ 1 ; k ¼ 3 ; 4 and hðkÞ¼ 1 ; k ¼ 1 ; 2

: :

0 elsewhere 0 elsewhere