Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 72

which is described by the unit-impulse response

n uðnÞ

hðnÞ¼ð0:25Þ

determine whether this system is stable.
Solution:
Using Equation (3.18) , we have

ð0:25Þ

S ¼ N
k ¼ N

jhðkÞj ¼ N
k ¼ N

k uðkÞ

Applying the definition of the unit-step function uðkÞ¼1 for k 0, we have

N

k

¼ 1 þ 0:25 þ 0:25 2 þ /

S ¼ ð0:25Þ

k ¼0

Using the formula for a sum of the geometric series (see Appendix F),
N

1
1 a ;

a k

¼

k ¼0

where a ¼ 0:25 < 1, we conclude

1 0:25 ¼ 4

1

S ¼ 1 þ 0:25 þ 0:25 2 þ / ¼

3 < N

Since the summation is a finite number, the linear system is stable.

3.5 DIGITAL CONVOLUTION
Digital convolution plays an important role in digital filtering. As we verified in the last section,
a linear time-invariant system can be represented using a digital convolution sum. Given a linear time-
invariant system, we can determine its unit-impulse response hðnÞ , which relates the system input and
output. To find the output sequence yðnÞ for any input sequence xðnÞ , we write the digital convolution
shown in Equation (3.15) as

yðnÞ¼ N

k¼ N hðkÞxðn kÞ
¼ / þ hð 1 Þxðn þ 1 Þþhð 0 ÞxðnÞþhð 1 Þxðn 1 Þþhð 2 Þxðn 2 Þþ /

(3.19)

The sequences hðkÞ and xðkÞ in Equation (3.19) are interchangeable. Hence, we have an alternative
form:

yðnÞ¼ N

k¼ N xðkÞhðn kÞ
¼ / þ xð 1 Þhðn þ 1 Þþxð 0 ÞhðnÞþxð 1 Þhðn 1 Þþxð 2 Þhðn 2 Þþ /

(3.20)