Digital Signal Processing Systems, Basic Filtering Types, and Digital Filter Realizations - Digital Signal Processing: Fundamentals and Applications - page 170

If we know the transfer function HðzÞ and z-transform of the input XðzÞ , we are able to determine the
system response yðnÞ by finding the inverse z-transform of the output YðzÞ :
yðnÞ¼Z 1 fYðzÞg (6.11)

EXAMPLE 6.7
Given a transfer function depicting a DSP system

z þ 1
z 0:5

HðzÞ¼

determine
a. the impulse response hðnÞ,

b. step response yðnÞ, and

n uðnÞ.

c. system response yðnÞ if the input is given as xðnÞ¼ð0:25Þ

Solution:
a. The transfer function can be rewritten as
H ð z z ¼ z þ 1

zðz 0:5Þ ¼

A
z þ

B
z 0:5

where

z¼0 ¼2 and B ¼ z¼0:5 ¼ 3

z þ 1
ðz 0:5Þ

z þ 1
z

A ¼

Thus we have

H ð z z ¼ z þ 3

z 0:5

and

z ¼2 þ

z þ

3
z 0:5

3z
z 0:5

HðzÞ¼

By taking the inverse z-transform as shown in Equation (6.8) , we yield the impulse response

n uðnÞ

hðnÞ¼2dðnÞþ3ð0:5Þ

z
z 1 , we can determine the z-transform of the step

b. For the step input xðnÞ¼uðnÞ and its z-transform X ðzÞ¼

response as

z þ 1
z 0:5

z
z 1

Y ðzÞ¼HðzÞX ðzÞ¼

Applying the partial fraction expansion leads to
Y ð z z ¼

z þ 1
ðz 0:5Þðz 1Þ ¼

A
z 0:5 þ

B
z 1