Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Z Domain Analysis
H z)
=
( z + 0.8)( z
2
2
z ( z −1)(
z + 1)
,
−1.38593z
+ 0.9604)( z + 1.64545z
+ 0.9025) , find the zeros and poles.
(
2
2
Re-writing in polar form as:
2
π π
z ( z −1)(
z − exp
j j)(
z − exp
− j )
2 2
H(
z)
=
π
π
5
( z + 0.8) z − 0.98exp
j
z − 0.98exp
− j
z − 0.95exp
j π
z − 0.95exp
−
4
4
6
5
j π
6
The zeros and poles are easily to be found and described as:
5 zeros: ⎛ p ⎞
π 5
0, 0 ,1, exp⎜
± j ⎟ , and 5 poles: − 0.8,
0.98exp
± j , 0.95exp
± j π . .
⎝ 2 ⎠ 4 6
Any zeros or poles at the origin only produce a time advance or delay. A minimum-delay system requires:
The number of poles = the number of zeros
If zeros are more than poles, a system becomes non-causal. Some poles are needed to be added to change into causal.
Example 5.7: If a transfer function H
1
− 2rz
cosθ
+ r
−2
( z)
=
=
2
2
−1
2 −2
z
1−
2rz
z
cosθ
+ r z
, ,
It can be found
2
y[
n]
= 2r
cosq y[
n −1]
− r y[
n − 2] + x[
n − 2]
.
or
2
h[
n]
= 2r
cosq
h[
n −1]
− r h[
n − 2] + d[
n − 2]
Assuming the system is causal, h [ n]
= 0 when n < 0
h[0]
= 2r
cosq
h[
−1]
− r
h[1]
= 2r
cosq
h[0]
− r
2
2
h[
−2]
+ d[
−2]
= 0
h[
−1]
+ d[
−1]
= 0.
i.e., from the impulse response it can be seen that in the system, there is time delay of 2 samples. However, alternatively,
if the z-transform is given by
H
z
1
− 2rz
cosq + r 1−
2rz
cosq
+ r z ,
2
( z)
=
=
2
2
−1
2 −2
z
74
Download free ebooks at bookboon.com