Digital Signal Processing Reference
In-Depth Information
The
M
roots of the numerator polynomial
c
1
c
2
…
c
M
are the
zeros
, and the
N
roots of the denominator polynomial
d
1
d
2
…
d
N
are the
poles
of the system
function.
The poles of the system function define the
stability
of the system in the
complex
z
-plane. If all the poles of the system function lie inside the unit
circle, hence satisfying
<
1, for
k =
1, 2
, … N
, then the system is
uncondi-
tionally stable
. For an example system function
d
k
(
z
−
2
)
Hz
()
=
zz
(
−
05
. )(
z
−
08
. )
the poles, or the roots of the denominator polynomial, are located at
z =
0,
0.5 and 0.8. Since all the poles lie inside the unit circle, the system
H
(
z
) is
unconditionally stable.
2.1.2
System Frequency Response
H
(
e
j
ω
)
The frequency response of the system is very important to define the practical
property of the system, such as low-pass or high-pass filtering. It can be
obtained by considering the system function
H
(
z
) on the unit circle shown
in Figure 2.1, which corresponds to the
z =
e
j
ω
circle. Hence, from Equation 2.7,
the frequency response is:
j
ω
He
()
=
Hz
()
(2.10)
ze
j
ω
=
j
ω
Unit circle
|z|
= 1
σ
ω
FIGURE 2.1
Complex
z-
plane with the unit circle.
