Digital Signal Processing Reference
In-Depth Information
As an example of each expression, consider the three forms of a transfer
function that have a second-order numerator and third-order denominator:
2
6
⋅
(
s
+
0
66667
)
H
a
(
s
)
=
(2.4a)
3
2
s
+
3
1650
⋅
s
+
5
0081
⋅
s
+
4
0001
6
⋅
(
s
+
j
0
81650
)(
s
−
j
0
81650
)
H
b
(
s
)
=
(2.4b)
(
s
+
1
5975
)
⋅
(
s
+
0
7837
+
j
1
3747
)
⋅
(
s
+
0
7837
−
j
1
3747
)
2
6
.
0
⋅
(
s
+
0
.
66667
)
H
c
(
s
)
=
(2.4c)
2
(
s
+
1
.
5975
)
⋅
(
s
+
1
.
5675
⋅
s
+
2
.
5040
)
2.1.2 Pole-Zero Plots and Transfer Functions
When the quadratic form of the transfer function is used, it is easy to generate the
pole-zero plot for a particular transfer function. The pole-zero plot simply plots
the roots of the numerator (zeros) and the denominator (poles) on the complex
s-
plane. As an example, the pole-zero plot for the sample transfer function given in
(2.4) is shown in Figure 2.2.
Figure 2.2
Pole-zero plot for (2.4).
A pole is traditionally represented by an
X
and a zero by an
O
. If the transfer
function is odd, the first-order pole or zero will be located on the real axis. All
poles and zeros from the quadratic factors are symmetrically located pairs in the
complex plane on opposite sides of the real axis. The gain of the transfer function
