Digital Signal Processing Reference
In-Depth Information
must be indicated on the plot or the information would be incomplete. Note that
there are only two zeros shown, but there is one located at infinity. We can verify
this by observing that if we were to allow |
s
| to approach infinity, |
H
(
s
)| would
approach zero. Transfer functions always have the same number of poles and
zeros, but some exist at infinity.
Conversely, we can also determine a filter's transfer function from the pole-
zero plot. In general, any critical frequency (pole or zero) is specified by
indicating the real (σ) and imaginary (ω) component. The transfer function would
then include a factor of [
s
− (σ +
j
ω)]. If the critical frequency is complex, we can
combine the two complex conjugate factors into a single quadratic factor by
multiplying them as shown in (2.5):
2
2
2
[
s
−
(
σ
+
j
ω
)]
⋅
[
s
−
(
σ
−
j
ω
)]
=
s
−
2
⋅
σ
⋅
s
+
(
σ
+
ω
)
(2.5)
Example 2.1 Generating a Transfer Function from a Pole-Zero Plot
Problem:
Assume that a pole-zero plot shows poles at (−3 ±
j
2) and (−4.5)
and zeros at (−5 ±
j
1) and (−1). Determine the transfer function if its gain is 1.0 at
s
= 0.
Solution:
Using the technique of (2.5), the complex conjugate poles and
zeros can be combined into quadratic factors as indicated. The first-order factors
are handled directly and the gain is included in the numerator. The easiest method
to use when given a gain requirement of 1.0 at
s
= 0 is to prepare each factor
independently to have a gain of 1 at that frequency as shown in the first transfer
function. Then the set of constants can be combined as shown in the second
equation.
2
(
s
+
10
⋅
s
+
26
)
(
s
+
1
)
4
.
5
13
H
(
s
)
=
⋅
⋅
⋅
2
26
1
(
s
+
4
.
5
)
(
s
+
6
⋅
s
+
13
)
2
2
.
25
⋅
(
s
+
1
)
⋅
(
s
+
10
⋅
s
+
26
)
=
2
(
s
+
4
.
5
)
⋅
(
s
+
6
⋅
s
+
13
)
2.1.3 Normalized Transfer Functions
In this chapter we concentrate on developing what is referred to as a normalized
transfer function. A normalized lowpass transfer function is one in which the
passband edge radian frequency is set to 1 rad/sec. Of course, this seems a rather
unusual frequency, since seldom would a lowpass filter be required to have such a
low frequency. However, the technique actually allows the filter designer
considerable latitude in designing filters because a normalized transfer function
