Digital Signal Processing Reference
In-Depth Information
the transfer function description is the predominant method used in filter design,
and we will perform most of our filter design using it.
Figure 2.1
The filter as a system.
2.1.1 Transfer Function Characterization
The transfer function
H
(
s
) for a filter system can be characterized in a number of
ways. As shown in (2.1),
H
(
s
) is typically represented as the ratio of two
polynomials in
s
where in this case the numerator polynomial is order
m
and the
denominator is a polynomial of order
n
.
G
represents an overall gain constant that
can take on any value.
m
m
−
1
m
−
2
G
⋅
[
s
+
a
⋅
s
+
a
⋅
s
+
+
a
⋅
s
+
a
]
m
−
1
m
−
2
1
0
H
(
s
)
=
(2.1)
n
n
−
1
n
−
2
[
s
+
b
⋅
s
+
b
⋅
s
+
+
b
⋅
s
+
b
]
n
−
1
n
−
2
1
0
Alternately, the polynomials can be factored to give a form as shown in (2.2).
In this representation, the numerator and denominator polynomials have been
separated into first-order factors. The
z
s represent the roots of the numerator and
are referred to as the zeros of the transfer function. Similarly, the
p
s represent the
roots of the denominator and are referred to as the poles of the transfer function.
G
⋅
[(
s
+
z
)
⋅
(
s
+
z
)
(
s
+
z
)
⋅
(
s
+
z
)]
0
1
m
−
2
m
−
1
H
(
s
)
=
(2.2)
[(
s
+
p
)
⋅
(
s
+
p
)
(
s
+
p
)
⋅
(
s
+
p
)]
0
1
n
−
2
n
−
1
Most of the poles and zeros in filter design will be complex valued and will
occur as complex conjugate pairs. In this case, it will be more convenient to
represent the transfer function as a ratio of quadratic terms that combine the
individual complex conjugate factors as shown in (2.3). The first-order factors that
are included will be present only if the numerator or denominator polynomial
orders are odd. We will be using this form for most of the analog filter design
material.
2
2
G
⋅
[(
s
+
z
)
⋅
(
s
+
a
⋅
s
+
a
)
(
s
+
a
⋅
s
+
a
)]
0
01
02
q
1
q
2
H
(
s
)
=
(2.3)
2
2
[(
s
+
p
)
⋅
(
s
+
b
⋅
s
+
b
)
(
s
+
b
⋅
s
+
b
)]
0
01
02
r
1
r
2
