H
hp
s
1
s 1
s
1
s
s
s 1
:
6:2:13
Similarly, we can calculate H
hp
s for higher order filters. We can show that the
denominator polynomials of H(s) and H
hp
s are the same, but the numerator becomes
s
L
for the Lth-order highpass filters. Thus H
hp
s has an additional Lth-order zero at the
origin, and has identical poles s
k
as given in (6.2.10).
Transfer functions of bandpass filters can be obtained from the corresponding low-
pass filters by replacing s with s
2
V
2
m
=BW. That is,
H
bp
s Hs
s
s
2
V
2
m
BW
,
6:2:14
where V
m
is the center frequency of the bandpass filter and BW is its bandwidth. As
illustrated in Figure 5.3 and defined in (5.1.10) and (5.1.11), the center frequency is
defined as
V
m
V
a
V
b
p
,
6:2:15
where V
a
and V
b
are the lower and upper cut-off frequencies. The filter bandwidth is
defined by
BW V
b
À V
a
:
6:2:16
Note that for an Lth-order lowpass filter, we obtain a 2Lth-order bandpass filter
transfer function.
For example, consider L 1. From Table 6.2 and (6.2.14), we have
H
bp
s
1
s 1
s
s
2
V
2
m
BW
BWs
s
2
BWs V
2
m
:
6:2:17
In general, H
bp
s has L zeros at the origin and L pole-pairs.
Bandstop filter transfer functions can be obtained from the corresponding highpass
filters by replacing s in the highpass filter transfer function with s
2
V
2
m
=BWs. That
is,
H
bs
s H
hp
s
s
s
2
V
2
m
BWs
,
6:2:18
where V
m
is the center frequency defined in (6.2.15) and BW is the bandwidth defined in
(6.2.16).
254
DESIGN AND IMPLEMENTATION OF IIR FILTERS