d
p
10
A
p
=20
À 1
10
A
p
=20
1
5:1:8
and
d
s
10
ÀA
s
=20
:
5:1:9
Example 5.1: Consider a filter specified as having a magnitude response in the
passband within Æ0:01. That is, d
p
0:01. From (5.1.6), we have
A
p
20 log
10
1:01
0:99
0:1737 dB:
When the minimum stopband attenuation is given as d
s
0:01, we have
A
s
À20 log
10
0:01 40 dB:
The transition band is the area between the passband edge frequency !
p
and the
stopband edge frequency !
s
. The magnitude response decreases monotonically from the
passband to the stopband in this region. Generally, the magnitude in the transition band
is left unspecified. The width of the transition band determines how sharp the filter is. It
is possible to design filters that have minimum ripple over the passband, but a certain
level of ripple in this region is commonly accepted in exchange for a faster roll-off of
gain in the transition band. The stopband is chosen by the design specifications.
Generally, the smaller d
p
and d
s
are, and the narrower the transition band, the more
complicated (higher order) the designed filter becomes.
An example of a narrow bandpass filter is illustrated in Figure 5.3. The center
frequency !
m
is the point of maximum gain (or maximum attenuation for a notch
filter). If a logarithm scale is used for frequency such as in many audio applications, the
center frequency at the geometric mean is expressed as
!
m
!
a
!
b
p
,
5:1:10a
H(w)
w
1
2
1
w
a
w
m
w
b
Figure 5.3 Magnitude response of bandpass filter with narrow bandwidth
INTRODUCTION TO DIGITAL FILTERS
187