Digital Signal Processing Reference
In-Depth Information
1
ju=u
c
þ
1
HðjuÞ¼
The magnitude response is determined by
1
s
jHðjuÞj ¼
(8.4)
u
u
c
2
1
þ
Similarly, we verify the gains at
u ¼
0,
u ¼ u
c
,
u ¼
100
u
c
,
u ¼
10
;
000
u
c
to be 1
;
1
=
p
,0
:
0995,
and 0.01, respectively. The filter gain at the cutoff frequency
u
c
equals 1
=
p
, and the DC gain is 1. The
gain approaches zero when
u ¼þ
N
. We notice that filter gains do not change but that the filter
frequency is scaled up by a factor of
u
c
. This verifies that the prototype transformation converts the
lowpass prototype to the analog lowpass filter with the specified cut-off frequency of
u
c
without an
effect on the filter gain.
This first-order prototype function is used here for illustrative purposes. We will obtain general
functions for Butterworth and Chebyshev lowpass prototypes in Section 8.3.
The highpass, bandpass, and bandstop filters using the specified lowpass prototype transformation
can be easily verified. We review them in
Figures 8.3, 8.4 and 8.5
, respectively.
The transformation from the lowpass prototype to the highpass filter
H
HP
ðsÞ
with a cutoff
frequency
u
c
radians/second is given in
Figure 8.3
, where
s ¼ u
c
=s
in the lowpass prototype
transformation.
The transformation of the lowpass prototype function to a bandpass filter with a center frequency
u
0
, a lower cutoff frequency
u
l
, and an upper cutoff frequency
u
h
in the passband is depicted in
2
2
Figure 8.4
,
where
s ¼ðs
0
Þ=ðsWÞ
is substituted into the lowpass prototype.
þ u
p
u
l
u
h
while the passband bandwidth is given by
W ¼ u
h
u
l
. Similarly, the transformation from the
lowpass prototype to a bandstop (band reject) filter is illustrated in
Figure 8.5
with
s ¼ sW=ðs
2
0
Þ
þ u
substituted into the lowpass prototype.
Hv
P
()
Hj
HP
()
s
c
s
v
0
1
0
c
HsHs
HP
()
()
/
P
s
s
c
FIGURE 8.3
Analog lowpass prototype transformation to the highpass filter.
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