Digital Signal Processing Reference
In-Depth Information
to reduce the order
N
of the filter. The elliptic filters allow ripples in both the
pass and stop bands to derive a filter with the lowest order
N
among the four
implementations. M
ATLAB
instructions to design the four implementations
are also presented in Section 7.3.
In Section 7.4, we covered three transformations for converting a highpass
filter to a lowpass filter, a lowpass to a bandpass filter, and a bandstop to a lowpass
filter. Using these transformations, we were able to map the specifications of
any type of the frequency-selective filters in terms of a normalized lowpass
filter. After designing the normalized lowpass filter using the design algorithms
covered in Section 7.3, the transfer function of the lowpass filter is transformed
back into the original domain of the frequency-selective filter.
Problems
7.1
Determine the impulse response of an ideal bandpass filter and an ideal
bandstop filter. In each case, assume a gain of
A
within the pass bands and
cut off frequencies of
ω
c1
and
ω
c2
.
7.2
Derive and sketch the location of the poles for the Butterworth filters of
orders
N
=
12 and 13 in the complex s-plane.
7.3
Show that a lowpass Butterworth filter with an odd value of order
N
will
always have at least one pole on the real axis in the complex s-plane.
7.4
Show that all complex poles of the lowpass Butterworth filter occur in
conjugate pairs.
7.5
Show that the
N
th -order Type I Chebyshev polynomial
T
N
(
ω
) has
N
simple
roots in the interval [
−
1
,
1], which are given by
(2
n
+
1)
π
2
N
ω
n
=
cos
0
≤
n
≤
−
1
.
N
7.6
Show that the roots of the characteristic equation
1
+ ε
2
T
N
(j
/
s
)
=
0
for the Type II Chebyshev filter are the inverse of the roots of the charac-
teristic equation
1
+ ε
2
T
N
(
s
/
j)
=
0
for the Type I Chebyshev filter.
7.7
Design a Butterworth lowpass filter for the following specifications:
pass band (0
≤ω≤
10 radians/s)
0
.
9
≤
H
(
ω
)
≤
1;
stop band (
ω >
20 radians/s)
H
(
ω
)
≤
0
.
10
,
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