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2.161 Signal Processing: Continuous and Discrete
Fall
200
8
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1
Massachusetts
Institute
of
Technology
Department
of
Mechanical
Engineering
2.161
Signal
Processing
 Continuous
and
Discrete
Fall
Term
2008
Lecture
7
1
Reading:
ã
Class
handout:
Introduction
to
Continuous
Time
Filter
Design
.
Butterworth
Filter
Design
Example
(Same
problem
as
in
the
Class
Handout).
Design
a
Butterworth
lowpass
ﬁlter
to
meet
the
power
gain
speciﬁcations
shown
below:
At
the
two
critical
frequencies
1
= 0
.
9
−→
= 0
.
3333
1 +
2
1
= 0
.
05
−→
λ
= 4
.
358
1 +
λ
2
Then
log(
λ/
)
N
≥
= 3
.
70
log(Ω
r
/
Ω
c
)
1
copyright
c D.Rowell
2008
7–1
we
therefore
select
N=4.
The
4
poles
(
p
1
...p
4
)
lie
on
a
circle
of
radius
r
= Ω
c
−
1
/N
= 13
.
16
and
are
given
by

p
n

= 13
.
16
p
n
=
π
(2
n
+ 3)
/
8
for
n
= 1
...
4,
giving
a
pair
of
complex
conjugate
pole
pairs
p
1
,
4
=
−
5
.
04
±
j
12
.
16
p
2
,
3
=
−
12
.
16
±
j
5
.
04
The
transfer
function,
normalized
to
unity
gain,
is
29993
H
(
s
) =
(
s
2
+ 10
.
07
s
+
173
.
2)(
s
2
+ 24
.
32
s
+
173
.
2)
and
the
ﬁlter
Bode
plots
are
shown
below.
Bode Diagram
−101 23
1010101010Frequency (rad/sec)
2
Chebyshev
Filters
The
order
of
a
ﬁlter
required
to
met
a
lowpass
speciﬁcation
may
often
be
reduced
by
relaxing
the
requirement
of
a
monotonically
decreasing
power
gain
with
frequency,
and
allowing
−150 −100 −50 0 50
M a g n i t u d e ( d B )
−360 −270 −180 −90 0
P h a s e ( d e g )
7–2
“ripple”
to
occur
in
either
the
passband
or
the
stopband.
The
Chebyshev
ﬁlters
allow
these
conditions:
Type
1

H
(
j
Ω)

2
=
1 +
�
2
T
N
1
2
(Ω
/
Ω
c
)
(1)
1
Type
2

H
(
j
Ω)

2
=
1 +
�
2
(
T
N
2
(Ω
r
/
Ω
c
)
/T
N
2
(Ω
r
/
Ω))
(2)
Where
T
N
(
x
)
is
the
Chebyshev
polynomial
of
degree
N
.
Note
the
similarity
of
the
form
of
the
Type
1
power
gain
(Eq.
(1))
to
that
of
the
Butterworth
ﬁlter,
where
the
function
T
N
(Ω
/
Ω
c
)
has
replaced
(Ω
/
Ω
c
)
N
.
The
Type
1
ﬁlter
produces
an
allpole
design
with
slightly
diﬀerent
pole
placement
from
the
Butterworth
ﬁlters,
allowing
resonant
peaks
in
the
passband
to
introduce
ripple,
while
the
Type
2
ﬁlter
introduces
a
set
of
zeros
on
the
imaginary
axis
above
Ω
r
,
causing
a
ripple
in
the
stopband.
The
Chebyshev
polynomials
are
deﬁned
recursively
as
follows
T
0
(
x
)
= 1
T
1
(
x
)
=
x
T
2
(
x
)
= 2
x
2
−
1
T
3
(
x
)
= 4
x
3
−
3
x
.
.
.
T
N
(
x
) = 2
xT
N
−
1
(
x
)
−
T
N
−
2
(
x
)
,
N >
1
(3)
with
alternate
deﬁnitions
T
N
(
x
)
=
cos(
N
cos
−
1
(
x
))
(4)
=
cosh(
N
cosh
−
1
(
x
))
(5)
The
Chebyshev
polynomials
have
the
minmax
property:
Of
all
polynomials
of
degree
N
with
leading
coeﬃcient
equal
to
one,
the
polynomial
T
N
(
x
)
/
2
N
−
1
has
the
smallest
magnitude
in
the
interval
x
This
“minimum
maximum”
amplitude
is
2
1
−
N
.
  ≤
1
.
In
lowpass
ﬁlters
given
by
Eqs.
(13)
and
(14),
this
property
translates
to
the
following
characteristics:
Filter
PassBand
Characteristic
StopBand
Characteristic
Butterworth
Maximally
ﬂat
Maximally
ﬂat
Chebyshev
Type
1
Ripple
between
1
and
1
/
(1
+
�
2
)
Maximally
ﬂat
Chebyshev
Type
2
Maximally
ﬂat
Ripple
between
1
and
1
/
(1
+
λ
2
)
7–3