1.
1
Analog
Filters
Analog Filters
(I)
Introduction
Yogananda Isukapalli

2.
Analog Filter Specifications
w’
p w’
s w’
Analog Frequency
A
m
p
l
I
t
u
d
e
ds
0
1+dp
1
1-dp
|
)
(
| w¢
j
H
Pass band Stop band
,
for
1
)
(
1 p
p
p j
H w
w
d
w
d ¢
£
¢
+
£
¢
£
-
,
for
)
( ¥
£
¢
£
¢
£
¢ w
w
d
w s
s
j
H
,
)
(
log
20
,
)
1
(
log
20
10
10
dB
dB
s
s
p
p
d
a
d
a
-
=
-
-
=
Where ap and as are the peak pass band ripple and
minimum stop band attenuation in dB respectively
2

3.
3
Analog
Filter
x(t) y(t)
Review :
å
å =
=
+
=
M
k
k
k
k
N
k
k
k
k
dt
t
y
d
b
dt
t
x
d
a
t
y
0
0
)
(
)
(
)
(
Example : a1= 1 and all other “a’s” and
“b’s” are zero.
dt
t
dx
t
y
)
(
)
( = Ideal Differentiator
Example : Second order Analog Filter
2
2
2
1
1
0 )
(
)
(
dt
y
d
b
dt
dy
b
dt
dx
a
t
x
a
t
y +
+
+
=

4.
4
Transfer Function :
å
å
=
=
-
= M
i
i
i
N
i
i
i
s
b
s
a
s
H
0
0
1
)
(
2
1
1
.
2
1
)
(
2
1
1
)
(
2
)
(
)
(
:
+
+
=
+
+
=
-
+
=
s
s
s
H
s
s
s
H
dt
dy
dt
dx
t
x
t
y
Example
Pole: s = -1/2 Zero: s = -1
Gain = 1/2
Im(s)
Re(s)
S- plane
pole zero plot

5.
5
Impulse Response :
0
0
t
1
0
1
1
1
1
0
0
)
(
1
)
(
:
)
(
)
(
³
<
-
=
-
=
=
t
t
b
e
b
a
t
h
s
b
a
s
H
Example
s
H
L
t
h
Stable Analog Filter :
Define h(t) ® 0 as t ® ¥
Implies that all poles must lie to the left
of the imaginary axis :
Im(s)
Re(s)
S- plane
Unstable
Re(s)> 0
Stable
Re(s)< 0
| | | | | | | | | | | | | | | | | | | | | | / / / / / / / / / / / / / / / / / / /

6.
6
Stability and Pole Locations :
Analog Filters with M poles:
t
p
m
t
p
t
p
m
m
M
N
m
e
g
e
g
e
g
t
h
p
s
g
p
s
g
p
s
g
s
H
p
s
p
s
p
s
z
s
z
s
z
s
G
s
H
+
+
+
=
-
+
+
-
+
-
=
-
-
-
-
-
-
=
.........
)
(
.......
)
(
)
.....(
).........
)(
(
)
.....(
).........
)(
(
)
(
2
1
2
1
2
2
1
1
2
1
2
1
Partial Fraction (no repeated poles)
Since stable means h(t) ® 0 as t ® ¥
It follows that : pi < 0
If pi is a complex pole :
1
|
|
0
)
Re(
)
(
)
Im(
)
Im(
)
Re(
=
<
=
t
p
j
i
t
p
j
t
p
i
i
i
i
i
e
p
e
e
g
t
h
Follows:

7.
7
Analog filters are stable when their poles
are in the left half of the s-plane, where as
digital filters are stable when their poles are
confined within the unit circle.
z- plane
Im(s)
Re(s)
S- plane
Im(z)
Re(z)
Stable Stable
Analog Filter : Magnitude
Function :
Magnitude Square Function :
)
(
)
(
|
)
(
)
( w
w
w w ¢
<
¢
=
=
¢ ¢
= j
H
j
H
s
H
j
H j
s
w
w ¢
=
-
=
¢ j
s
s
H
s
H
j
H |
)
(
)
(
)
(
2

8.
8
6
2
2
2
/
1
2
2
2
2
2
1
4
|
)
(
|
:
)
1
(
1
|
)
(
|
1
1
|
)
(
|
1
1
)
(
)
(
1
1
)
(
)
(
:
1
1
)
(
1
1
)
(
:
w
w
w
w
w
w
w
w
w
¢
+
¢
+
=
¢
¢
+
=
¢
¢
+
=
¢
¢
+
=
-
-
=
-
+
-
=
-
+
=
¢
=
j
H
j
H
j
H
s
H
s
H
s
s
H
s
H
Follows
s
s
H
s
s
H
j
s
Example
Example
Find H(s) or H(-s)

9.
9
)
1
)(
1
(
)
2
(
)
(
)
1
)(
1
(
)
2
(
)
(
)
1
)(
1
(
)
1
(
)
1
)(
1
(
)
1
(
)
1
)(
1
(
)
2
)(
2
(
1
4
|
)
(
|
)
(
)
(
:
2
2
2
3
2
3
3
3
6
2
2
2
2
s
s
s
s
s
H
s
s
s
s
s
H
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
j
H
s
H
s
H
Follows
+
-
-
-
=
-
+
+
+
+
=
+
-
+
=
+
+
+
-
=
-
+
-
+
-
=
-
-
=
-
=
¢
¢
=
-
w
w
H(s) : poles in the left - half of the s - plane
H(-s) : poles in the right - half of the s - plane