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Design a digital IIR filter by using
bilinear transformation
IIR digital filter is derived from analog filter
Steps:1. Filter Design Specifications
2. Analog Filter Design
3. Digital Filters from Analog Prototypes
From the given values design
a prototype lowpass filter
4
Design IIR Digital Filter with
the following specifications
H(z) ???
H(p)
Convert the lowpass prototype
filter to the wanted analog filter
H(S)
Design an analog filter with
the following specifications.
H(s) ???
2tan
2T
2 1
1
zs
T z
AnalogDigital
From the given values design
a prototype lowpass filter
5
Design IIR Digital Filter with
the following specific values.
H(z) ???
H(p)
Convert the lowpass prototype
filter to the wanted analog filter
H(S)
Design an analog filter with
the following specific values.
H(s) ???
2tan
2T
2 1
1
zs
T z
AnalogDigital
6
digital filter frequency
analog filter frequency
analog LP prototype filter frequency
7
IIR Filters
-Design a digital lowpass filter to satisfy the following:
(1) monotonic stopband and passband(Butterworth)
(2) -3dB cutoff frequencies of θp= 0.5π rad
(3) magnitude down at least -15dB at 0.75π rad
(4) Max magnitude =0dBTs=1sec
Design this IIR filter by using bilinear transformation
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| ( ) |dBH j
0dB
15dB
0.5p
0.75s 3dB
digital filter
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0.5 , 0.75
0.52 tan 2
2
0.752 tan 4.82
2
p s
p
s
2 1
1
zs
T z
2tan
2
2tan
2
T
T
: analog frequency
: digital frequency
change
change
10
| ( ) |dBH j
0dB
15dB
2p
3dB
analog filter
4.82s
11
3
15
p
s
A
A
to find LP prototype filter:
:
4.82 = , = 1 , = 2.41
2
p p sp s
p p p
normalization
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3dB
0
15dB
LP prototype analog filter
1p 2.41s
| ( ) |dBH j
13
0.1 0.1(3)
0.1
2
0.1(15)
2
For Butterworth filter:
ripple factor:
10 1 10 1 1
10 1log
: N = 2 log( )
10 1log
1 = =2
2log(2.41)
p
s
A
A
s
order
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(
1/
( 2
(2)1/2
1
( (2) 2
(2)1/2
2
1,2
: = e ,
1 e
1 e
1 1P = ± j
2 2
2k +N 1)j
N 2N
k
2+ 1)j
2
2 + 1)j
2
Poles P k = 1, 2
P
P
15
To restore the magnitude scale
2
(0/20)
2
H(p) = 1.414 1
10 1
11
1
1H(p) =
1.414 1
o
o
o
H
P P
H(p = 0)
H
H
P P
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2 2
2
1 2
21
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restor the frequency:
1 4H(S) =H(p) |
2.828 41.414 1
2 2
1 2H(z) =H(s) |
3.414 0.5857
p
S
z
z
to
Sp
S SS S
Z Z
Z
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1 2
2
1 2
2
1 2H(z)
3.414 0.5857
1 2H( )=H(z) |
3.414 0.5857
:
|H( ) |
j
j j
jz e
Z Z
Z
e e
e
check
plot
18
IIR Filters
-The specified magnitude response of a maximally flat
bandpass digital filter has:
- maximum value of 1.0 in its passband
- cutoff frequencies θ1 = 0.4π and θ 2 = 0.5π.
- the magnitude at cutoff frequencies to be no less
than 0.93
- the stopband frequency θs= 0.7π
-the magnitude at stopband frequency to be no
more than 0.004Ts=1sec
Design this IIR filter by using bilinear transformation
0.4 0.5 0.7
s
| ( ) |H j
1 2
digital filter
s
20
1 2
1
2
0.4 , 0.5 , 0.7
0.42 tan 1.453
2
0.52 tan 2
2
0.72 tan 3.925
2
s
s
2 1
1
zs
T z
2tan
2
2tan
2
T
T
: analog frequency
: digital frequency
change
change
211.453 2 3.925
s
analog filter
| ( ) |H j
22
20log 0.93 0.63
20log 0.004 48
p
s
A dB
A dB
2 1
1 2
bandwidth: 0.547 rad/s
Center freqency: =1.705 rad/so
B
to find LP prototype filter:
:
= , = 1 , =p p s
p s
p p p
normalization
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1.705 3.925 1.705
0.547 1.705 3.925
5.82
5.82
=
3.9250.6743 rad/sec
5.82
o okk
o k
o s os
o s
s
s
ss
p
sp
s
jj
B
j jj
B
j j
24
0
LP prototype analog filter
1p 5.84s
0.63dB
48dB
| ( ) | |dB
H j
25
0.1 0.1(0.63)
0.1
2
0.1(48)
2
For Butterworth filter:
ripple factor:
10 1 10 1 0.395
10 1log
: N = 2 log( )
10 1log
0.395 = =4
2log(5.82)
p
s
A
A
s
order
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(
1/
( 4
(4)1/4
1
( (2) 4
(4)1/4
2
( (3) 4
(4)1/4
3
( (4) 4
(4)1/4
4
1,4
: = e , 3,4
0.395 e
0.395 e
0.395 e
0.395 e
P = 0.4827 ±
2k +N 1)j
N 2N
k
2+ 1)j
2
2 + 1)j
2
2 + 1)j
2
2 + 1)j
2
Poles P k = 1,2,
P
P
P
P
2,3
j1.1654
P = 1.1654 ± j0.4827
27
the transfer function of the normalized prototype filter
4 3 2
1H(p) =
3.296 5.4325 5.24475 2.5317P P P P
28
To restore the magnitude scale
4 3 2
(0/20)
(5/20)
4 3 2
H(p) = 3.296 5.4325 5.24475 2.5317
10 1
=10 12.5317
2.5317
2.5317H(p) =
3.296 5.4325 5.24475 2.5317
o
o
o
H
P P P P
H(p = 0)
H
H
P P P P
29
2 2
2 2 2 2
0
1 1.705
0.547
12
1
restor the frequency:
1 1 1.705
0.547
H(S) =H(p) | ...
H(z) =H(s) | ...
S
S
z
z
to
S Sp
B S S
30
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DIGITAL SPECTRAL TRANSFORMATION
Digital LP prototype filter