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COMM 602
Digital Signal Processing
Lecture 9
Dr. Engy Aly MaherSpring 2020
Bilinear Transformation
See Text book
For the derivation
Of this equation
Page 714
2tan
2
T
Bilinear Transformation
• Derivation of:
(1)
Bilinear Transformation
Bilinear Transformation
2tan
2
T
Properties of LTIS Using Z-Transform
Properties of LTIS Using Z-Transform
(-ve)
(+ve)
Properties of LTIS Using Z-Transform
S-Plane
LHS RHSJ-axis
Z-Plane
Unit Circle z =1z = -1
Properties of Bilinear Transformation
• Accordingly, we may state the properties of bilinear
transformation:
(1) The left-half s-plane is mapped onto the interior of the unit circle
in the z-plane.
(2) The entire axis of the s-plane is mapped onto one complete
revolution of the unit circle in the z-plane
(3) The right-half of the s-plane is mapped onto the exterior of the
unit circle in the z-plane.
Then if the analog filter is stable and causal, the resulting digital filter
is also causal and stable.
j
Steps of Bilinear Transformation
)(H
H(z)
2tan
2 cc
T
c
Example 1
)(H
For T=2
c
c
2tan
2 cc
T
Example 1
H H(z)
Example 1
245.0
ny
1z
nx
1z
509.0
1
1
509.01
1245.0
z
zzH
Example 2
5.0
3
ssH
1
1
1
12
z
z
Ts
5.0
1
12
3
1
1
z
z
T
zH
1
1
1
1
15.01
12
13
zz
z
T
zzH
Convert the following analog filter system function into a digital
IIR filter by means of the bilinear transformation method:
Solution
Example 2
1
1
1
1
15.01
12
13
zz
z
T
zzH
11
1
15.012
13
zzT
zzH
1
1
25.05.0
2
13
zTT
zzH
1
1
1
1
1
1
5.02
25.0
15.02
13
z
z
z
T
T
T
zzH
s
Example 2
1
1
1
1
1
1
5.02
25.0
15.02
13
z
z
z
T
T
T
zzH
s
25.1
3
5.02
3
T
333.0
5.02
25.0
T
T
1
1
333.01
12
z
zzH
For T=2
Transformation of LPF to Other Filters
c
c
20
20
lu 0
lu
Example 3
Use the bilinear Transformation method to:
=0.55
Ts=2
2tan
2
T
cs
cc
cc
c
c
Example 3
cc
c
c
c
c
• Design a 3rd order Butterworth LPF to have a cutoff frequency
of 2.513 rad/sec using bilinear transformation.
solution
T=2
c cc
c c
Example 4
Example 4
Example 4
For and , the analog filter order N is
determined by the equation:
ps
1
pAjH 1 sAjH 2
p
s
p
s
A
A
N
log2
1
1log
2
2
ss
pp
A
A
1
Butterworth Filter Order
• Using bilinear transformation, design IIR filter that satisfies these
specifications: LPF with maximum ripples allowed in the
passband =0.293, the maximum ripples allowed in the stopband
=0.1, passband edge frequency= 0.125*pi, and stopband edge
frequency=0.5*pi.
solution
p
pA
sA
1
)(12
5.0tan
2tan
)(1989.02
125.0tan
2tan
rad
rad
ss
p
p
The pre-warped analog frequencies are:
The required amplitudes are:
1.0
707.0293.011
ss
pp
A
A
s
pc
Example 5
• The required order is:
p
s
p
s
A
A
N
log2
1
1log
2
2
423.1
1989.0
1log2
1707.0
115.0log
2
2
N
Let N = 2
Example 5
c
c
cc
c
c
c
c
c
c
c
c
pc
Example 5
c
cc
c
c
c
c
c
c
cc
c c
cc
Example 5
c c
Thank you