### Lecture09_IIR - Computer Engineering

```Hossein Sameti
Department of Computer Engineering
Sharif University of Technology
FIR
IIR
Achieving a linear phase is
always possible
Difficult to control the
linear-phase property.
Almost no particular
technique is available.
Can be unstable
Filter order: less
Always stable
Filter order: higher
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
2

We focus on IIR filters with a rational
transfer function:
P( z )
H ( z) 
Q( z )
P and Q are polynomials in z.
N
P( z )   a(n) z
n0
n
M
Q( z )   b(n) z n
n0
Filter Design: To determine the values of a(n)
and b(n) such that specs given to us are met.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
3
IIR Filter Design
Optimization
techniques
Pole-zero
placement
Impulse
Invariance
Bilinear
Transformation
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
4
1.
2.
3.
4.
A set of specs for the digital (discrete-time) filter is
given.
We transform the specs from the D.T. to C.T. (zs)
Design a C.T. IIR filter : H a (s )
sz H a (s)  H ( z )
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
5
 The
art of CT IIR filter design is
 Many CT IIR methods have relatively
closed-form design formulas.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
6
1) Causal/stable analog filter should be transformed to a
causal stable DT filter.
H a ( s)  H ( z )
Causal and stable
Causal and stable
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
7
2) jΏ axis in the s-plane (CT) needs to be
transformed to the unit circle in the z-domain.
* Needed to translate the specs from discrete to analog domain
3) Rational transfer function in the s-domain should be
transformed into a rational transfer function in the zdomain.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
8
Proposal:
H d ( z )  H a ( s) s  2 ( 1 z 1 )
T: arbitrary parameter
T 1 z 1
• Does this transformation satisfy the desirable properties that we
just discussed?
• Does a rational analog filter lead to a rational digital filter?
2 1 z  1
s (
)
1
T 1 z
A rational analog filter translated
into a rational digital filter.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
9
•
Is jΏ axis in the s-plane (CT) transformed to the unit
circle in the z-domain?
Let: z  e j
s
 : 0  

 : 0  
 : 0  

 : 0  
2

j tan
T
2
2 1 e  j
s (
)
T 1 e  j
   j

s
j
2
j
2

j
2
2e
(e e
)
j
j
T  j

e 2 (e 2 e 2 )
2

 tan
T
2
 0
Mehrdad Fatourechi, Electrical and
Computer Engineering, University of
British Columbia, Summer 2011
10
• Does a causal and stable analog filter lead to a causal and
stable digital filter?
• We need to show that LHP in the s-domain is mapped into
inside the unit circle in the z-domain.
2 1 z 1
s (
)
T 1 z 1
j
2 z 1
2
re
1
s (
)
s  ( j )
T z 1
T re 1
2
r 2 1
2r sin 
r  1   0
s (
j
)
2
2
T 1 r  2r cos 1 r  2r cos


r  1   0
11
H ()
1
0.89
0.178
0.2
0.3


• Using bilinear transformation, design a low-pass IIR
filter that satisfies the above spec.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
12
1.
2.
3.
4.
A set of specs for the digital (discrete-time) filter is
given.
We transform the specs from the D.T. to C.T. (zs)
Design a C.T. IIR filter : H a (s )
sz H a (s)  H ( z )
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
13
T 1
2

  tan
T
2
  2 tan

2
 p  0.2   p 2 tan 0.2 0.6498
2
0.3
s  0.3  s 2 tan
1.02
2
14
H a ()
1
0.89
0.178
0.6498 1.02

Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
15

Butterworth filter:
H c ( s) 
a0
a N s N  a N 1s N 1  ...  a1s  a0
2
H c () 
Monotonic in
stop-band and
pass-band
1
 
1  
 c



2N
Cut-off
frequency
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
16
1
2
H c () 
 
1  
 c



2N
s  j  H c ( s ) H c (  s )  H c (  )
H c ( s) H c ( s) 
Poles of
2
1
 s 

1  
 j c 
H c ()
2
:
2N
1
2N
sk  (1) ( j c )   c e
 j 
( 2 k  N 1) 

 2N 
k  0,1, ..., 2 N  1
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
17
Poles of    − = | Ω |2 :
1
2N
sk  (1) ( j c )   c e
 j 
( 2 k  N 1) 

 2N 
k  0,1, ..., 2 N  1
•
•
•
•
•
•
2N poles equally spaced in angle on a circle of radius  c in the s-plane
Poles are symmetric with respect to the imaginary axis
No poles on the imaginary axis
Two poles on the real axis only if N is odd
The poles are spaced by /N radians
To construct a stable and causal H(s), we choose all the poles on the LHP
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
18
Chebyshev: ripple
in either pass-band
or stop-band
Elliptic: ripple in
both pass-band or
stop-band
See Appendix B
in the textbook for
related formulae.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
19
H a ()
1
0.89
0.178
0.6498 1.02

- Which one would you choose?
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
20
H a ()
1
• Butterworth filter of order N:
2
H a () 
1
 
1  
 c



0.178
2N
2
0.6498

1
 0.6498

1  

c 


0.6498 1.02
H a () 0.6498  0.89
H a ( )
0.89
2N
 (0.89) 2
H a () 1.02  0.178
H a ( )
2
1.02

1
 1.02 

1  
 c 
2N
 (0.178) 2
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
21
1
2
H a () 
 
1  
 c



H a () 0.6498  0.89
2N
N  5.305
N 6
2N  12
N 6
angle
H a () 1.02  0.178
 c  0.766
360
 30
12
s  s1 , s 2 , s 3 , s 4 , s 5 , s 6
22
K
H a (s) 
( s  s1 )( s  s 2 )(s  s3 )(s  s4 )( s  s5 )( s  s6 )
H a (  0)  1
1
2 1 z
s (
)

1
T 1 z
K  0.20238
T 1
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
23
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
24
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
25
H ()
1
0.7
0.2
0.2
0.5


• Using bilinear transformation, design a low-pass IIR
filter that satisfies the above spec.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
26
T 1
2

  tan
T
2
  2 tan

2
 p  0.2   p 2 tan 0.2 0.6498
2
s  0.5   2 tan 0.5 2
s
2
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
27
H a ()
1
0.7
0.2
0.6498
2

Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
28
• Butterworth filter of order N:
1
2
H a () 
 
1  
 c



2N
H a () 0.6498  0.7
H a ( )
2
0.6498

1
 0.6498

1  
 c 
2N
 (0.7)
2
H a () 2  0.2
H a ( )
2
2

1
 2 
1   
 c 
2N
 (0.2) 2
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
29
1
2
H a () 
 
1  
 c



2N
H a () 0.6498  0.7
H a () 2  0.2
N  1.3957,  c  0.6465
N 2
N 2
 c  0.6437
2N  4
angle
360
 90
4
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
30
s1  0.4549 j 0.4549
s2  0.4549 j 0.4549
0.64372
H a ( s) 
( s  s1 )( s  s 2 )
2 1 z 1
s (
)
1
T 1 z
...
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
31
Frequency Transformation of
Low-pass IIR Filters
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
32
1.
2.
3.
4.
5.
6.
Start with spec. in D.T. For HPF.
Translate the filter specs from D.T. to C.T.  specs
of a HPF in C.T. (using bilinear transform.)
Translate specs of C.T. HPF to C.T. LPF
Design the LPF (Butterworth)
Transform C.T. LPF  C.T. HPF
Transform C.T. HPF  D.T. HPF (using bilinear transform.)
C.T. LPF
Ha(s1)
C/C
C/D
C.T. HPF
Ha’(s2)
D.T. HPF
Hd(z)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
33
LPF
H a ( s1 )
HPF
1
H a' (s2 )
Proposal: H ( s2 )  H a ( s1 )
k
a'
s1
s2
2
k: positive
constant
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
34

Is jΏ1 axis in the s1-plane (CT) transformed to the jΏ2
axis in the s2-plane (CT)?
j1  j 2
k
s1  j1 
s2
k
s2   j
  2  j 2
1
k
s2 
j 1
k
s2   j
1
k
 2  0,  2  
1
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
35
• Does a causal and stable analog LPF filter lead to a causal and
stable analog HPF?
• Does LHP in the s1-domain (CT) map into the
LHP in the s2-domain (CT)?
• It is easy to prove the above statement.
• It is also easy to show that a rational transfer function is
mapped into another rational transfer function.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
36
k
2  
1
ha (t ) real
H a ' (2 )  H a (1 )
1
k
2
Frequency response is symmetric.
H a (1 )  H a (1 )
H a ' ( 2 )  H a (1 )
H a ' (2 )  H a (1 )
1
k
2
k
1
2
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
37
H a' (2 )  H a (1 )
2
1 
k
2
1
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
38
*Assumption: prototype lowpass filter has band edge frequency
Type of Transformation
Lowpass
Highpass
Bandpass
Bandstop
Transformation
s
p
' p
s
s
 p ' p
s
s 2  l u
s  p
s(u  l )
s  p
s(u  l )
s 2  l u
p
Band edge frequencies
of the new filter
' p
' p
u , l
u , l
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
39
1.
2.
3.
4.
5.
6.
Start with spec. in D.T. For HPF.
Translate the filter specs from D.T. to C.T.  specs
of a HPF in C.T. (bilinear transformation)
Translate specs of C.T. HPF to C.T. LPF
Design the LPF (Butterworth)
Transform C.T. LPF  C.T. HPF
Transform C.T. HPF  D.T. HPF (bilinear transformation)
C.T. LPF
Ha(s1)
C/C
C/D
C.T. HPF
Ha’(s2)
D.T. HPF
Hd(z)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
40
1.
2.
Start with spec. in D.T. For HPF.
Translate the filter specs from D.T. HPF to D.T. LPF
(using the transformation discussed shortly)
3.
4.
Design the LPF : (a) translate the DT LPF specs 
CT LPF specs; (b) Design CT LPF;(c) Transform CT
LPF to DT LPF.
Transform D.T. LPF  D.T. HPF
C.T. LPF
Ha(s)
C/D
D/D
D.T.LPF
H(z1)
D.T. HPF
Hd(z2)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
41
Proposal: H d ( z2 )  H ( z1) z    z2
1
1z2
 1
 : real
It can be shown that this transformation has the 3
properties that we usually investigate for transformations:
1- A rational transfer function is transformed to a rational
transfer function.
2- Unit circle in one domain is mapped into the unit circle
in the other domain.
3- Inside of the unit circle in one domain is mapped to the
inside of the unit circle in the other domain.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
42
Proposal:
H d ( z2 )  H ( z1) z    z2
1
1z2
 1
 : real
Proof of the second property: Unit circle in one domain is
mapped into the unit circle in the other domain.
z2  e
j 2
j 2
  z2


e
z1  

1z2
1e j 2
e j 2 (1e j 2 )
z1  
z1  1
(1e j 2 )
e j 2 1e j 2
1e j 2
z1  1
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
43
Proposal:
H d ( z2 )  H ( z1) z    z2
1
e
j1
j 2


e

1e j2
1z2
 1
 : real
1 
f (2 ,  )
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
44
1.
2.
Start with spec. in D.T. For HPF.
Translate the filter specs from D.T. HPF to D.T. LPF
(using the transformation discussed earlier)
3.
4.
Design the LPF : (a) translate the DT LPF specs 
CT LPF specs; (b) Design CT LPF;(c) Transform CT
LPF to DT LPF.
Transform D.T. LPF  D.T. HPF
C.T. LPF
Ha(s)
C/D
D/D
D.T.LPF
H(z1)
D.T. HPF
Hd(z2)
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
45



Transform   to
is a lowpass system function
is a new system function that can have either
lowpass, highpass, bandpass, or bandstop
characteristics
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
46
Transformations from a lowpass digital filter prototype of cutoff frequency
to highpass, bandpass, and bandstop filters
Mehrdad Fatourechi, Electrical and
Computer Engineering, University of
British Columbia, Summer 2011
47

Suppose we have designed a filter that has met the
following specs:

We have designed a Chebyshev lowpass filter with the
following system function:
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
48
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
49

To transfer this filter to a highpass filter with passband
edge frequency of  p  0.6 :

This results in the following high-pass filter:
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
50
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
51



IIR filters generally have lower order compared to FIR
filters, however, linear-phase cannot be guaranteed.
The most popular technique is the transformation
technique, although other methods such as pole-zero
placement also exist.
Using transformation techniques, a low-pass prototype
filter can be transformed into HP, BP and BS filters.
Hossein Sameti, Dept. of Computer Eng.,
Sharif University of Technology
52
```