Design of IIR filters are described

1. AGC
DSP
Professor A G Constantinides
1
IIR Digital Filter Design
Standard approach
(1) Convert the digital filter specifications
into an analogue prototype lowpass filter
specifications
(2) Determine the analogue lowpass filter
transfer function
(3) Transform by replacing the
complex variable to the digital transfer
function
)
(s
Ha
)
(z
G
)
(s
Ha

2. AGC
DSP
Professor A G Constantinides
2
IIR Digital Filter Design
 This approach has been widely used for
the following reasons:
(1) Analogue approximation techniques
are highly advanced
(2) They usually yield closed-form
solutions
(3) Extensive tables are available for
analogue filter design
(4) Very often applications require
digital simulation of analogue systems

3. AGC
DSP
Professor A G Constantinides
3
IIR Digital Filter Design
 Let an analogue transfer function be
where the subscript “a” indicates the
analogue domain
 A digital transfer function derived from
this is denoted as
)
(
)
(
)
(
s
D
s
P
s
H
a
a
a 
)
(
)
(
)
(
z
D
z
P
z
G 

4. AGC
DSP
Professor A G Constantinides
4
IIR Digital Filter Design
 Basic idea behind the conversion of
into is to apply a mapping from the
s-domain to the z-domain so that essential
properties of the analogue frequency
response are preserved
 Thus mapping function should be such that
 Imaginary ( ) axis in the s-plane be
mapped onto the unit circle of the z-plane
 A stable analogue transfer function be
mapped into a stable digital transfer
function
)
(s
Ha
)
(z
G

j

5. AGC
DSP
Professor A G Constantinides
5
IIR Digital Filter: The bilinear
transformation
 To obtain G(z) replace s by f(z) in H(s)
 Start with requirements on G(z)
G(z) Available H(s)
Stable Stable
Real and Rational in z Real and Rational
in s
Order n Order n
L.P. (lowpass) cutoff L.P. cutoff T
c

c


6. AGC
DSP
Professor A G Constantinides
6
IIR Digital Filter
 Hence is real and rational in z of
order one
 i.e.
 For LP to LP transformation we require
 Thus
)
(z
f
d
cz
b
az
z
f



)
(
1
0 

 z
s 0
0
)
1
( 


 b
a
f
1





 z
j
s 0
)
1
( 





 d
c
j
f
1
1
.
)
(









z
z
c
a
z
f

7. AGC
DSP
Professor A G Constantinides
7
IIR Digital Filter
 The quantity is fixed from
 ie on
 Or
 and






c
a c
cT 


2
tan
.
)
(
1
:
T
j
c
a
z
f
z
C c









2
tan
.
T
j
c
a
j c
c









1
1
1
1
.
2
tan
























z
z
T
s
c
c


8. AGC
DSP
Professor A G Constantinides
8
Bilinear Transformation
 Transformation is unaffected by scaling.
Consider inverse transformation with scale
factor equal to unity
 For
 and so
s
s
z



1
1
o
o j
s 



2
2
2
2
2
)
1
(
)
1
(
)
1
(
)
1
(
o
o
o
o
o
o
o
o
z
j
j
z



















1
0 

 z
o

1
0 

 z
o

1
0 

 z
o


9. AGC
DSP
Professor A G Constantinides
9
Bilinear Transformation
 Mapping of s-plane into the z-plane

10. AGC
DSP
Professor A G Constantinides
10
Bilinear Transformation
 For with unity scalar we have
or
)
2
/
tan(
1
1 


j
e
e
j j
j




 


j
e
z 
)
2
/
tan(



11. AGC
DSP
Professor A G Constantinides
11
Bilinear Transformation
 Mapping is highly nonlinear
 Complete negative imaginary axis in the
s-plane from to is
mapped into the lower half of the unit
circle in the z-plane from to
 Complete positive imaginary axis in the
s-plane from to is mapped
into the upper half of the unit circle in
the z-plane from to


 0


0

 


1


z 1

z
1

z 1


z

12. AGC
DSP
Professor A G Constantinides
12
Bilinear Transformation
 Nonlinear mapping introduces a
distortion in the frequency axis called
frequency warping
 Effect of warping shown below

13. AGC
DSP
Professor A G Constantinides
13
Spectral Transformations
 To transform a given lowpass
transfer function to another transfer
function that may be a lowpass,
highpass, bandpass or bandstop filter
(solutions given by Constantinides)
 has been used to denote the unit
delay in the prototype lowpass filter
and to denote the unit delay in the
transformed filter to avoid
confusion
)
(z
GL
)
ˆ
(z
GD
1
ˆ
z
1

z
)
(z
GL
)
ˆ
(z
GD

14. AGC
DSP
Professor A G Constantinides
14
Spectral Transformations
 Unit circles in z- and -planes defined
by
,
 Transformation from z-domain to
-domain given by
 Then
ẑ
ẑ

j
e
z  ̂
ˆ j
e
z 
)
ˆ
(z
F
z 
)}
ˆ
(
{
)
ˆ
( z
F
G
z
G L
D 

15. AGC
DSP
Professor A G Constantinides
15
Spectral Transformations
 From , thus ,
hence
 Therefore must be a stable allpass
function
)
ˆ
(z
F
z  )
ˆ
(z
F
z 











1
if
,
1
1
if
,
1
1
if
,
1
)
ˆ
(
z
z
z
z
F
)
ˆ
(
/
1 z
F
1
,
ˆ
ˆ
1
)
ˆ
(
1
1
*















 


 

L
z
z
z
F

16. AGC
DSP
Professor A G Constantinides
16
Lowpass-to-Lowpass
Spectral Transformation
 To transform a lowpass filter with a
cutoff frequency to another lowpass filter
with a cutoff frequency , the
transformation is
 On the unit circle we have
which yields
)
(z
GL
)
ˆ
(z
GD
c

c
̂







z
z
z
F
z
ˆ
ˆ
1
)
ˆ
(
1
1





ˆ
ˆ
1 j
j
j
e
e
e 





)
2
/
ˆ
tan(
1
1
)
2
/
tan( 


 









17. AGC
DSP
Professor A G Constantinides
17
Lowpass-to-Lowpass
Spectral Transformation
 Solving we get
 Example - Consider the lowpass digital
filter
which has a passband from dc to
with a 0.5 dB ripple
 Redesign the above filter to move the
passband edge to
 
 
2
/
)
ˆ
(
sin
2
/
)
ˆ
(
sin
c
c
c
c








)
3917
.
0
6763
.
0
1
)(
2593
.
0
1
(
)
1
(
0662
.
0
)
( 2
1
1
3
1









z
z
z
z
z
GL

25
.
0

35
.
0

18. AGC
DSP
Professor A G Constantinides
18
Lowpass-to-Lowpass
Spectral Transformation
 Here
 Hence, the desired lowpass transfer
function is
1934
.
0
)
3
.
0
sin(
)
05
.
0
sin(







1
1
1
ˆ
1934
.
0
1
1934
.
0
ˆ
)
(
)
ˆ
(







z
z
z
L
D z
G
z
G
0 0.2 0.4 0.6 0.8 1
-40
-30
-20
-10
0
/
Gain,
dB
G
L
(z) G
D
(z)

19. AGC
DSP
Professor A G Constantinides
19
Lowpass-to-Lowpass
Spectral Transformation
 The lowpass-to-lowpass transformation
can also be used as highpass-to-
highpass, bandpass-to-bandpass and
bandstop-to-bandstop transformations







z
z
z
F
z
ˆ
ˆ
1
)
ˆ
(
1
1

20. AGC
DSP
Professor A G Constantinides
20
Lowpass-to-Highpass
Spectral Transformation
 Desired transformation
 The transformation parameter is given by
where is the cutoff frequency of the
lowpass filter and is the cutoff frequency
of the desired highpass filter
1
1
1
ˆ
1
ˆ







z
z
z


 
 
2
/
)
ˆ
(
cos
2
/
)
ˆ
(
cos
c
c
c
c










c

c
̂

21. AGC
DSP
Professor A G Constantinides
21
Lowpass-to-Highpass
Spectral Transformation
 Example - Transform the lowpass filter
 with a passband edge at to a
highpass filter with a passband edge at
 Here
 The desired transformation is
)
3917
.
0
6763
.
0
1
)(
2593
.
0
1
(
)
1
(
0662
.
0
)
( 2
1
1
3
1









z
z
z
z
z
GL

25
.
0

55
.
0
3468
.
0
)
15
.
0
cos(
/
)
4
.
0
cos( 


 


1
1
1
ˆ
3468
.
0
1
3468
.
0
ˆ







z
z
z

22. AGC
DSP
Professor A G Constantinides
22
Lowpass-to-Highpass
Spectral Transformation
 The desired highpass filter is
1
1
1
ˆ
3468
.
0
1
3468
.
0
ˆ
)
(
)
ˆ
(








z
z
z
D z
G
z
G
0 0.2 0.4 0.6 0.8 
80
60
40
20
0
Normalized frequency
Gain,
dB

23. AGC
DSP
Professor A G Constantinides
23
Lowpass-to-Highpass
Spectral Transformation
 The lowpass-to-highpass transformation
can also be used to transform a
highpass filter with a cutoff at to a
lowpass filter with a cutoff at
 and transform a bandpass filter with a
center frequency at to a bandstop
filter with a center frequency at
c

c
̂
o
̂
o


24. AGC
DSP
Professor A G Constantinides
24
Lowpass-to-Bandpass
Spectral Transformation
 Desired transformation
1
ˆ
1
2
ˆ
1
1
1
1
ˆ
1
2
ˆ
1
2
1
2
1

















z
z
z
z
z









25. AGC
DSP
Professor A G Constantinides
25
Lowpass-to-Bandpass
Spectral Transformation
 The parameters and are given by
where is the cutoff frequency of the
lowpass filter, and and are the
desired upper and lower cutoff frequencies of
the bandpass filter
 
  )
2
/
tan(
2
/
)
ˆ
ˆ
(
cot 1
2 c
c
c 


 

 
 
2
/
)
ˆ
ˆ
(
cos
2
/
)
ˆ
ˆ
(
cos
1
2
1
2
c
c
c
c








c

1
ˆc
 2
ˆc


26. AGC
DSP
Professor A G Constantinides
26
Lowpass-to-Bandpass
Spectral Transformation
 Special Case - The transformation can
be simplified if
 Then the transformation reduces to
where with denoting
the desired center frequency of the
bandpass filter
1
2 ˆ
ˆ c
c
c 

 

o

 ˆ
cos
 o
̂
1
1
1
1
ˆ
1
ˆ
ˆ 







z
z
z
z



27. AGC
DSP
Professor A G Constantinides
27
Lowpass-to-Bandstop
Spectral Transformation
 Desired transformation
1
ˆ
1
2
ˆ
1
1
1
1
ˆ
1
2
ˆ
1
2
1
2
1
















z
z
z
z
z









28. AGC
DSP
Professor A G Constantinides
28
Lowpass-to-Bandstop
Spectral Transformation
 The parameters and are given
by
where is the cutoff frequency of the
lowpass filter, and and are the
desired upper and lower cutoff
frequencies of the bandstop filter
c

 
1
ˆc
 2
ˆc

 
 
2
/
)
ˆ
ˆ
(
cos
2
/
)
ˆ
ˆ
(
cos
1
2
1
2
c
c
c
c








  )
2
/
tan(
2
/
)
ˆ
ˆ
(
tan 1
2 c
c
c 


 
