Introduction, concepts, and mathematics of IIR filters.ppt

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

AGC
DSP
2
 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

AGC
DSP
3
 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 

AGC
DSP
4
 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

AGC
DSP
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


AGC
DSP
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

AGC
DSP
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


AGC
DSP
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


AGC
DSP
9
 Mapping of s-plane into the z-plane

AGC
DSP
10
 For with unity scalar we have
or
)
2
/
tan(
1
1 


j
e
e
j j
j




 


j
e
z 
)
2
/
tan(



AGC
DSP
11
 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

AGC
DSP
12
 Nonlinear mapping introduces a
distortion in the frequency axis called
frequency warping
 Effect of warping shown below

AGC
DSP
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

AGC
DSP
14
 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 

AGC
DSP
15
 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

AGC
DSP
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( 


 









AGC
DSP
17
Lowpass-to-Lowpass
 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

AGC
DSP
18
Lowpass-to-Lowpass
 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)

AGC
DSP
19
Lowpass-to-Lowpass
 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

AGC
DSP
20
Lowpass-to-Highpass
 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
̂

AGC
DSP
21
Lowpass-to-Highpass
 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

AGC
DSP
22
Lowpass-to-Highpass
 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

AGC
DSP
23
Lowpass-to-Highpass
 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


AGC
DSP
24
Lowpass-to-Bandpass
1
ˆ
1
2
ˆ
1
1
1
1
ˆ
1
2
ˆ
1
2
1
2
1

















z
z
z
z
z









AGC
DSP
25
Lowpass-to-Bandpass
 The parameters and are given by
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


AGC
DSP
26
Lowpass-to-Bandpass
 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



AGC
DSP
27
Lowpass-to-Bandstop
1
ˆ
1
2
ˆ
1
1
1
1
ˆ
1
2
ˆ
1
2
1
2
1
















z
z
z
z
z









AGC
DSP
28
Lowpass-to-Bandstop
 The parameters and are given
by
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 


 


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Introduction, concepts, and mathematics of IIR filters.ppt