AGC
DSP
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 H a (s)
(3) Transform H a (s) by replacing the
complex variable to the digital transfer
function
G (z )
Professor A G Constantinides
1
AGC
DSP
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
Professor A G Constantinides
2
AGC
DSP
IIR Digital Filter Design


Let an analogue transfer function be
Pa ( s )
H a ( s) 
Da ( s )
where the subscript “a” indicates the
analogue domain
A digital transfer function derived from
this is denoted as
P( z )
G( z) 
D( z )
Professor A G Constantinides
3
AGC
DSP
IIR Digital Filter Design


Basic idea behind the conversion of H a (s)
into G (z ) 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
( j ) 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
Professor A G Constantinides
4
AGC
DSP
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
c
cT
Professor A G Constantinides
5
AGC
DSP
IIR Digital Filter



Hence f (z ) is real and rational in z of
order one
az  b
i.e.
f ( z) 
cz  d
For LP to LP transformation we require
s  0  z  1 f (1)  0  a  b  0
s   j  z  1 f (1)   j  c  d  0

Thus
a  z 1

f ( z )   .
 c  z 1
Professor A G Constantinides
6
AGC
DSP
IIR Digital Filter

The quantity a 
c

ie on

Or

and
C : z 1
cT  c
is fixed from
a
T

f ( z ) c   . j tan
2
c
a
cT

jc   . j tan
2
c




1
c
1 z


s
.
1

T


1

z
 tan c 

 
Professor A G Constantinides
  2 
7
AGC
DSP
Bilinear Transformation



Transformation is unaffected by scaling.
Consider inverse transformation with scale
factor equal to unity
For
z  1 s
1 s
s  o  jo
2
2
(1   o )  jo
(
1


)


2
o
o
z
z 
(1   o )  jo
(1   o ) 2  o2
and so
o  0  z 1
o  0  z 1
o  0  z 1
Professor A G Constantinides
8
AGC
DSP
Bilinear Transformation

Mapping of s-plane into the z-plane
Professor A G Constantinides
9
AGC
DSP
Bilinear Transformation

j
with unity scalar we have
 j
j  1  e  j  j tan( / 2)
1 e
For z  e
or
  tan( / 2)
Professor A G Constantinides
10
AGC
DSP
Bilinear Transformation



Mapping is highly nonlinear
Complete negative imaginary axis in the
s-plane from    to   0 is
mapped into the lower half of the unit
circle in the z-plane from z  1 to z  1
Complete positive imaginary axis in the
s-plane from   0 to    is mapped
into the upper half of the unit circle in
the z-plane from z  1 to z  1
Professor A G Constantinides
11
AGC
DSP
Bilinear Transformation


Nonlinear mapping introduces a
distortion in the frequency axis called
frequency warping
Effect of warping shown below
Professor A G Constantinides
12
AGC
DSP
Spectral Transformations


To transform GL (z ) a given lowpass
transfer function to another transfer
function GD (zˆ ) that may be a lowpass,
highpass, bandpass or bandstop filter
(solutions given by Constantinides)
1
has been used to denote the unit
z
delay in the prototype lowpass filter GL (z )
and zˆ 1 to denote the unit delay in the
transformed filter GD (zˆ ) to avoid
confusion
Professor A G Constantinides
13
AGC
DSP
Spectral Transformations



Unit circles in z- and ẑ -planes defined
by
z  e j zˆ  e j̂
,
Transformation from z-domain to
ẑ -domain given by
z  F (zˆ )
Then GD ( zˆ )  GL {F ( zˆ )}
Professor A G Constantinides
14
AGC
DSP
Spectral Transformations


From z  F (zˆ ) ,
hence
 1,

F ( zˆ )  1,
 1,

thus z  F (zˆ )
if z  1
,
if z  1
if z  1
Therefore 1 / F ( zˆ ) must be a stable allpass
function 1
L  1  * zˆ 
 ,   1
   


F ( zˆ )
 1
zˆ    Professor
 A G Constantinides
15
AGC
DSP
Lowpass-to-Lowpass
Spectral Transformation

To transform a lowpass filter GL (z ) with a
cutoff frequency  c to another lowpass filter
GD (zˆ ) with a cutoff frequency ̂ c , the
transformation is
1
1   zˆ

F ( zˆ ) zˆ  
On the unit circle we have
 jˆ
 j
e  e jˆ
1 e
which yields
z 1 

tan( / 2)   1    tan(ˆ / 2)
 1    Professor A G Constantinides
16
AGC
DSP
Lowpass-to-Lowpass
Spectral Transformation
sin ( c  ˆ c ) / 2 

sin ( c  ˆ c ) / 2 
 Example - Consider the lowpass digital
filter
0.0662(1  z 1 )3
GL ( z ) 
1
1
2
(1  0.2593 z )(1  0.6763 z  0.3917 z )
0.25
which has a passband from dc to
with a 0.5 dB ripple
 Redesign the above filter to move the
Professor A G Constantinides
0
.
35

17
passband edge to

Solving we get
DSP
Lowpass-to-Lowpass
Spectral Transformation


Here
sin(0.05 )
 
  0.1934
sin(0.3 )
Hence, the desired lowpass transfer
function is GD ( zˆ )  GL ( z )
zˆ  0.1934
z 
1
1
1 0.1934 zˆ 1
0
Gain, dB
AGC
-10
G (z)
G (z)
L
D
-20
-30
-40
0
0.2
0.4
0.6
/
0.8
1
Professor A G Constantinides
18
AGC
Lowpass-to-Lowpass
Spectral Transformation
DSP

The lowpass-to-lowpass transformation
1
1   zˆ

1
z 

F ( zˆ ) zˆ  
can also be used as highpass-tohighpass, bandpass-to-bandpass and
bandstop-to-bandstop transformations
Professor A G Constantinides
19
AGC
DSP
Lowpass-to-Highpass
Spectral Transformation

Desired transformation
z

1
1
zˆ  

1   zˆ 1
The transformation parameter
cos ( c  ˆ c ) / 2 
 
cos ( c  ˆ c ) / 2 
is given by
where c is the cutoff frequency of the
lowpass filter and ̂c is the cutoff frequency
of the desired highpass filter Professor A G Constantinides
20
AGC
DSP
Lowpass-to-Highpass
Spectral Transformation
Example - Transform the lowpass filter
1 3
0.0662(1  z )
GL ( z ) 
(1  0.2593 z 1 )(1  0.6763 z 1  0.3917 z 2 )




with a passband edge at 0.25 to a
0.55edge at
highpass filter with a passband
Here   cos( 0.4 ) / cos( 0.15 )  0.3468
The desired transformation is
1
z
ˆ  0.3468
1
z 
1
1  0.3468 zˆ Professor A G Constantinides
21
DSP
Lowpass-to-Highpass
Spectral Transformation

The desired highpass filter is
GD ( zˆ )  G ( z ) z
1
zˆ 1 0.3468

10.3468 zˆ 1
0
20
Gain, dB
AGC
40
60
80
0
0.2
0.4
0.6
0.8
Normalized frequency

Professor A G Constantinides
22
AGC
Lowpass-to-Highpass
Spectral Transformation
DSP


The lowpass-to-highpass transformation
can also be used to transform a
highpass filter with a cutoff at c to a
lowpass filter with a cutoff at ̂c
and transform a bandpass filter with a
center frequency at  o to a bandstop
filter with a center frequency at ̂ o
Professor A G Constantinides
23
AGC
DSP
Lowpass-to-Bandpass
Spectral Transformation

Desired transformation
2 1   1
zˆ 
zˆ 
 1
 1
1
z 
  1 2 2 1
zˆ 
zˆ  1
 1
 1
2
Professor A G Constantinides
24
AGC
Lowpass-to-Bandpass
Spectral Transformation
DSP

 and  are given by
cos (ˆ c 2  ˆ c1 ) / 2

cos (ˆ c 2  ˆ c1 ) / 2
The parameters
  cot (ˆ c 2  ˆ c1 ) / 2 tan(c / 2)
where c is the cutoff frequency of the
lowpass filter, and ˆ c1 and ˆ c 2 are the
desired upper and lower cutoff frequencies of
the bandpass filter
Professor A G Constantinides
25
AGC
Lowpass-to-Bandpass
Spectral Transformation
DSP


Special Case - The transformation can
be simplified if c  ˆ c 2  ˆ c1
Then the transformation reduces to
1
z
1
1 ˆ  
z   zˆ
1
1   zˆ
where   cos ˆ o with ̂ o denoting
the desired center frequency of the
bandpass filter
Professor A G Constantinides
26
AGC
Lowpass-to-Bandstop
Spectral Transformation
DSP

Desired transformation
2 1 1  
zˆ 
zˆ 
1 
1 
1
z 
1   2 2 1
zˆ 
zˆ  1
1 
1 
2
Professor A G Constantinides
27
AGC
Lowpass-to-Bandstop
Spectral Transformation
DSP

The parameters  and  are given
by
cos (ˆ c 2  ˆ c1 ) / 2

cos (ˆ c 2  ˆ c1 ) / 2
  tan(ˆ c 2  ˆ c1 ) / 2 tan(c / 2)
where c is the cutoff frequency of the
lowpass filter, and ˆ c1 and ˆ c 2 are the
desired upper and lower cutoff
frequencies of the bandstop
filter
Professor A G Constantinides
28