Transcript DSP & Digital Filters - Imperial College London

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
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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
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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
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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
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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
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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
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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
.
 tan  cT   1  z 1

 
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  2 
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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
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AGC
DSP
Bilinear Transformation

Mapping of s-plane into the z-plane
Professor A G Constantinides
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AGC
DSP
Bilinear Transformation

j
with unity scalar we have
 j
1

e
j 
 j tan( / 2)
 j
1 e
For z  e
or
  tan( / 2)
Professor A G Constantinides
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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
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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
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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
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AGC
DSP
Spectral Transformations



Unit circles in z- and zˆ -planes defined
j
jˆ
by
z e
zˆ  e
,
Transformation from z-domain to
zˆ -domain given by
z  F (zˆ)
Then GD ( zˆ)  GL{F ( zˆ)}
Professor A G Constantinides
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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
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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 ˆ 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



1


tan( / 2)  
tan(ˆ / 2)

 1    Professor A G Constantinides
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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

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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
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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
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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 ˆ c is the cutoff frequency
of the desired highpass filter Professor A G Constantinides
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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
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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
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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 ˆ c
and transform a bandpass filter with a
center frequency at  o to a bandstop
filter with a center frequency at ˆ o
Professor A G Constantinides
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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
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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
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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 ˆ o denoting
the desired center frequency of the
bandpass filter
Professor A G Constantinides
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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
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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
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