2-digital filters (iir).ppt
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AGC
DSP
AGC
DSP
Professor A G Constantinides
IIR Digital Filter DesignIIR Digital Filter Design
Standard approach
(1) Convert the digital flter specifcations
into an analogue prototype lowpassflter specifcations
(2) Determine the analogue lowpassflter transer unction
(!) "ransorm #y replacing thecomple$ varia#le to the digital transerunction
)(sHa
)(zG
)(sHa
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AGC
DSP
AGC
DSP
Professor A G Constantinides
IIR Digital Filter DesignIIR Digital Filter Design
"his approach has #een widely usedor the ollowing reasons%
(1) Analogue appro$imation
techni&ues are highly advanced(2) "hey usually yield closed'ormsolutions
(!) $tensive ta#les are availa#le or
analogue flter design() *ery oten applications re&uiredigital simulation o analogue systems
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AGC
DSP
AGC
DSP
Professor A G Constantinides
IIR Digital Filter DesignIIR Digital Filter Design
+et an analogue transer unction#e
where the su#script ,a- indicates
the analogue domain A digital transer unction derived
rom this is denoted as
)(
)()(
sD
sPsH
a
aa =
)(
)()(
zD
zPzG =
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AGC
DSP
AGC
DSP
Professor A G Constantinides
IIR Digital Filter DesignIIR Digital Filter Design .asic idea #ehind the conversion o
into is to apply a mapping rom thes'domain to thez'domain so that essentialproperties o the analogue re&uency
response are preserved "hus mapping unction should #e such that
/maginary ( ) a$is in the s'plane #emapped onto the unit circle o thez'plane
A sta#le analogue transer unction #emapped into a sta#le digital transerunction
)(sHa)(zG
j
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AGC
DSP
AGC
DSP
Professor A G Constantinides
IIR Digital Filter: The bilinearIIR Digital Filter: The bilinear
transformationtransformation
"o o#tain G(z)replace s#y f(z) inH(s)
Start with re&uirements on G(z)G(z) Availa#le H(s)
Sta#le Sta#le
0eal and 0ational inz 0eal and0ational in s
rder n rder n
+P (lowpass) cuto3 +P cuto3 Tcc
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AGC
DSP
AGC
DSP
Professor A G Constantinides
IIR Digital FilterIIR Digital Filter
4ence is real and rational inzo order one
ie
5or +P to +P transormation were&uire
"hus
)(zf
dcz
bazzf
++=)(
10 == zs 00)1( =+= baf
1 == zjs 0)1( == dcjf
1
1.)(
+
=
z
z
c
azf
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AGC
DSP
AGC
DSP
Professor A G Constantinides
IIR Digital FilterIIR Digital Filter
"he &uantity is f$ed rom
ie on
r
and
ca
ccT
2
tan.)(1: T
j
c
azfzC
c
==
2tan. T
jc
aj cc
=
1
1
1
1.
2
tan
+
=
z
z
Ts
c
c
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Bilinear TransformationBilinear Transformation "ransormation is una3ected #y scaling
Consider inverse transormation withscale actor e&ual to unity
5or
and so
ssz
+=11
oo js +=22
222
)1(
)1(
)1(
)1(
oo
oo
oo
oo zj
jz
+++=
++=
10 == zo10 > zo
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Bilinear TransformationBilinear Transformation 6apping o s'plane into thez'plane
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Bilinear TransformationBilinear Transformation
5or with unity scalarwehave
or
)2/tan(1
1
je
ejj
j
=+=
j
ez=
)2/tan(=
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Bilinear TransformationBilinear Transformation
6apping is highly nonlinear Complete negative imaginary a$is in
the s'plane rom to is mapped
into the lower hal o the unit circle inthez'plane rom to
Complete positive imaginary a$is inthe s'plane rom to is
mapped into the upper hal o the unitcircle in thez'plane rom to
= 0=
0= =
1=z 1=z
1=z 1=z
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Bilinear TransformationBilinear Transformation 7onlinear mapping introduces a
distortion in the re&uency a$iscalled frequency warping
3ect o warping shown #elow
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Spectral TransformationsSpectral Transformations"o transorm a given lowpass
transer unction to another transer
unction that may #e a lowpass8highpass8 #andpass or #andstop flter(solutions given #y Constantinides)
has #een used to denote the unit
delay in the prototype lowpass flterand to denote the unit delay in
the transormed flter to avoidconusion
)(zGL
)(zGD
1
z
1
z )(zGL
)(zGD
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Spectral TransformationsSpectral Transformations 9nit circles inz' and 'planes
defned #y
8
"ransormation romz'domain to
'domain given #y
"hen
z
z
jez= jez=
)(zFz=
)(!)( zFGzG LD =
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Spectral TransformationsSpectral Transformations 5rom 8 thus 8
hence
"hereore must #e a sta#leallpass unction
)(zFz= )(zFz=
>
1if"1
1if"1
1if"1
)(
z
z
z
zF
)(/1 zF
1"
1
)
(
1
1
#
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-LowpassLowpass-to-Lowpass
Spectral TransformationSpectral Transformation "o transorm a lowpass flter with a
cuto3 re&uency to another lowpassflter with a cuto3re&uency 8 the transormation is
n the unit circle we have
which yields
)(zGL
)(zGDc
c
== z zzFz
1
)(11
1 j
jj
eee
=
)2/tan(
1
1)2/tan(
+=
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-LowpassLowpass-to-Lowpass
Spectral TransformationSpectral Transformation Solving we get
$ample' Consider the lowpass digital
flter
which has a pass#and rom dctowith a :;d. ripple
0edesign the a#ove flter to move the
pass#and edge to
( )( )2/)(sin
2/)(sin
cc
cc
+=
)3917.06763.01)(2593.01(
)1(0662.0)(
211
31
++=
zzz
zzGL
25.0
35.0
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-LowpassLowpass-to-Lowpass
Spectral TransformationSpectral Transformation 4ere
4ence8 the desired lowpass transer
unction is
1934.0)3.0sin(
)05.0sin( ==
1
11
1934.01
1934.0)()(
++==
z
zzLD
zGzG
0 0.2 0.4 0.6 0.8 1-40
-30
-20
-10
0
/
Gain,
dB G
L(z) G
D(z)
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-LowpassLowpass-to-Lowpass
Spectral TransformationSpectral Transformation"he lowpass'to'lowpass
transormation
can also #e used as highpass'to'
highpass8 #andpass'to'#andpassand#andstop'to'#andstoptransormations
==
z
z
zFz
1
)(
11
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-HighpassLowpass-to-Highpass
Spectral TransformationSpectral Transformation Desired transormation
"he transormation parameter is given #y
where is the cuto3 re&uency o thelowpass flter and is the cuto3 re&uencyo the desired highpass flter
1
11
1
+
+=z
zz
( )
( )2/)($os
2/)($os
cc
cc
+=
c
c
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-HighpassLowpass-to-Highpass
Spectral TransformationSpectral Transformation $ample'"ransorm the lowpass
flter
with a pass#and edge at to ahighpass flter with a pass#and edge
at 4ere
"he desired transormation is
)3917.06763.01)(2593.01(
)1(0662.0)(
211
31
++=
zzz
zzGL
25.055.0
3468.0)15.0$os(/)4.0$os( ==
1
11
3468.01
3468.0
=
z
zz
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-HighpassLowpass-to-Highpass
Spectral TransformationSpectral Transformation
"he desired highpass flter is
1
11
3468.01
3468.0)()(
==
z
z
zD
zGzG
0 0.2 0.4 0.6 0.8
80
6040
20
0
Normalized frequency
Gain,
dB
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-HighpassLowpass-to-Highpass
Spectral TransformationSpectral Transformation"he lowpass'to'highpass
transormation can also #e used to
transorm a highpass flter with acuto3 at to a lowpass flter with acuto3 at
and transorm a #andpass flter with acenter re&uency at to a #andstopflter with a center re&uency at
cc
oo
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-BandpassLowpass-to-Bandpass
Spectral TransformationSpectral Transformation Desired transormation
11
2
1
1
1
11
2
12
12
1
++
+
+
++=
zz
zzz
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-BandpassLowpass-to-Bandpass
Spectral TransformationSpectral Transformation "he parameters and are given #y
where is the cuto3 re&uency o thelowpass flter8 and and are thedesired upper and lower cuto3 re&uencieso the #andpass flter
( ) )2/tan(2/)($ot 12 ccc =
( )
( )2/)($os
2/)($os
12
12
cc
cc
+
=
c1c 2c
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-BandpassLowpass-to-Bandpass
Spectral TransformationSpectral Transformation Special Case'"he transormation
can #e simplifed i"hen the transormation reduces to
where withdenoting the desired centerre&uency o the #andpass flter
12 ccc =
o $os= o
1
111
1
=z
zzz
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-BandstopLowpass-to-Bandstop
Spectral TransformationSpectral Transformation Desired transormation
1
1
2
1
1
11
12
12
12
1
+
+
+
+++=
zz
zz
z
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AGC
DSP
AGC
DSP
Professor A G Constantinides
Lowpass-to-BandstopLowpass-to-Bandstop
Spectral TransformationSpectral Transformation"he parameters and are given
#y
where is the cuto3 re&uency othe lowpass flter8 and and arethe desired upper and lower cuto3re&uencies o the #andstop flter
c
1c 2c
( )
( )2/)($os
2/)($os
12
12
cc
cc
+=
( ) )2/tan(2/)(tan 12 ccc =
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