2-Digital Filters (IIR)
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Transcript of 2-Digital Filters (IIR)
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8/2/2019 2-Digital Filters (IIR)
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AGC
DSP
Professor A G Constantinides Professor A G 11
IIR Digital Filter Design
Standard approach
(1) Convert the digital filter specifications
into an analogue prototype lowpass filterspecifications
(2) Determine the analogue lowpass filter
transfer function(3) Transform by replacing thecomplex variable to the digital transferfunction
)(sHa
)(zG
)(sHa
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AGC
DSP
Professor A G Constantinides Professor A G 22
IIR Digital Filter Design
n Let an analogue transfer function be
where the subscript a indicates theanalogue domain
n
A digital transfer function derived fromthis is denoted as
)(
)()(
sD
sPsH
a
aa =
)(
)()(
zD
zPzG =
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AGC
DSP
Professor A G Constantinides Professor A G 33
IIR Digital Filter Design
n Basic idea behind the conversion ofinto is to apply a mapping from thes-domain to the z-domain so that essentialproperties of the analogue frequencyresponse are preserved
n Thus mapping function should be such thatn Imaginary ( ) axis in the s-plane be
mapped onto the unit circle of the z-planen A stable analogue transfer function be
mapped into a stable digital transfer
function
)(sHa)(zG
j
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AGC
DSP
Professor A G Constantinides Professor A G 44
IIR Digital Filter: The bilineartransformation
n To obtain G(z) replace sby f(z) in H(s)
n Start with requirements on G(z)
G(z) Available H(s)
Stable Stable
Real and Rational in z Real and Rational ins
Order n Order n
L.P. (lowpass) cutoff L.P. cutoff Tcc
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AGC
DSP
Professor A G Constantinides Professor A G 55
IIR Digital Filtern
Hence is real and rational in zoforder one
n i.e.
n For LP to LP transformation we require
n Thus
)(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
Professor A G Constantinides Professor A G 66
IIR Digital Filtern
The quantity is fixed from
n ie on
n Or
n and
ca
ccT
2
tan.)(1:T
j
c
azfzC
c
==
2tan.
Tj
c
aj cc
=
1
1
1
1.
2
tan
+
=
z
z
Ts
c
c
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AGC
DSP
Professor A G Constantinides Professor A G 77
Bilinear Transformationn Transformation is unaffected by scaling.
Consider inverse transformation with scalefactor equal to unity
n For
n 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
Professor A G Constantinides Professor A G 88
Bilinear Transformation
n Mapping ofs-plane into the z-plane
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AGC
DSP
Professor A G Constantinides Professor A G 99
Bilinear Transformationn
For with unity scalarwe have
or
)2/tan(1
1
je
ejj
j
=+=
j
ez=
)2/tan(=
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AGC
DSP
Professor A G Constantinides Professor A G 1010
Bilinear Transformation
n Mapping is highly nonlinearn Complete negative imaginary axis in the
s-plane from to is mapped
into the lower half of the unit circle inthe z-plane from to
n Complete positive imaginary axis in the
s-plane from to is mappedinto the upper half of the unit circle inthe z-plane from to
= 0=
0= =
1=z 1=z
1=z 1=z
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AGC
DSP
Professor A G Constantinides Professor A G 1111
Bilinear Transformationn Nonlinear mapping introduces a
distortion in the frequency axis calledfrequency warpingn Effect of warping shown below
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AGC
DSP
Professor A G Constantinides Professor A G 1212
Spectral Transformations
n To transform a given lowpasstransfer function to another transfer
function that may be a lowpass,highpass, bandpass or bandstop filter(solutions given by Constantinides)
n
has been used to denote the unitdelay in the prototype lowpass filterand to denote the unit delay in
the transformed filter to avoid
confusion
)(zGL
)(zGD
1
z
1
z )(zGL
)(zGD
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AGC
DSP
Professor A G Constantinides Professor A G 1313
Spectral Transformations
n Unit circles in z- and -planes definedby
,
n Transformation from z-domain to
-domain given by
n Then
z
z
jez= jez=
)(zFz=
)}({)( zFGzG LD =
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AGC
DSP
Professor A G Constantinides Professor A G 1414
Spectral Transformations
n From , thus ,hence
n Therefore must be a stable allpassfunction
)(zFz= )(zFz =
>
1if,1
1if,1
1if,1
)(
z
z
z
zF
)(/1 zF
1,
1
)
(
1
1
*
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AGC
DSP
Professor A G Constantinides Professor A G 1515
Lowpass-to-LowpassSpectral Transformationn To transform a lowpass filter with a
cutoff frequency to another lowpass filterwith a cutoff frequency , the
transformation is
n On 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
Professor A G Constantinides Professor A G 1616
Lowpass-to-LowpassSpectral Transformationn Solving we get
n Example - Consider the lowpass digital
filter
which has a passband from dc towith a 0.5 dB ripple
n Redesign the above filter to move the
passband 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
Professor A G Constantinides Professor A G 1717
Lowpass-to-LowpassSpectral Transformationn Here
n Hence, the desired lowpass transfer
function 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
Professor A G Constantinides Professor A G 1818
Lowpass-to-Lowpass
Spectral Transformationn The lowpass-to-lowpass transformation
can also be used as highpass-to-highpass, bandpass-to-bandpass andbandstop-to-bandstop transformations
==
z
z
zFz
1
)(
11
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AGC
DSP
Professor A G Constantinides Professor A G 1919
Lowpass-to-HighpassSpectral Transformationn Desired transformation
n The transformation parameter is given by
where is the cutoff frequency of thelowpass filter and is the cutoff frequency of
the desired highpass filter
1
11
1
+
+=z
zz
( )
( )2/)(cos
2/)(cos
cc
cc
+=
c
c
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AGC
DSP
Professor A G Constantinides Professor A G 2020
Lowpass-to-HighpassSpectral Transformationn Example - Transform the lowpass filter
n with a passband edge at to ahighpass filter with a passband edge at
n Here
n The desired transformation is
)3917.06763.01)(2593.01(
)1(0662.0)(
211
31
+
+=
zzz
zzGL
25.055.0
3468.0)15.0cos(/)4.0cos( ==
1
11
3468.01
3468.0
=
z
zz
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AGC
DSP
Professor A G Constantinides Professor A G 2121
Lowpass-to-HighpassSpectral Transformation
n The desired highpass filter is
1
11
3468.01
3468.0)()(
=
=
z
z
zD
zGzG
0 0.2 0.4 0.6 0.8
80
60
40
20
0
Normalized frequency
an,
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AGC
DSP
Professor A G Constantinides Professor A G 2222
Lowpass-to-Highpass
Spectral Transformationn The lowpass-to-highpass transformation
can also be used to transform a
highpass filter with a cutoff at to alowpass filter with a cutoff at
n and transform a bandpass filter with a
center frequency at to a bandstopfilter with a center frequency at
cc
o
o
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AGC
DSP
Professor A G Constantinides Professor A G 2323
Lowpass-to-Bandpass
Spectral Transformationn Desired transformation
11
2
1
1
1
11
2
12
12
1
++
+
+
++=
zz
zzz
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AGC
DSP
Professor A G Constantinides Professor A G 2424
Lowpass-to-Bandpass
Spectral Transformationn The parameters and are given by
where is the cutoff frequency of thelowpass filter, and and are thedesired upper and lower cutoff frequencies ofthe bandpass filter
( ) )2/tan(2/)(cot 12 ccc =
( )
( )2/)(cos
2/)(cos
12
12
cc
cc
+
=
c1c 2c
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AGC
DSP
Professor A G Constantinides Professor A G 2525
Lowpass-to-Bandpass
Spectral Transformationn Special Case - The transformation can
be simplified ifn Then the transformation reduces to
where with denotingthe desired center frequency of thebandpass filter
12 ccc =
o cos= o
1
111
1
=z
zzz
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AGC
DSP
Professor A G Constantinides Professor A G 2626
Lowpass-to-Bandstop
Spectral Transformationn Desired transformation
1
1
2
1
1
11
12
12
12
1
+
+
+
+++=
zz
zz
z
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AGC
DSP
Professor A G ConstantinidesProfessor A G 2727
Lowpass-to-Bandstop
Spectral Transformationn The parameters and are given
by
where is the cutoff frequency ofthe lowpass filter, and and arethe desired upper and lower cutofffrequencies of the bandstop filter
c
1c 2c
( )
( )2/)(cos
2/)(cos
12
12
cc
cc
+
=
( ) )2/tan(2/)(tan 12 ccc =