2-digital filters (iir).ppt

5/18/2018 2-Digital filters (IIR).ppt

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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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"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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+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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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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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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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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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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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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Bilinear TransformationBilinear Transformation 6apping o s'plane into thez'plane


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Bilinear TransformationBilinear Transformation

5or with unity scalarwehave

or

)2/tan(1

1

je

ejj

j

=+=

j

ez=

)2/tan(=


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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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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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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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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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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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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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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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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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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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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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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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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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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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Lowpass-to-BandpassLowpass-to-Bandpass


11

2

1

1

1

11

2

12

12

1

++

+

+

++=

zz

zzz


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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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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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Lowpass-to-BandstopLowpass-to-Bandstop


1

1

2

1

1

11

12

12

12

1

+

+

+

+++=

zz

zz

z


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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 =

2-digital filters (iir).ppt

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