2-digital filters (iir)

8/2/2019 2-Digital Filters (IIR)

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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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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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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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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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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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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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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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Bilinear Transformation

n Mapping ofs-plane into the z-plane


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Bilinear Transformationn

For with unity scalarwe have

or

)2/tan(1

1

je

ejj

j

=+=

j

ez=

)2/tan(=


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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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Bilinear Transformationn Nonlinear mapping introduces a

distortion in the frequency axis calledfrequency warpingn Effect of warping shown below


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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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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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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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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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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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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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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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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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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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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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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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Lowpass-to-Bandpass

Spectral Transformationn Desired transformation

11

2

1

1

1

11

2

12

12

1

++

+

+

++=

zz

zzz


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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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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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Lowpass-to-Bandstop

Spectral Transformationn Desired transformation

1

1

2

1

1

11

12

12

12

1

+

+

+

+++=

zz

zz

z


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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 =

2-digital filters (iir)

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