ch7-1 rtentitivity

8/11/2019 ch7-1 rtentitivity

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1Copyright 2001, S. K. Mitra

Digital Filter DesignDigital Filter Design

Objective- Determination of a realizable

transfer function G(z)approximating agiven frequency response specification is an

important step in the development of a

digital filter

If an IIR filter is desired, G(z)should be a

stable real rational function

Digital filter design is the process of

deriving the transfer function G(z)


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Digital Filter SpecificationsDigital Filter Specifications

Usually, either the magnitudeand/or the

phase(delay) response is specified for the

design of digital filter for most applications

In some situations, the unit sample response

or the step responsemay be specified

In most practical applications, the problem

of interest is the development of a realizableapproximation to a given magnitude

response specification


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Digital Filter SpecificationsDigital Filter Specifications We discuss in this course only the

magnitude approximation problem

There are four basic types of ideal filterswith magnitude responses as shown below

1

0 c

c

HLP(ej )

0 c

c

1

HHP(ej )

11

c1 c1 c2 c2

HBP (ej )

1

c1 c1 c2 c2

HBS(ej )


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As the impulse response corresponding to

each of these ideal filters is noncausal and

of infinite length, these filters are notrealizable

In practice, the magnitude responsespecifications of a digital filter in the

passband and in the stopband are given with

some acceptable tolerances

In addition, a transition band is specified

between the passband and stopband


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Digital Filter SpecificationsDigital Filter Specifications For example, the magnitude response

of a digital lowpass filter may be given as

indicated below

)( !jeG


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As indicated in the figure, in the passband,

defined by , we require that

with an error , i.e.,

In the stopband, defined by , we

require that with an error ,i.e.,

1)( "!jeG

0)( "

!j

eG s#

p#$

p!%!%0

&%!%!s

ppj

p eG !%!#'%%#( ! ,1)(1

&%!%!#%! ssjeG ,)(


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- passband edge frequency

- stopband edge frequency - peak ripple valuein thepassband

- peak ripple valuein the stopband Since is a periodic function of !,

and of a real-coefficient digital

filter is an even function of ! As a result, filter specifications are given

only for the frequency range

p!

s!

s#

p#

)( !jeG)( !jeG

&%!%0


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Specifications are often given in terms ofloss function A in

dB

Peak passband ripple

dB

Minimum stopband attenuation

dB

)(log20)( 10!()! jeG

)1(log20 10 pp #(()*

)(log20 10 ss #()*


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Digital Filter SpecificationsDigital Filter Specifications Magnitude specifications may alternately be

given in a normalized form as indicated

below


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Here,the maximum value of the magnitude

in the passband is assumed to be unity

- Maximum passband deviation,

given by the minimum value of the

magnitude in the passband

- Maximum stopband magnitudeA1

21/1 +'


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For the normalized specification, maximumvalue of the gain function or the minimum

value of the loss function is0 dB

Maximum passband attenuation-

dB

For , it can be shown that

dB

210max 1log20 +')*

1..#p)21(log20 10max p#(("*


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Digital Filter SpecificationsDigital Filter Specifications In practice, passband edge frequency

and stopband edge frequency are

specified in Hz For digital filter design, normalized

bandedge frequencies need to be computed

from specifications in Hz using

TFF

F

F pT

p

T

p

p &)

&

)

/

)! 2

2

TF

F

F

F s

T

s

T

ss &)

&)

/)! 2

2

sFpF


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Example- Let kHz, kHz, andkHz

Then

7)pF 3)sF25)TF

&)00&)! 56.01025

)107(23

3

p

&)0

0&)! 24.0

1025

)103(23

3

s


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The transfer function H(z)meeting thefrequency response specifications should be

a causal transfer function For IIR digital filter design, the IIR transfer

function is a real rational function of :

H(z) must be a stable transfer function and

must be of lowest order Nfor reduced

computational complexity

Selection of Filter TypeSelection of Filter Type

1(z

NM

zdzdzdd

zpzpzppzH

N

N

M %

''''

'''') (((

(((,)(

2

2

1

10

22

110

!

!


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For FIR digital filter design, the FIRtransfer function is a polynomial inwith real coefficients:

For reduced computational complexity,degree Nof H(z)must be as small as

possible If a linear phase is desired, the filter

coefficients must satisfy the constraint:

1))

(

n

nznhzH0

][)(

][][ nNhnh ($)

1(z


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Advantages in using an FIR filter-

(1) Can be designed with exact linear phase,

(2) Filter structure always stable withquantized coefficients

Disadvantages in using an FIR filter- Orderof an FIR filter, in most cases, is

considerably higher than the order of an

equivalent IIR filter meeting the same

specifications, and FIR filter has thus higher

computational complexity


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Digital Filter Design:Digital Filter Design:

Basic ApproachesBasic Approaches

Most common approach to IIR filter design-(1) Convert the digital filter specifications

into an analog prototype lowpass filter

specifications

(2) Determine the analog lowpass filter

transfer function (3) Transform into the desired digital

transfer function

)(sa

)(zG

)(sHa

Di it l Filt D i


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Basic ApproachesBasic Approaches This approach has been widely used for the

following reasons:

(1) Analog approximation techniques arehighly advanced

(2) They usually yield closed-formsolutions

(3) Extensive tables are available foranalog filter design

(4) Many applications require digital

simulation of analog systems


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Basic ApproachesBasic Approaches An analog transfer function to be denoted as

where the subscript a specificallyindicates the analog domain

A digital transfer function derived fromshall be denoted as

)()()(sDsPsH

a

aa )

)(

)()(

zD

zPzG )

)(sa

Digital Filter DesignDigital Filter Design:


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Basic ApproachesBasic Approaches Basic idea behind the conversion of

into is to apply a mapping from the

s-domain to the z-domain so that essential

properties of the analog frequency response

are preserved

Thus mapping function should be such that

Imaginary ( ) axis in the s-plane bemapped onto the unit circle of the z-plane

A stable analog transfer function be mapped

into a stable digital transfer function

)(sHa)(zG

/j


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Basic ApproachesBasic Approaches

Three commonly used approaches to FIRfilter design-

(1)Windowed Fourier series approach(2)Frequency sampling approach

(3) Computer-based optimization methods

IIR Di it l Filt D i BiliIIR Di it l Filt D i Bili


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IIR Digital Filter Design: BilinearIIR Digital Filter Design: Bilinear

Transformation MethodTransformation Method Bilinear transformation-

Above transformation maps a single pointin the s-plane to a unique point in the

z-plane and vice-versa

Relation between G(z)and is then

given by

4467789 '() (

(

1

1

112

zz

Ts

)(sa

446

778

9

('

(

())

11

1

12)()(

zzTsa

sHzG


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Bilinear TransformationBilinear Transformation

Digital filter design consists of 3 steps:

(1) Develop the specifications of byapplying the inverse bilinear transformation

to specifications ofG(z)

(2) Design

(3) Determine G(z)by applying bilinear

transformation to As a result, the parameter Thas no effect on

G(z)and T= 2is chosen for convenience

)(sa

)(sa

)(sa


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Bilinear TransformationBilinear Transformation Inverse bilinear transformation for T= 2is

For

Thus,

s

sz

(

')1

1

oo js /':)

22

222

)1(

)1()1()1(

oo

oo

oo

oo zjjz

/':(/':');

/(:(/':')

10 )


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Mapping of s-plane into the z-plane


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Bilinear TransformationBilinear Transformation For with T= 2we have

or

)(

)(

1

1

2/2/2/

2/2/2/

!(!!(

!(!!(

!(

!(

'

()

'

()/jjj

jjj

j

j

eee

eee

e

ej

!) jez

)2/tan(!)/

)2/tan(2

)2/cos(2

)2/sin(2!)

!

!) jj


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Mapping is highly nonlinear

Complete negative imaginary axis in thes-

plane from to is mapped into

the lower half of the unit circle in the z-plane

from to Complete positive imaginary axis in the s-

plane from to is mapped into theupper half of the unit circle in the z-plane

from to

(>)/ 0)/

0)/ >)/

1()z 1)z

1)z 1()z


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Nonlinear mapping introduces a distortionin the frequency axis called frequency

warping Effect of warping shown below

Bili T f tiBili T f ti


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Steps in the design of a digital filter-(1) Prewarp to find their analogequivalents

(2) Design the analog filter

(3) Design the digital filter G(z)by applying

bilinear transformation to Transformation can be used only to design

digital filters with prescribed magnituderesponse with piecewise constant values

Transformation does not preserve phase

response of analog filter

),( sp !!),(

sp//

)(sa

)(sa

IIR Digital Filter Design UsingIIR Digital Filter Design Using


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Bilinear TransformationBilinear Transformation Example- Consider

Applying bilinear transformation to the above

we get the transfer function of a first-order

digital lowpass Butterworth filter

c

cassH /'

/))(

)1()1(

)1()()(

11

1

11

11 ((

(

('

(() '/'('/

))zz

zsHzG

c

c

z

zsa


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Rearranging terms we get

where

1

1

1

12

1)( (

(

(

'?()

z

zzG

*

*

)2/tan(1

)2/tan(1

1

1

cc

cc !'

!()/'

/()*



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Bilinear TransformationBilinear Transformation Example- Consider the second-order analognotch transfer function

for which

is called the notch frequency

If then

is the 3-dB notch bandwidth

22

22

)(o

oa

sBs

ssH

/''/')

0)( )/oa jH

1)()0( )>) jj aao/

2/1)()( 12 )/)/ jHjH aa12 /(/)B



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Bilinear TransformationBilinear Transformation Then

where

1

1

11)()(

(

(

'()

)zzsa

sHzG

22122

22122

)1()1(2)1()1()1(2)1(

((((

(/''/(('/'/''/((/')

zBzBzz

ooo

ooo

21

21

)1(2121

21

((

((

''('(?')

zzzz

**@@*

o

o

o !)/'/()@ cos

11

2

2)2/tan(1)2/tan(1

11

2

2

w

w

o

o

BB

BB

'()

'/'(/')*



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Bilinear TransformationBilinear Transformation Example- Design a 2nd-order digital notchfilter operating at a sampling rate of 400 Hz

with a notch frequency at 60 Hz, 3-dBnotch

bandwidth of 6 Hz

Thus

From the above values we get

&)&)! 3.0)400/60(2o&)&) 03.0)400/6(2wB

90993.0)*

587785.0)@



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IIR Digital Filter Design Usingg a e es g Us g

Bilinear TransformationBilinear Transformation Thus

The gain and phase responses are shown below

21

21

90993.01226287.11954965.01226287.1954965.0)( ((

((

'( '() zzzzzG

0 50 100 150 200-40

-30

-20

-10

0

Frequency, Hz

Gain,

dB

0 50 100 150 200-2

-1

0

1

2

Frequency, Hz

Phase,radians


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IIRIIR LowpassLowpass Digital Filter DesignDigital Filter Design


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IIRIIRLowpassLowpassDigital Filter DesignDigital Filter Design

Using Bilinear TransformationUsing Bilinear Transformation

Thus

We chooseN= 3

To determine we use

6586997.2)/1(log

)/1(log

10

110 ))k

kN

c/

22

2

11

)/(11)( +')//')/ Ncp

pa jH

IIRIIR LowpassLowpass Digital Filter DesignDigital Filter Design


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Using Bilinear TransformationUsing Bilinear Transformation We then get

3rd-order lowpass Butterworth transfer

function for is

Denormalizing to get we

arrive at

588148.0)(419915.1 )/)/ pc

1)/c

)1)(1(

1)(

2 ''')

ssssHan

4

6

78

9

) 588148.0)( s

HsH ana

588148.0)/c



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pp g gg g

Using Bilinear TransformationUsing Bilinear Transformation

Applying bilinear transformation to

we get the desired digital transfer function

Magnitude and gain responses of G(z)shown

below:

)(sHa

1

1

11)()(

(

(

'())zzsa szG

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

!/&

Magnitud

e

0 0.2 0.4 0.6 0.8 1-40

-30

-20

-10

0

!/&

Gain,dB

Design of IIRDesign of IIR HighpassHighpass,, BandpassBandpass,,


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es g og g passg p ,, a dpassp ,,

andand BandstopBandstopDigital FiltersDigital Filters First Approach-

(1) Prewarp digital frequency specificationsof desired digital filter to arrive at

frequency specifications of analog filter

of same type

(2) Convert frequency specifications of

into that of prototype analog lowpass filter

(3) Design analog lowpass filter

)(zGD)(sD

)(sHD

)(sLP

)(sLP


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D i f IIRD i f IIR Hi hHi h B dB d


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Design of IIRDesign of IIRHighpassHighpass,,BandpassBandpass,,

andandBandstopBandstopDigital FiltersDigital Filters

Second Approach-

(1) Prewarp digital frequency specifications

of desired digital filter to arrive at

frequency specifications of analog filterof same type

(2) Convert frequency specifications ofinto that of prototype analog lowpass filter

)(zGD)(sD

)(sLP

)(sD

D i f IIRD i f IIR Hi hHi h B dB d


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Design of IIRDesign of IIRHighpassHighpass,,BandpassBandpass,,

andandBandstopBandstopDigital FiltersDigital Filters

(3) Design analog lowpass filter

(4) Convert into an IIR digital

transfer function using bilinear

transformation

(5) Transform into the desired

digital transfer function We illustrate the first approach

)(sLP

)(sLP)(zGLP

)(zGLP

)(zGD

IIRIIR HighpassHighpass Digital Filter DesignDigital Filter Design


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IIRIIRHighpassHighpassDigital Filter DesignDigital Filter Design

Design of a Type 1 Chebyshev IIR digitalhighpass filter

Specifications: Hz, Hz, dB, dB, kHz

Normalized angular bandedge frequencies

700)pF 500)sF2)TF1)*p 32)*s

&)0&

)&

)! 7.02000

70022

T

pp

F

F

&)0&

)&

)! 5.0

2000

50022

T

ss

F

F


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IIRIIR HighpassHighpass Digital Filter DesignDigital Filter Design


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IIRHighpassg p Digital Filter Designg g

MATLAB code fragments used for the design[N, Wn] = cheb1ord(1, 1.9626105, 1, 32, s)

[B, A] = cheby1(N, 1, Wn, s);[BT, AT] = lp2hp(B, A, 1.9626105);

[num, den] = bilinear(BT, AT, 0.5);

0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10

0

!/&

Gain,

dB

IIRIIR BandpassBandpass Digital Filter DesignDigital Filter Design


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IIRIIR BandpassBandpassDigital Filter DesignDigital Filter Design

Design of a Butterworth IIR digital bandpassfilter

Specifications: , , , , dB, dB

Prewarping we get

&)! 45.01p &)! 65.02p&)! 75.02s&)! 3.01s 1)*p 40)*s

8540807.0)2/tan( 11 )!)/ pp

6318517.1)2/tan( 22 )!)/ pp

41421356.2)2/tan( 22 )!)/ ss

5095254.0)2/tan( 11 )!)/ ss

IIRIIR BandpassBandpass Digital Filter DesignDigital Filter Design


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pp g gg g

Width of passband

We therefore modify so that and

exhibit geometric symmetry withrespect to

We set For the prototype analog lowpass filter we

choose

777771.0 12 )/(/) ppwB

393733.1 212 )//)/ ppo

221

23010325.1 oss /B)//

1s/

2 s/

1s/

2o/

5773031.0 1)/s

1)/p

IIRIIRBandpassBandpassDigital Filter DesignDigital Filter Design


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pp g gg g

Using we get

Specifications of prototype analog

Butterworth lowpass filter:

, , dB,

dB

w

op

B//(//()/

22

3617627.2777771.05773031.0

3332788.0393733.1)

0()/s

1)/p 3617627.2)/s 1)*p40)*s

IIRIIRBandpassBandpassDigital Filter DesignDigital Filter Design


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p g g

MATLAB code fragments used for the design[N, Wn] = buttord(1, 2.3617627, 1, 40, s)

[B, A] = butter(N, Wn, s);[BT, AT] = lp2bp(B, A, 1.1805647, 0.777771);


0 0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10

0

!/&

Gain,

dB

IIRIIR BandstopBandstopDigital Filter DesignDigital Filter Design


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pp g gg g

Design of an elliptic IIR digital bandstop filter Specifications: , ,

, , dB, dB Prewarping we get

Width of stopband

&)! 45.01s &)! 65.02s

&)! 75.02p&)! 3.01p 1)*p 40)*s

,8540806.0 1)/s ,6318517.1 2)/s,5095254.0 1)/p 4142136.2 2)/p

777771.0 12 )/(/) sswB393733.1 12

2 )//)/ sso

212 230103.1 opp /B)//

IIRIIR BandstopBandstop Digital Filter DesignDigital Filter Design


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IIRIIRBandstopBandstopDigital Filter DesignDigital Filter Design

We therefore modify so that andexhibit geometric symmetry with respect to

We set

For the prototype analog lowpass filter wechoose

Using we get

1 p/ 1 p/ 2 p/

2

o/577303.0 1)/p

1)/s

22

/(/

/

/)/ o

w

s

B

0.42341263332787.0393733.1

777771.05095254.0)

(

0)/

p

IIRIIRBandstopBandstopDigital Filter DesignDigital Filter Design


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pp g gg g

MATLAB code fragments used for the design[N, Wn] = ellipord(0.4234126, 1, 1, 40, s);

[B, A] = ellip(N, 1, 40, Wn, s);[BT, AT] = lp2bs(B, A, 1.1805647, 0.777771);


0 0.2 0.4 0.6 0.8 1-50

-40

-30-20

-10

0

!/&

Gain,

dB

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