6666a92d8fdd2Week13.pdf digital image processing

Digital image processing

1. Digital Signal Processing
Spring 2024
1Zulaikha Kiran, 2024

2. Week 13
Material taken from:
Discrete time Signal Processing by Oppenheim and Schafer –3
rd
Edition
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3. Filter Design
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4. •The design of filters involves the following stages:
•the specification of the desired properties of the system
•Dependent on application
•the approximation of the specifications using a causal discrete timesystem
•the realization of the system.
•Dependent on available technologies
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5. •Inapracticalsetting,thedesiredfilterisgenerallyimplementedwith
digitalhardwareandoftenusedtofilterasignalthatisderivedfroma
continuous-timesignalbymeansofperiodicsamplingfollowedby
A/Dconversion.
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6. Low pass filter tolerance scheme
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7. Discrete Time IIR Filters from
Continuous Time Filters
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8. Why use continuous time filter as starting
point?
•The art of continuous-time filter design is highly advanced
•Many designs are simple
•Approximations that work for continuous time, do not directly work
for discrete time design, since frequency response is periodic
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9. Design Procedure
•Thedesignprocedureforthediscrete-timesystembeginsfromasetof
discrete-timespecifications.
•Weassumethatthesespecificationshavebeenappropriatelydetermined.
•Thespecificationsforthecontinuous-timefilterareobtainedbya
transformationofthespecificationsforthedesireddiscrete-timefilter.
•ThesystemfunctionHc(s)orimpulseresponsehc(t)ofthecontinuous-time
filteristhenobtainedthroughoneoftheestablishedapproximation
methodsusedforcontinuous-timefilterdesign.
•Next,thesystemfunctionH(z)orimpulseresponseh[n]forthediscrete-
timefilterisobtainedbyapplyingatransformationtoHc(s)orhc(t)
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10. •Intransformations,wegenerallyrequirethattheessentialproperties
ofthecontinuous-timefrequencyresponsebepreservedinthe
frequencyresponseoftheresultingdiscrete-timefilter.
•Wewanttheimaginaryaxisofthes-planetomapontotheunitcircleofthez-
plane.
•Astablecontinuous-timefiltershouldbetransformedtoastablediscrete-
timefilter.
•Ifthecontinuous-timesystemhaspolesonlyinthelefthalfofthes-plane,thenthe
discrete-timefiltermusthavepolesonlyinsidetheunitcircleinthez-plane
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11. Impulse Invariance
•Intheconceptofimpulseinvariance,adiscrete-timesystemis
definedbysamplingtheimpulseresponseofacontinuous-time
system
•providesadirectmeansofcomputingsamplesoftheoutputofabandlimited
continuous-timesystemforbandlimitedinputsignals
•iftheoverallobjectiveistosimulateacontinuous-timesystemina
discrete-timesetting,wemightdesignthediscrete-timesystemsuch
thatitsimpulseresponsecorrespondstosamplesofthecontinuous-
time
•ORitmightbedesirabletomaintain,inadiscretetimesetting,certain
time-domaincharacteristicsofwell-developedcontinuous-timefilters
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12. •in the context of filter design,
•impulseinvarianceisamethodforobtainingadiscrete-timesystemwhose
frequencyresponseisdeterminedbythefrequencyresponseofa
continuous-timesystem.
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13. Impulse Invariance
•the impulse response of the discrete-time filter is chosen
proportional to equally spaced samples of the impulse response of
the continuous-time filter
•where T
drepresents a sampling interval
•the parameter Td has no role whatsoever in the design process or the
resulting discrete-time filter
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14. •The frequency response of the discrete-time filter is related to the
frequency response of the continuous-time filter by
•If
•Then
•i.e., the discrete-time and continuous-time frequency responses are
related by a linear scaling of the frequency axis, namely, ω = ΩT
dfor
|ω| < π.
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15. •anypracticalcontinuous-timefiltercannotbeexactlybandlimited,and
consequently,interferencebetweensuccessivespectraoccurs,causing
aliasing
•However,ifthecontinuous-timefilterapproacheszeroathighfrequencies,
thealiasingmaybenegligiblysmall,andausefuldiscrete-timefiltercan
resultfromsamplingtheimpulseresponseofacontinuous-timefilter
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16. Process
•forthedesignofadiscrete-timefilterwithgivenfrequencyresponse
specifications,thediscrete-timefilterspecificationsarefirst
transformedtocontinuous-timefilterspecificationsusing
•we obtain the specifications on Hc(jΩ) by applying the relation
•Afterobtainingacontinuous-timefilterthatmeetsthese
specifications,thecontinuoustimefilterwithsystemfunctionHc(s)is
transformedtothedesireddiscrete-timefilterwithsystemfunction
H(z)
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17. •in the transformation back to discrete-time frequency, H(e
jω
) will be
related to H
c(jΩ) through
•which again applies the transformation to the frequency axis.
•As a consequence, the “sampling” parameter T
dcannot be used to
control aliasing
•if Td is made smaller, then the cutoff frequency of the continuous-
time filter must increase in proportion
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18. •consider the system function of a causal continuous-time filter
expressed in terms of a partial fraction expansion
•Corresponding impulse response is
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19. Zulaikha Kiran, 2024 19

20. •a pole at s = s
kin the s-plane transforms to a pole at z = e
SkTd
in the z-plane
and the coefficients in the partial fraction expansions of H
c(s) and H(z) are
equal, except for the scaling multiplier T
d
•If the continuous-time causal filter is stable (pole in the left half plane),
then the discrete-time causal filter is also stable (pole inside the unit circle)
•itisimportanttorecognizethattheimpulseinvariancedesignprocedure
doesnotcorrespondtoasimplemappingofthes-planetothez-plane
•thezerosinthediscrete-timesystemwillnotingeneralbemappedinthesameway
thepolesaremapped
•SeeExample2Chapter7Oppenheim
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21. •In the impulse invariance design procedure, the relationship between
continuous time and discrete-time frequency is linear; consequently,
except for aliasing, the shape of the frequency response is preserved.
•Theimpulseinvariancetechniqueisappropriateonlyforbandlimited
filters;highpassorbandstopcontinuous-timefilters,forexample,
wouldrequireadditionalbandlimitingtoavoidseverealiasing
distortionifimpulseinvariancedesignisused
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22. Zulaikha Kiran, 2024 22j  0
Im(z)
Re(z)
unit-circleT
 T



23. Zulaikha Kiran, 2024 23j  0
Im(z)
Re(z)
unit-circleT
 T



24. Zulaikha Kiran, 2024 24j  0
Re(z)
unit-circleT
 T



25. Bilinear Transformation
•Algebraic transformation b/w variables ‘s’and ‘z’
•One-one mapping between the s and z-planes
•Given a C.T. prototype filter H
c(s), the corresponding D.T. filter H(z) is
•TheparameterT
dhasnoinfluencewhatsoeverinthefilterdesign,itiscancelled
outaswestartwithdigitalspecsandreturntodigitalfilterwhilepassingby
analogfilter
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1
21
()
1
d
z
s
Tz




 1
1
21
( ) ( ( ))
1
c
d
z
H z H
Tz






26. 1 ( / 2)
1 ( / 2)
d
d
Ts
z
Ts


 Substitute sj   1 2 2
1 2 2
dd
dd
T j T
z
T j T


  

  
//
// 
(1) If ＜0, then |z|＜1 for any
similarly, if ＞0, then |z|＞1 for all 
(2) If ＝0, then12

12
d
d
jT
z
jT



/
/ 1z 12

12
j d
d
jT
e
jT



/
/ 21
1
j
j
d
e
s
Te







  
 

/2
/2
2 sin / 222
tan / 2
2 cos / 2
j
j
dd
ej j
sj
T e T








    
 2
tan( )
2
d
T

 2arctan( / 2)
d
T
Since ＝0
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27. Zulaikha Kiran, 2024 272
tan( )
2
d
T

 2arctan( / 2)
d
T

28. •Whole of left-half s-plane mapped to inside the unit circle in z-plane
•The whole of imaginary axis on s-plane mapped to unit circle, no
aliasing problem
•0≤Ω≤∞maps to 0≤??????≤??????
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29. •The bilinear transformation
avoids the problem of
aliasing encountered with the
use of impulse invariance,
because it maps the entire
imaginary axis of the s-plane
onto the unit circle in the z-
plane.
•The price paid for this,
however, is the nonlinear
compression of the
frequency axis
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30. •the design of discrete-time
filters using the bilinear
transformation is useful only
when the nonlinear
compression of the frequency
axis can be tolerated or
compensated for
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31. Zulaikha Kiran, 2024 31

32. Filter Design Comparisons
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33. Butterworth Filter
•Monotonic in both passbandand stopband.
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34. ChebyshevType I
•Equiripplein passband, Monotonic in stopband
•Where the Chebyshevpolynomial is given by
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1
|)(|
22
2
cN
c
V
jH


 )coscos()(
1
xNxV
N

 c

11

35. ChebyshevType II
•Monotonic in passband, Equiripplein stopband
•Where the Chebyshevpolynomial is given by
Zulaikha Kiran, 2024 35)coscos()(
1
xNxV
N

 122
2
)]/([1
1
|)(|



cN
c
V
jH

1 c


36. Elliptic Filter
•Equiripplein both passbandand
stopband
•where U
N(Ω) is a Jacobianelliptic
function
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1
|)(|
22
2


N
c
U
jH


37. •Equationsmaybeobtainedfordiscrete-timeChebyshevandelliptic
filters.However,thedetailsofthedesigncomputationsforthese
commonlyusedclassesoffiltersarebestcarriedoutbycomputer
programs.
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38. Ex: 7-4 Design Comparisons
Butterworth Design11
10
20log (1 ) 0 or equivalently 0
pp
   22
10
20log (1 ) 0.3 or equivalently 0.0339
pp
    10
20log ( ) 30 or equivalently 0.0316
ss
  
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39. Frequency Transformations of LowpassIIR Filters to Other Filters
•Design a frequency-normalized prototype LPF
•Use algebraic transformation to derive the desired filter from prototype LPF
Given a lowpasssystem function 
lp
HZ
Transform it to a new system function Hz 
11
Z G z

  

11lp
Z G z
H z H Z



Constraints on the transformation: 
1
Gz

must be a rational function of1
z

The inside of the unit circle of the Z-plane must map to the inside of the unit circle of the z-plane
The unit circle of the Z-plane must map onto the unit circle of the z-plane
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40. Zulaikha Kiran, 2024 40

41. End
41Zulaikha Kiran, 2024