1. Chapter 7. Filter Design Techniques
7.1 Introduction - Digital Filter Design
7.2 IIR Filter Design by Impulse Invariance
7.3 IIR Filter Design by Bilinear Transformation
7.4 FIR Filter Design by Windowing
7.5 Kaiser Window based FIR Filter Design
7.6 Approximation based Optimal FIR Filter Design
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2. 1. Introduction -- Digital Filter Design
(1) Frequency selective filters : spectral shapers
lowpass highpass bandpass bandstop
(2)Filter Design Techniques
IIR : - mapping from analog filters
- impulse invariance
FIR: - windowing
- equiripple design
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3. In some IIR filter design
(3)Filter Specification(LPF)
In some IIR filter design 1 2 3 1 2 3
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4. 2. IIR Filter Design by Impulse Invariance
(1) Design Concept
- Utilize existing analog filter design technique
- Convert analog impulse response into digital impulse
response h[n] by taking samples
- Take
Then by fundamental sampling property, we get
- If analog filter were bandlimited,
Then
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5. - But the issue is that the above assumption is not likely,
(2) Aliasing Problem
- But the issue is that the above assumption is not likely,
so aliasing is inevitable in reality
aliasing
Therefore this design technique is useful only when
designing a narrowband sharp lowpass filters
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6. (3) Parameter Conversion
- Let the analog filter has the partial-fraction expansion
- After sampling, h [ n ] = T h
(nT ) = d c d
Therefore,
pole at in is mapped to
Pole at
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7. * Impulse Invariance Filter Design Procedure
1) Given specification in domain.
2) Convert it into specification in domain
3) Design analog filter meeting the specification 
4) Convert it into digital filter function H(z)
by putting
[ 5) Implement it in 2nd order cascade form]
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8. Design Example 1 2
* Choose Td=1 3
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9. * You plot the pole locations in the z-plans!4
* You plot the pole locations in the z-plans!
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10. 3. IIR Filter Design by Bilinear Transformation
(1) Design Concept
- s-plane to z-plane conversion
any mapping than maps stable region is s-plane (left half plane)
to stable region in z-plane (inside u.c) ? or
bilinear transform!
* Td inserted for convention may put to any convenient value
for practical use.

11. (2) Properties

12. * IIR Filter Design Procedure
Given specification in digital domain
Convert it into analog filter specification
Design analog filter (Butterworth, Chebyshov, elliptic):H(s)
Apply bilinear transform to get H(z) out of H(s) 1 2 3 4 3 2 4 1

13. Design Example (Butterworth Filter)
Given specification 1 2
Specification Conversion
(Set Td=1)
Butterworth filter design 3 N c j H 2 ) / ( 1 | W + =

14. Bilinear Transform 4

15. *Comparison of Butterworth, Chebyshev, elliptic filters
-Filter equations 1
Butterworth filter B
Chebyshev filter (type I) C 1
Chebyshev polynomial

16. *Comparison of Butterworth, Chebyshev, elliptic filters (Cont’d)
Chebyshev filter (type II) 1
Elliptic filter E 1
Jacobian elliptic function

17. *Comparison of Butterworth, Chebyshev, elliptic filters: Example
-Given specification
-Order
Butterworth Filter : N=14. ( max flat)
Chebyshev Filter : N= ( Cheby 1, Cheby 2)
Elliptic Filter : N= ( equiripple) B C E

18. -Pole-zero plot (analog)B C1 C2 E
-Pole-zero plot (digital) B C1 C2 E
(14)
(8)

19. -Magnitude
-Group delay C1 B B E C1 E C2 C2

20. 4. FIR Filter Design by Windowing
(1) Design Concept
- Given a desired frequency response
evaluate
- Then, is
the
desired
filter
coefficients.
However, is
infinitely
long, so
not
practical.
Therefore, take a
finite
segment of , or
such
that
the
resulting
frequency
spectrum
fall in
the
given
specification.
This process of getting out of is
called Windowing

21. (2) Rectangular Windowing

22. W ( e j ( w ) )

24. But M cannot improve this. (due to Gibb’s phenomena).
(3) Design Point e ( w ) :
depends on
the
attenuation of
the
peak
sidelobe of W ( e j w ).
But M cannot improve this. (due to Gibb’s phenomena).
Therefore, once a specific window is given, is fixed. e ( w ) D w :
depends on
the
width of
main
lobe of W ( e j w ).
This can be improved by increasing M.

25. (4) Commonly Used Windows
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26. BGL/SNU

27. Frequency Spectrum of Windows
(a) Rectangular, (b) Bartlett, (c) Hanning, (d) Hamming, (e) Blackman , (M=50)
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28. 5. Kaiser Window based FIR Filter Design
(1) Design Concept ( I
: 0th order modified Bessel function)
targets at limited duration in time and energy concentration at
low frequency
- compromisable. (choose appropriate )
Performance comparable to Hamming window ( when )
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29. Frequency Spectrum of Kaiser Window
(a) Window shape, M=20, (b) Frequency spectrum, M=20, (c) beta=6
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30. (2) Determination of Filter Order ( Kaiser, 1974 )① ② ③
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31. - Design Example : ① ② ③ ④
(Note)

32. BGL/SNU

33. 6. Approximation based Optimal FIR Filter Design
(1) Design Concept
- Linear phase filters possess the property
- More Generally, constant delay filters have the expression
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34. - Approximation error
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35. - Approximation ( Chebyshev)
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36. (2) Type I Lowpass Filter case
- desired :
- approximation
- weighting
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37. - Error function

38. (3) Type II Lowpass Filter case
- desired (original) :
- approximation
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39. - Weighting (modified)
- Desired (modified)
- error function
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40. BGL/SNU

41. (4) Alternation Theorem
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44. (5) Parks-McClellan Algorithm① ② ③ ④
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45. Remez Exchange Algorithm (1934) (multiple exchange)⑤
Remez Exchange Algorithm (1934) (multiple exchange) ① ②
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46. ③
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47. ④ ⑤ ⑥
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48. Selection of new extrema
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49. Remez Exchange Algorithm
Initial guess of
(L+2) extremal frequencies
Remez Exchange Algorithm
Calculate the optimum
 On extremal set
Interpolate through (L+1)
Points to obtain Ae(ej)
Best approximation
Calculate error E()
And find local maxima
Where | E()|>= 
Unchanged
Check whether the
Extremal points changed
Changed
More than
(L+2) extrema?
Yes
Retain (L+2)
Largest extrema No
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50. Design Examples ①
Kaiser, (1974)
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51. DSP Sample & Hold Comp. Reconst. filter②
DSP
Sample
& Hold
Comp.
Reconst.
filter
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52. BGL/SNU

53. H.W. of Chapter 7 Due 11/17 (Mon.)
[1] Design a IIR lowpass filter whose specification is the same as that given in Example 7.3 (page 454) except the passband and stopband edges are shifted to 0.7pi and 0.8pi respectively, using bilinear transform technique.
Text : [2] [3] 7.3    [4] 7.17   
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54. H.W. of Chapter 7 Due 11/24 (Mon.) [1] Design a Length-21 FIR Filter
Use the MATLAB command h=remez(20, [0,0.4,0.5,1], [1,1,0,0]) to design a length-21 filter with a passband from 0 to wp=0.4pi and a stopband from ws=0.5pi to pi with a desired response of 1 in the passband and zero in the stopband. Plot the impulse response, the zero locations, and the amplitude response. How many “ripples” are there? How many extremal frequencies are there (places where the ripples are the same maximum size)? How many “small ripples” are there that do not give extremal frequencies, and if there are any, are they in the passband or stopband? Are there zeros that do not contribute directly to a ripple? Most zero pairs off the unit circle in the z-plane cause a maximum-size ripple in the passband or stopband. Some cause only a “small ripple,” and some cause no ripple.
Text : [2] 7.28     [3] 7.23   [6] 7.36
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