Warping Concept (iir filters-bilinear transformation method)

A good description on warping effect.

Uet Peshwar Atd campus.
03335297070

1. Part 3: IIR Filters — Bilinear Transformation
Method

Tutorial ISCAS 2007

Copyright © 2007 Andreas Antoniou
Victoria, BC, Canada
Email: aantoniou @ieee.org

July 24, 2007

Frame #1 Slide # 1 A. Antoniou Part3: IIR Filters - Bilinear Transformation Method

2. e Aprocedure for the design of IIR filters that would satisfy
arbitrary prescribed specifications will be described.

3. e Aprocedure for the design of IIR filters that would satisfy
arbitrary prescribed specifications will be described.

e The method is based on the bilinear transformation and it
can be used to design lowpass (LP), highpass (HP),
bandpass (BP), and bandstop (BS), Butterworth,
Chebyshev, Inverse-Chebyshev, and Elliptic filters.

Note: The material for this module is taken from Antoniou,
Digital Signal Processing: Signals, Systems, and Filters,
Chap. 12.)

4. Given an analog filter with a continuous-time transfer function
H,(s), a digital filter with a discrete-time transfer function Hp(z)
can be readily deduced by applying the bilinear transformation
as follows:

Hp(Z) = Ha(s) (A)
$(55)

5. The bilinear transformation method has the following important
features:

e A stable analog filter gives a stable digital filter.

6. The bilinear transformation method has the following important
features:

e A stable analog filter gives a stable digital filter.

e The maxima and minima of the amplitude response in the
analog filter are preserved in the digital filter.
As a consequence,
— the passband ripple, and
— the minimum stopband attenuation
of the analog filter are preserved in the digital filter.

7. Amplitude Response

Gain, dB

Frequency, radis

8. Amplitude Response

T
23

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rot
055 0€ 035 CT 07

In
015 92 035 C3 235 0e ou
Frequency, radis

T

005 01

c

ozs

3

9. If we let w and Q represent the frequency variable in the analog
filter and the derived digital filter, respectively, then Eq. (A), i.e.,

Hp(Z) = Ha(s) (A)

z-1

s=# (#1)

gives the frequency response of the digital filter as a function of
the frequency response of the analog filter as

Hp(el27) = Hao)

: 2 (z-1
provided that s = 7 6 7 3)
F 2 (e274 2 QT
or jo = (Sr) or o= > lan (B)

10. e ForQ <0.3/T

wa

and, as a result, the digital filter has the same frequency
response as the analog filter over this frequency range.

11. e ForQ <0.3/T

wa

and, as a result, the digital filter has the same frequency
response as the analog filter over this frequency range.

e For higher frequencies, however, the relation between w
and Q becomes nonlinear, and distortion is introduced in
the frequency scale of the digital filter relative to that of the
analog filter.

This is known as the warping effect.

13. The warping effect changes the band edges of the digital filter
relative to those of the analog filter in a nonlinear way, as
illustrated for the case of a BS filter:

40)

.
7
x
“
35 M
1
:
30 ï
‘
| Digital + Analog
25 filter N ı 2 filter
a ,
3 1
á 20
a

14. e From Eq. (B), i.e.,

2 QT
= + tan — B
o=- tan (B)
a frequency w in the analog filter corresponds to a
frequency @ in the digital filter and hence

2 oT
A = tan! —
T

2

15. e From Eq. (B), i.e.,

2 QT
= + tan — B
o=- tan (B)
a frequency w in the analog filter corresponds to a
frequency @ in the digital filter and hence

2 oT
= tan! —
T 2

e fo, @o,..., @;,... are the passband and stopband edges
in the analog filter, then the corresponding passband and
stopband edges in the derived digital filter are given by

2 4 oT A
i= = — =1,2,...
Qi 7 tan > | 1,2,

16. e lf prescribed passband and stopband edges 221,
Qo, ..., Q;,... are to be achieved, the analog filter must be
prewarped before the application of the bilinear
transformation to ensure that its band edges are given by

2 QT
o = 7 ian Z-

17. e If prescribed passband and stopband edges Q4,
Qe,..., Qi,... are to be achieved, the analog filter must be
prewarped before the application of the bilinear
transformation to ensure that its band edges are given by
_ 2 la QT
Hop à
e Then the band edges of the digital filter would assume their
prescribed values Q; since
2 _1 oT

Q; = —tan —
T 2

= Ztan-' Là À ni
=T 2 T 2

=; for i=1,2,...

18. Consider a normalized analog LP filter characterized by Hy(s)
with an attenuation

An(o) = 20log TANGO)!

(also known as loss) and assume that

0 <An(w) <A, for 0 < |w| < a

An(@) = Aa for wa < |o| < 00

Note: The transfer functions of analog LP filters are reported in
the literature in normalized form whereby the passband edge is
typically of the order of unity.

19. 3
=<

20. A denormalized LP, HP, BP, or BS filter that has the same
passband ripple and minimum stopband attenuation as a given
normalized LP filter can be derived from the normalized LP filter
through the following steps:

1. Apply the transformation s = fx(S)

Hx(S) = Hn(s)

s=fy (8)

where fx(5) is one of the four standard analog-filters
transformations, given by the next slide.

21. A denormalized LP, HP, BP, or BS filter that has the same
passband ripple and minimum stopband attenuation as a given

normalized LP filter can be derived from the normalized LP filter
through the following steps:

1. Apply the transformation s = fx(S)
Hx(S) = Hn(s) -
s=fx(5)
where fx(5) is one of the four standard analog-filters
transformations, given by the next slide.
2. Apply the bilinear transformation to Hy (S), i.e.,

Hp(Z) = CIA

+)

22. Standard forms of fx (S)

23. e The digital filter designed by this method will have the
required passband and stopband edges only if the
parameters À, wo, and B of the analog-filter transformations
and the order of the continuous-time normalized LP
transfer function, Hy(s), are chosen appropriately.

24. e The digital filter designed by this method will have the
required passband and stopband edges only if the
parameters À, wo, and B of the analog-filter transformations
and the order of the continuous-time normalized LP
transfer function, Hy(s), are chosen appropriately.

e This is obviously a difficult problem but general solutions
are available for LP, HP, BP, and BS, Butterworth,
Chebyshev, inverse-Chebyshev, and Elliptic filters.

25. An outline of the methodology for the derivation of general
solutions for LP filters is as follows:

1. Assume that a continuous-time normalized LP transfer
function, Hy(s), is available that would give the required
passband ripple, Ap, and minimum stopband attenuation
(loss), Aa.

Let the passband and stopband edges of the analog filter
be wp and wa, respectively.

26. A(w)

A

Op We

|
|
I
|
I
|
|
|
|
|
|
|
|
|
1 A
Va

Attenuation characteristic of continuous-time normalized LP filter

27. 2. Apply the LP-to-LP analog-filter transformation to Hy(s) to
obtain a denormalized discrete-time transfer function
H,p(S).

28. 2. Apply the LP-to-LP analog-filter transformation to Hy(s) to
obtain a denormalized discrete-time transfer function
H,p(S).

3. Apply the bilinear transformation to H.p(S) to obtain a
discrete-time transfer function Hp(z).

29. 2. Apply the LP-to-LP analog-filter transformation to Hy(s) to
obtain a denormalized discrete-time transfer function
H,p(S).

3. Apply the bilinear transformation to H.p(S) to obtain a
discrete-time transfer function Hp(z).

4. At this point, assume that the derived discrete-time transfer
function has passband and stopband edges that satisfy the
relations

A <Q) and <a

where $2, and @, are the prescribed passband and
stopband edges, respectively.

In effect, we assume that the digital filter has passband
and stopband edges that satisfy or oversatisfy the required
specifications.

30. A(2)

|
it
I I
| |
| |
| |
| |
| |
| |
| |
| |
! y!
tif it 2
Bp Sp La Qa
Attenuation characteristic of required LP digital filter

31. 5. Solve for 2, the parameter of the LP-to-LP analog-filter
transformation.

32. 5. Solve for 2, the parameter of the LP-to-LP analog-filter
transformation.

6. Find the minimum value of the ratio wp/wa for the
continuous-time normalized LP transfer function.

The ratio wp/wa is a fraction less than unity and it is a
measure of the steepness of the transition characteristic. It
is often referred to as the selectivity of the filter.

The selectivity of a filter dictates the minimum order to
achieve the required specifications.

Note: As the selectivity approaches unity, the filter-order
tends to infinity!

33. 7. The same methodology is applicable for HP filters, except
that the LP-HP analog-filter transformation is used in Step
2.

34. 7. The same methodology is applicable for HP filters, except
that the LP-HP analog-filter transformation is used in Step
2.

8. The application of this methodology yields the formulas
summarized by the table shown in the next slide.

35. op > Ko
Wa
LP
h= pT
~ 2tan(@,T/2)
Wn, Y
Wa ~ Ko
HP
2wp tan(QpT /2)
a JS
tan(2,T/2)
where Ky = P

~ tan(@,T/2)

36. e The table of formulas presented is applicable to all the
classical types of analog filters, namely,

Butterworth

— Chebyshev
Inverse-Chebyshev
Elliptic

37. e The table of formulas presented is applicable to all the
classical types of analog filters, namely,

Butterworth

— Chebyshev
Inverse-Chebyshev
Elliptic

e Formulas that can be used to design digital versions of
these filters will be presented later.

38. An outline of the methodology for the derivation of general
solutions for BP filters is as follows:

1. Assume that a continuous-time normalized LP transfer
function, Hy(s), is available that would give the required
passband ripple, Ap, and minimum stopband attenuation,
Aa.

Let the passband and stopband edges of the analog filter
be wp and wa, respectively.

39. 2. Apply the LP-to-BP analog-filter transformation to Hy(s) to
obtain a denormalized discrete-time transfer function
Hgp(S).

40. 2. Apply the LP-to-BP analog-filter transformation to Hy(s) to
obtain a denormalized discrete-time transfer function
Hgp(S).

3. Apply the bilinear transformation to Hgp(S) to obtain a
discrete-time transfer function Hp(z).

41. 4. Atthis point, assume that the derived discrete-time transfer
function has passband and stopband edges that satisfy the

relations

Qnt S21 Apo = Apo
and

lar > Lar Nao < Qae
where

Qp1 and Q,2 are the actual lower and upper passband
edges,

Qp1 and Qp2 are the prescribed lower and upper passband
edges,

Qai and 4 are the actual lower and upper stopband
edges,

Lp: and Qpe are the prescribed lower and upper stopband
edges, respectively.

42. A(2)

Bi-els-2-2=- 2...
©

12 |

Lal Sp 1 Qp2 La

Attenuation characteristic of required BP digital filter

43. 5. Solve for B and wo, the parameters of the LP-to-BP
analog-filter transformation.

44. 5. Solve for B and wo, the parameters of the LP-to-BP
analog-filter transformation.

6. Find the minimum value for the selectivity, i.e., the ratio
@p/@a, for the continuous-time normalized LP transfer
function.

45. 5. Solve for B and wo, the parameters of the LP-to-BP
analog-filter transformation.

6. Find the minimum value for the selectivity, i.e., the ratio
@p/@a, for the continuous-time normalized LP transfer
function.

7. The same methodology can be used for the design of BS
filters except that the LP-to-BS transformation is used in
Step 2.

46. 5. Solve for B and wo, the parameters of the LP-to-BP
analog-filter transformation.

6. Find the minimum value for the selectivity, i.e., the ratio
@p/@a, for the continuous-time normalized LP transfer
function.

7. The same methodology can be used for the design of BS
filters except that the LP-to-BS transformation is used in
Step 2.

8. The application of this methodology yields the formulas
summarized in the next two slides.

47. w 29 ía
Ir
Kı fKe>K
ap mp, [K c= Ks
Wa Ko if Ke < Kg
2Ka
B--1
Ty
QT ET ST, QoeT
where Ky, = tan —2 tan —— Kg = tan “2 tan =
0 Q La T/2
Kc = tan LarT tan Lael K, = Katana T/2) _
_ 2 Kg - tan? (2 T/2)
Katan(Qg2T /2)

* tan? (Qa 7/2) — Ke

48. 2./Kg
Wo =
7
To,
— if Ko > Kg
ps >!"
Wa .
— IK, K;
K, ¡Ac < Kg
2Kawp
a 2
po T GT GT. OT
where Ka = tan —2 tan 5 Kg = tan “2 tan 5
? QT Ka tan(Qa1 7/2
Kc = tan AT sp E ka AAA
_ 2 Ks -tan (Rai T/2)
Katan(Q22 7/2)

2 tan (Gap 7/2) — Ke

49. e The formulas presented apply to any type of normalized
analog LP filter with an attenuation that would satisfy the
following conditions:

0 < Aníw) <A, for O< lo] < wp

An(w) > Aa for wa< |w| < ©

50. e The formulas presented apply to any type of normalized
analog LP filter with an attenuation that would satisfy the
following conditions:

0 < Aníw) <A, for O< lo] < wp
An(w) > Aa for wa< |w| < ©

e However, the values of the normalized passband edge, wp,
and the required filter order, n, depend on the type of filter.

51. e The formulas presented apply to any type of normalized
analog LP filter with an attenuation that would satisfy the
following conditions:

0 < Aníw) <A, for O< lo] < wp

An(w) > Aa for wa< |w| < ©

e However, the values of the normalized passband edge, wp,
and the required filter order, n, depend on the type of filter.

e Formulas for these parameters for Butterworth, Chebyshev,
and Elliptic filters are presented in the next three slides.

52. LP K=K
r

HP K=—
Ko

BP k= |
Ko

;

BS K=|'2
K

log D

= og

if Kc > KB
if Kc < Kg

if Kc = Kg
if Kc < Kg

_ 100-142 =
07

Wp = (1009-14p _ 1)'/2n

53. LP K=K
1
HP K=—
Ko
BP K- Ky if Ke = Ke
Ka if Kc < Kg
to.
K if Ke > Kg
BS K=(4
K if Ko < Kp
cosh~' /D 100.14 _ 4
n2 dk OT
cosh '(1/K) 10% — 1

Wp =1

54. k @p
LP K=K Ro
1 1
HP K=—
Ko [Ro
Bp x | fkozks VK
Ko ifKe<Ks VR
1 1
— ifKko>Ke —
BS K= . VR
— IK K =>
Kı I Ko < Ke KR
cosh"* VD 100-1Aa _ 4

cosh (1/K) 1001, — 1

55. An HP filter that would satisfy the following specifications is
required:

Ap=10B, A=45dB %=35 rad/s,

a = 1.5 rad/s, ws = 10 rad/s.

Design a Butterworth, a Chebyshev, and then an Elliptic digital
filter.

56. Solution
Filter type n Op À

Butterworth 5 0.873610 5.457600
Chebyshev 1.0 6.247183
Elliptic 3 0.509526 3.183099

>

57. ap ‘ssoj puegsseg 2

NS Elliptic

70
60F
0
0
0
0
10F
0

58. Design an Elliptic BP filter that would satisfy the following
specifications:

Ap =1dB, Aa=45dB, p1=900 rad/s, Qp2 = 1100 rad/s,
Qu = 800 rad/s, Qa = 1200 rad/s, ws = 6000 rad/s.

Solution
k = 0.515957
@p = 0.718302
n=4
wo = 1,098.609
B = 371.9263

59. Ap so] puegsseg

10

2000
2, rad/s

1500

=

A

4 1000

1

500

900 1100 1200

800

60. Design a Chebyshev BS filter that would satisfy the following
specifications:

Ap =0.5dB, Aa=40dB, £p1 = 350rad/s, Qpe = 700 rad/s,

Lar = 430 rad/s, Lao = 600 rad/s, ws = 3000 rad/s.

Solution
@p = 1.0
n=5
wo = 561, 4083
B = 493, 2594

61. 60

50

40

30

Loss, dB
Passband loss, dB

20

10 0.5

2, rad/s
350 430 600 700

62. A DSP software package that incorporates the design
techniques described in this presentation is D-Filter. Please see

http://www.d-filter.ece.uvic.ca

for more information.

63. e Adesign method for IIR filters that leads to a complete
description of the transfer function in closed form either in
terms of its zeros and poles or its coefficients has been
described.

64. e Adesign method for IIR filters that leads to a complete
description of the transfer function in closed form either in
terms of its zeros and poles or its coefficients has been
described.

e The method requires very little computation and leads to
very precise optimal designs.

65. e Adesign method for IIR filters that leads to a complete
description of the transfer function in closed form either in
terms of its zeros and poles or its coefficients has been
described.

e The method requires very little computation and leads to
very precise optimal designs.

e Itcan be used to design LP, HP, BP, and BS filters of the
Butterworth, Chebyshev, Inverse-Chebyshev, Elliptic types.

66. e Adesign method for IIR filters that leads to a complete
description of the transfer function in closed form either in
terms of its zeros and poles or its coefficients has been
described.

e The method requires very little computation and leads to
very precise optimal designs.

e Itcan be used to design LP, HP, BP, and BS filters of the
Butterworth, Chebyshev, Inverse-Chebyshev, Elliptic types.

e All these designs can be carried out by using DSP software
package D-Filter.

67. e A. Antoniou, Digital Signal Processing: Signals, Systems,
and Filters, Chap. 15, McGraw-Hill, 2005.

e A. Antoniou, Digital Signal Processing: Signals, Systems,
and Filters, Chap. 15, McGraw-Hill, 2005.

68. This slide concludes the presentation.
Thank you for your attention.