This document describes the bilinear transformation method for designing IIR filters. It begins by explaining that an analog filter design can be transformed into a digital filter design using the bilinear transformation. It then discusses the warping effect caused by this transformation at higher frequencies. To address this, the document presents the concept of prewarping analog filters so that the desired digital filter frequency responses are achieved after transformation. Finally, it outlines the general design procedure for lowpass filters using this method.
2. Introduction
t A procedure for the design of IIR filters that would satisfy
arbitrary prescribed specifications will be described.
Frame # 2 Slide # 2 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
3. Introduction
t A procedure for the design of IIR filters that would satisfy
arbitrary prescribed specifications will be described.
t 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.)
Frame # 2 Slide # 3 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
4. Introduction Cont’d
Given an analog filter with a continuous-time transfer function
HA (s ), a digital filter with a discrete-time transfer function HD (z )
can be readily deduced by applying the bilinear transformation
as follows:
H D (z ) = H A (s ) (A)
2 z −1
s= T z +1
Frame # 3 Slide # 4 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
5. Introduction Cont’d
The bilinear transformation method has the following important
features:
t A stable analog filter gives a stable digital filter.
Frame # 4 Slide # 5 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
6. Introduction Cont’d
The bilinear transformation method has the following important
features:
t A stable analog filter gives a stable digital filter.
t 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.
Frame # 4 Slide # 6 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
9. The Warping Effect
If we let ω and represent the frequency variable in the analog
filter and the derived digital filter, respectively, then Eq. (A), i.e.,
H D (z ) = H A (s ) (A)
2
s= T z −1
z +1
gives the frequency response of the digital filter as a function of
the frequency response of the analog filter as
H D (e j T
) = HA (j ω)
2 z −1
provided that s =
T z +1
2 ej T
−1 2 T
or jω = or ω = tan (B)
T ej T +1 T 2
Frame # 7 Slide # 9 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
10. The Warping Effect Cont’d
··· 2 T
ω= tan (B)
T 2
t For < 0.3/T
ω≈
and, as a result, the digital filter has the same frequency
response as the analog filter over this frequency range.
Frame # 8 Slide # 10 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
11. The Warping Effect Cont’d
··· 2 T
ω= tan (B)
T 2
t For < 0.3/T
ω≈
and, as a result, the digital filter has the same frequency
response as the analog filter over this frequency range.
t For higher frequencies, however, the relation between ω
and 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.
Frame # 8 Slide # 11 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
13. The Warping Effect Cont’d
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
35
30
Digital Analog
25 filter filter
Loss, dB
20
15
10
5
0
0 1 2 3 4 5
ω, rad/s
Frame # 10 Slide # 13 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
14. Prewarping
t From Eq. (B), i.e.,
2 T
ω= tan (B)
T 2
a frequency ω in the analog filter corresponds to a
frequency in the digital filter and hence
2 ωT
= tan−1
T 2
Frame # 11 Slide # 14 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
15. Prewarping
t From Eq. (B), i.e.,
2 T
ω= tan (B)
T 2
a frequency ω in the analog filter corresponds to a
frequency in the digital filter and hence
2 ωT
= tan−1
T 2
t If ω1 , ω2 , . . . , ωi , . . . 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 ωi T
i = tan−1 i = 1, 2, . . .
T 2
Frame # 11 Slide # 15 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
16. Prewarping Cont’d
t If prescribed passband and stopband edges ˜ 1 ,
˜ 2 , . . . , ˜ i , . . . 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 ˜ iT
ωi = tan
T 2
Frame # 12 Slide # 16 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
17. Prewarping Cont’d
t If prescribed passband and stopband edges ˜ 1 ,
˜ 2 , . . . , ˜ i , . . . 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 ˜ iT
ωi = tan
T 2
t Then the band edges of the digital filter would assume their
prescribed values i since
2 ωi T
i = tan−1
T 2
2 T 2 ˜ iT
= tan−1 · tan
T 2 T 2
= ˜i for i = 1, 2, . . .
Frame # 12 Slide # 17 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
18. Design Procedure
Consider a normalized analog LP filter characterized by HN (s )
with an attenuation
1
AN (ω) = 20 log
|HN (j ω)|
(also known as loss) and assume that
0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp
AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞
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.
Frame # 13 Slide # 18 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
19. Design Procedure Cont’d
A(ω)
Aa
Ap
ω
ωp ωc ωa
Frame # 14 Slide # 19 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
20. Design Procedure
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 )
¯
H X (s ) = H N (s )
¯
s =fX (s )
¯
where fX (s ) is one of the four standard analog-filters
transformations, given by the next slide.
Frame # 15 Slide # 20 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
21. Design Procedure
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 )
¯
H X (s ) = H N (s )
¯
s =fX (s )
¯
where fX (s ) is one of the four standard analog-filters
transformations, given by the next slide.
¯
2. Apply the bilinear transformation to HX (s ), i.e.,
¯
H D (z ) = H X (s ) z −1
¯ 2
s= T z +1
Frame # 15 Slide # 21 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
22. Design Procedure Cont’d
¯
Standard forms of fX (s )
X ¯
fX (s )
LP ¯
λs
HP ¯
λ/s
ω02
BP 1 ¯
s+
B s¯
Bs¯
BS
¯
s 2 + ω0
2
Frame # 16 Slide # 22 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
23. Design Procedure Cont’d
t The digital filter designed by this method will have the
required passband and stopband edges only if the
parameters λ, ω0 , and B of the analog-filter transformations
and the order of the continuous-time normalized LP
transfer function, HN (s ), are chosen appropriately.
Frame # 17 Slide # 23 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
24. Design Procedure Cont’d
t The digital filter designed by this method will have the
required passband and stopband edges only if the
parameters λ, ω0 , and B of the analog-filter transformations
and the order of the continuous-time normalized LP
transfer function, HN (s ), are chosen appropriately.
t This is obviously a difficult problem but general solutions
are available for LP, HP, BP, and BS, Butterworth,
Chebyshev, inverse-Chebyshev, and Elliptic filters.
Frame # 17 Slide # 24 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
25. General Design Procedure for LP Filters
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, HN (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 ωp and ωa , respectively.
Frame # 18 Slide # 25 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
26. Design of LP Filters Cont’d
A(ω)
Aa
Ap
ω
ωp ωc ωa
Attenuation characteristic of continuous-time normalized LP filter
Frame # 19 Slide # 26 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
27. Design of LP Filters Cont’d
2. Apply the LP-to-LP analog-filter transformation to HN (s ) to
obtain a denormalized discrete-time transfer function
¯
HLP (s ).
Frame # 20 Slide # 27 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
28. Design of LP Filters Cont’d
2. Apply the LP-to-LP analog-filter transformation to HN (s ) to
obtain a denormalized discrete-time transfer function
¯
HLP (s ).
¯
3. Apply the bilinear transformation to HLP (s ) to obtain a
discrete-time transfer function HD (z ).
Frame # 20 Slide # 28 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
29. Design of LP Filters Cont’d
2. Apply the LP-to-LP analog-filter transformation to HN (s ) to
obtain a denormalized discrete-time transfer function
¯
HLP (s ).
¯
3. Apply the bilinear transformation to HLP (s ) to obtain a
discrete-time transfer function HD (z ).
4. At this point, assume that the derived discrete-time transfer
function has passband and stopband edges that satisfy the
relations
˜ p ≤ p and a ≤ ˜a
where ˜ p and ˜ a 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.
Frame # 20 Slide # 29 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
30. Design of LP Filters Cont’d
A(Ω)
Aa
Ap
Ω
~ ~
Ωp Ωp Ωa Ωa
Attenuation characteristic of required LP digital filter
Frame # 21 Slide # 30 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
31. Design of LP Filters Cont’d
5. Solve for λ, the parameter of the LP-to-LP analog-filter
transformation.
Frame # 22 Slide # 31 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
32. Design of LP Filters Cont’d
5. Solve for λ, the parameter of the LP-to-LP analog-filter
transformation.
6. Find the minimum value of the ratio ωp /ωa for the
continuous-time normalized LP transfer function.
The ratio ωp /ωa 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!
Frame # 22 Slide # 32 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
33. Design of LP Filters Cont’d
7. The same methodology is applicable for HP filters, except
that the LP-HP analog-filter transformation is used in Step
2.
Frame # 23 Slide # 33 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
34. Design of LP Filters Cont’d
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.
Frame # 23 Slide # 34 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
35. Formulas for LP and HP Filters
ωp
≥ K0
ωa
LP
ωp T
λ=
2 tan( ˜ p T /2)
ωp 1
≥
ωa K0
HP
2ωp tan( ˜ p T /2)
λ=
T
tan( ˜ p T /2)
where K0 =
tan( ˜ a T /2)
Frame # 24 Slide # 35 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
36. Design of LP Filters Cont’d
t The table of formulas presented is applicable to all the
classical types of analog filters, namely,
– Butterworth
– Chebyshev
– Inverse-Chebyshev
– Elliptic
Frame # 25 Slide # 36 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
37. Design of LP Filters Cont’d
t The table of formulas presented is applicable to all the
classical types of analog filters, namely,
– Butterworth
– Chebyshev
– Inverse-Chebyshev
– Elliptic
t Formulas that can be used to design digital versions of
these filters will be presented later.
Frame # 25 Slide # 37 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
38. General Design Procedure for BP Filters
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, HN (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 ωp and ωa , respectively.
Frame # 26 Slide # 38 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
39. Design of BP Filters Cont’d
2. Apply the LP-to-BP analog-filter transformation to HN (s ) to
obtain a denormalized discrete-time transfer function
¯
HBP (s ).
Frame # 27 Slide # 39 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
40. Design of BP Filters Cont’d
2. Apply the LP-to-BP analog-filter transformation to HN (s ) to
obtain a denormalized discrete-time transfer function
¯
HBP (s ).
¯
3. Apply the bilinear transformation to HBP (s ) to obtain a
discrete-time transfer function HD (z ).
Frame # 27 Slide # 40 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
41. Design of BP Filters Cont’d
4. At this point, assume that the derived discrete-time transfer
function has passband and stopband edges that satisfy the
relations
˜
p1 ≤ p1
˜
p2 ≥ p2
and
a1 ≥ ˜ a1 a2 ≤ ˜ a2
where
– p 1 and p 2 are the actual lower and upper passband
edges,
– ˜ p 1 and ˜ p 2 are the prescribed lower and upper passband
edges,
– a1 and a2 are the actual lower and upper stopband
edges,
– ˜ p 1 and ˜ p 2 are the prescribed lower and upper stopband
edges, respectively.
Frame # 28 Slide # 41 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
42. Design of LP Filters Cont’d
A(Ω)
Aa Aa
Ap
Ω
~ ~
Ωa1 Ωp1 Ωp2 Ωa2
Ωa1 ~ ~ Ωa2
Ωp1 Ωp2
Attenuation characteristic of required BP digital filter
Frame # 29 Slide # 42 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
43. Design of LP Filters Cont’d
5. Solve for B and ω0 , the parameters of the LP-to-BP
analog-filter transformation.
Frame # 30 Slide # 43 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
44. Design of LP Filters Cont’d
5. Solve for B and ω0 , the parameters of the LP-to-BP
analog-filter transformation.
6. Find the minimum value for the selectivity, i.e., the ratio
ωp /ωa , for the continuous-time normalized LP transfer
function.
Frame # 30 Slide # 44 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
45. Design of LP Filters Cont’d
5. Solve for B and ω0 , the parameters of the LP-to-BP
analog-filter transformation.
6. Find the minimum value for the selectivity, i.e., the ratio
ωp /ωa , 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.
Frame # 30 Slide # 45 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
46. Design of LP Filters Cont’d
5. Solve for B and ω0 , the parameters of the LP-to-BP
analog-filter transformation.
6. Find the minimum value for the selectivity, i.e., the ratio
ωp /ωa , 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.
Frame # 30 Slide # 46 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
47. Formulas for the design of BP Filters
√
2 KB
ω0 =
T
ωp K1 if KC ≥ KB
BP ≥
ωa K2 if KC < KB
2KA
B=
T ωp
˜ p2T ˜ p 1T ˜ p 1T ˜ p2T
where KA = tan − tan KB = tan tan
2 2 2 2
˜ a1 T ˜ a2 T KA tan( ˜ a1 T /2)
KC = tan tan K1 =
2 2 KB − tan2 ( ˜ a1 T /2)
KA tan( ˜ a2 T /2)
K2 =
tan2 ( ˜ a2 T /2) − KB
Frame # 31 Slide # 47 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
48. Formulas for the Design of BS Filters
√
2 KB
ω0 =
T
⎧
⎪1
⎪
⎨ if KC ≥ KB
ωp K2
BS ≥
ωa ⎪ 1
⎪
⎩ if KC < KB
K1
2KA ωp
B=
T
˜ p2T ˜ p 1T ˜ p 1T ˜ p2T
where KA = tan − tan KB = tan tan
2 2 2 2
˜ a1 T ˜ a2 T KA tan( ˜ a1 T /2)
KC = tan tan K1 =
2 2 KB − tan2 ( ˜ a1 T /2)
KA tan( ˜ a2 T /2)
K2 =
tan2 ( ˜ a2 T /2) − KB
Frame # 32 Slide # 48 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
49. Formulas for ωp and n
t The formulas presented apply to any type of normalized
analog LP filter with an attenuation that would satisfy the
following conditions:
0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp
AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞
Frame # 33 Slide # 49 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
50. Formulas for ωp and n
t The formulas presented apply to any type of normalized
analog LP filter with an attenuation that would satisfy the
following conditions:
0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp
AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞
t However, the values of the normalized passband edge, ωp ,
and the required filter order, n , depend on the type of filter.
Frame # 33 Slide # 50 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
51. Formulas for ωp and n
t The formulas presented apply to any type of normalized
analog LP filter with an attenuation that would satisfy the
following conditions:
0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp
AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞
t However, the values of the normalized passband edge, ωp ,
and the required filter order, n , depend on the type of filter.
t Formulas for these parameters for Butterworth, Chebyshev,
and Elliptic filters are presented in the next three slides.
Frame # 33 Slide # 51 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
52. Formulas for Butterworth Filters
LP K = K0
1
HP K =
K0
K1 if KC ≥ KB
BP K =
K2 if KC < KB
⎧
⎪1
⎪ if KC ≥ KB
⎨
K2
BS K =
⎪1
⎪
⎩ if KC < KB
K1
log D 100.1Aa − 1
n≥ , D=
2 log(1/K ) 100.1Ap − 1
ωp = (100.1Ap − 1)1/2n
Frame # 34 Slide # 52 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
53. Formulas for Chebyshev Filters
LP K = K0
1
HP K =
K0
K1 if KC ≥ KB
BP K =
K2 if KC < KB
⎧
⎪ 1 if K ≥ K
⎪
⎨ C B
K2
BS K =
⎪1
⎪
⎩ if KC < KB
K1
√
cosh−1 D 100.1Aa − 1
n≥ , D=
cosh−1 (1/K ) 100.1Ap − 1
ωp = 1
Frame # 35 Slide # 53 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
54. Formulas for Elliptic Filters
k ωp
√
LP K = K0 K0
1 1
HP K = √
K0 K0
√
K1 if KC ≥ KB K
BP K = √ 1
K2 if KC < KB K2
⎧
⎪1
⎪ 1
⎨ if KC ≥ KB √
K2 K2
BS K =
⎪1
⎪ 1
⎩ if KC < KB √
K1 K1
√
cosh−1 D 100.1Aa − 1
n≥ , D=
cosh−1 (1/K ) 100.1Ap − 1
Frame # 36 Slide # 54 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
55. Example – HP Filter
An HP filter that would satisfy the following specifications is
required:
Ap = 1 dB, Aa = 45 dB, ˜ p = 3.5 rad/s,
˜ a = 1.5 rad/s, ωs = 10 rad/s.
Design a Butterworth, a Chebyshev, and then an Elliptic digital
filter.
Frame # 37 Slide # 55 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
56. Example – HP Filter Cont’d
Solution
Filter type n ωp λ
Butterworth 5 0.873610 5.457600
Chebyshev 4 1.0 6.247183
Elliptic 3 0.509526 3.183099
Frame # 38 Slide # 56 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
62. D-Filter
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.
Frame # 44 Slide # 62 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
63. Summary
t A design 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.
Frame # 45 Slide # 63 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
64. Summary
t A design 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.
t The method requires very little computation and leads to
very precise optimal designs.
Frame # 45 Slide # 64 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
65. Summary
t A design 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.
t The method requires very little computation and leads to
very precise optimal designs.
t It can be used to design LP, HP, BP, and BS filters of the
Butterworth, Chebyshev, Inverse-Chebyshev, Elliptic types.
Frame # 45 Slide # 65 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
66. Summary
t A design 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.
t The method requires very little computation and leads to
very precise optimal designs.
t It can be used to design LP, HP, BP, and BS filters of the
Butterworth, Chebyshev, Inverse-Chebyshev, Elliptic types.
t All these designs can be carried out by using DSP software
package D-Filter.
Frame # 45 Slide # 66 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
67. References
t A. Antoniou, Digital Signal Processing: Signals, Systems,
and Filters, Chap. 15, McGraw-Hill, 2005.
t A. Antoniou, Digital Signal Processing: Signals, Systems,
and Filters, Chap. 15, McGraw-Hill, 2005.
Frame # 46 Slide # 67 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
68. This slide concludes the presentation.
Thank you for your attention.
Frame # 47 Slide # 68 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method