Jul 24, 2007 ... The method is based on the bilinear transformation and it can be used to ... can
be readily deduced by applying the bilinear transformation.
Part 3: IIR Filters – Bilinear Transformation Method Tutorial ISCAS 2007
Copyright © 2007 Andreas Antoniou Victoria, BC, Canada Email:
[email protected] July 24, 2007
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Introduction t A procedure for the design of IIR filters that would satisfy
arbitrary prescribed specifications will be described.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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.)
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Part3: IIR Filters – Bilinear Transformation Method
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: HD (z ) = HA (s )
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A. Antoniou
s = T2
z −1 z +1
(A)
Part3: IIR Filters – Bilinear Transformation Method
Introduction Cont’d The bilinear transformation method has the following important features: t A stable analog filter gives a stable digital filter.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Introduction Cont’d
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Part3: IIR Filters – Bilinear Transformation Method
Introduction Cont’d
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Part3: IIR Filters – Bilinear Transformation Method
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., HD (z ) = HA (s ) (A) s = T2
z −1 z +1
gives the frequency response of the digital filter as a function of the frequency response of the analog filter as
HD (e j T ) = HA (j ω) 2 provided that s = T 2 jω = T
or Frame # 7
Slide # 9
z −1 z +1
e j T − 1 e j T + 1 A. Antoniou
or ω =
2 T tan T 2
(B)
Part3: IIR Filters – Bilinear Transformation Method
The Warping Effect Cont’d
···
ω=
2 T tan T 2
t For < 0.3/T
(B)
ω≈
and, as a result, the digital filter has the same frequency response as the analog filter over this frequency range.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
The Warping Effect Cont’d
···
ω=
2 T tan T 2
t For < 0.3/T
(B)
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
The Warping Effect Cont’d 6.0 T=2
4.0
ω 2.0
0.1π
|HA(jω)|
0.2π
0.3π
0.4π
0.5π Ω, rad/s
|HD(e jΩT)|
Frame # 9 Slide # 12
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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 filter
Loss, dB
25
Analog filter
20 15 10 5 0 0
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1
2
A. Antoniou
ω, rad/s
3
4
5
Part3: IIR Filters – Bilinear Transformation Method
Prewarping t From Eq. (B), i.e.,
T 2 tan T 2
ω=
(B)
a frequency ω in the analog filter corresponds to a frequency in the digital filter and hence =
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ωT 2 tan−1 T 2
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Prewarping t From Eq. (B), i.e.,
T 2 tan T 2
ω=
(B)
a frequency ω in the analog filter corresponds to a frequency in the digital filter and hence =
ωT 2 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 i =
Frame # 11 Slide # 15
2 ωi T tan−1 T 2 A. Antoniou
i = 1, 2, . . .
Part3: IIR Filters – Bilinear Transformation Method
Prewarping Cont’d t If prescribed passband and stopband edges ˜ 1,
˜ i , . . . are to be achieved, the analog filter must be ˜ 2, . . . , prewarped before the application of the bilinear transformation to ensure that its band edges are given by ωi =
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A. Antoniou
˜ iT 2 tan T 2
Part3: IIR Filters – Bilinear Transformation Method
Prewarping Cont’d t If prescribed passband and stopband edges ˜ 1,
˜ i , . . . are to be achieved, the analog filter must be ˜ 2, . . . , prewarped before the application of the bilinear transformation to ensure that its band edges are given by ωi =
˜ iT 2 tan T 2
t Then the band edges of the digital filter would assume their
prescribed values i since 2 ωi T tan−1 T 2 ˜ 2 T T 2 i · tan tan−1 = T 2 T 2
i =
˜i = Frame # 12 Slide # 17
for i = 1, 2, . . .
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Design Procedure Consider a normalized analog LP filter characterized by HN (s ) with an attenuation
AN (ω) = 20 log
1 |HN (j ω)|
(also known as loss) and assume that 0 ≤ AN (ω) ≤ Ap
AN (ω) ≥ Aa
for 0 ≤ |ω| ≤ ωp
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.
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Part3: IIR Filters – Bilinear Transformation Method
Design Procedure Cont’d
A(ω)
Aa Ap
ω ωp
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A. Antoniou
ωc
ωa
Part3: IIR Filters – Bilinear Transformation Method
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¯ ) HX (s¯ ) = HN (s ) s =fX (s¯ )
where fX (s¯ ) is one of the four standard analog-filters transformations, given by the next slide.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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¯ ) HX (s¯ ) = HN (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., HD (z ) = HX (s¯ ) 2 z −1 s¯ = T
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A. Antoniou
z +1
Part3: IIR Filters – Bilinear Transformation Method
Design Procedure Cont’d Standard forms of fX (s¯ ) X
fX (s¯ )
LP
λs¯
HP BP
1 B
BS
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A. Antoniou
λ/s¯
ω2 s¯ + 0 s¯
B s¯ s¯ 2 + ω02
Part3: IIR Filters – Bilinear Transformation Method
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.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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
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
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
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¯ ).
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Part3: IIR Filters – Bilinear Transformation Method
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 ).
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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 ˜a ˜ p ≤ p and a ≤ ˜ p and ˜ a are the prescribed passband and where 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
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
Design of LP Filters Cont’d 5. Solve for λ, the parameter of the LP-to-LP analog-filter transformation.
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Part3: IIR Filters – Bilinear Transformation Method
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!
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Formulas for LP and HP Filters
LP
ωp ≥ K0 ωa ωp T ˜ p T /2) 2 tan(
λ=
HP
1 ωp ≥ ωa K0 λ=
where
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˜ p T /2) 2ωp tan( T
K0 =
A. Antoniou
˜ p T /2) tan( ˜ a T /2) tan( Part3: IIR Filters – Bilinear Transformation Method
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
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
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.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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¯ ).
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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
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 ˜ p 1 p 2 ≥ ˜ p2 p 1 ≤ and
˜ a1 a 1 ≥
˜ a2 a 2 ≤
where – p 1 and p 2 are the actual lower and upper passband edges, ˜ p 2 are the prescribed lower and upper passband ˜ p 1 and – edges, – a1 and a2 are the actual lower and upper stopband edges, ˜ p 2 are the prescribed lower and upper stopband ˜ p 1 and – edges, respectively. Frame # 28 Slide # 41
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Design of LP Filters Cont’d
A(Ω)
Aa
Aa Ap
Ω ~ Ωa1
Ωa1
Ωp1
~ Ωp1
~ Ωa2
Ωp2 ~ Ωa2 Ωp2
Attenuation characteristic of required BP digital filter Frame # 29 Slide # 42
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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
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
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
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
Formulas for the design of BP Filters
BP
√ 2 KB ω0 = T K1 if KC ≥ KB ωp ≥ ωa K2 if KC < KB
B= where
Frame # 31 Slide # 47
2KA T ωp
˜ p 1T ˜ p2T ˜ p2T ˜ p 1T − tan tan KB = tan 2 2 2 2 ˜ a2 T ˜ a1 T ˜ a1 T /2) KA tan( tan KC = tan K1 = ˜ a1 T /2) 2 2 KB − tan2 ( ˜ KA tan(a2 T /2) K2 = ˜ a2 T /2) − KB tan2 (
KA = tan
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Formulas for the Design of BS Filters
BS
√ 2 KB ω0 = T ⎧ 1 ⎪ ⎪ if KC ≥ KB ⎨ ωp K2 ≥ 1 ωa ⎪ ⎪ ⎩ if KC < KB K1 2KA ωp T ˜ p 1T ˜ p2T ˜ p2T ˜ p 1T − tan tan KA = tan KB = tan 2 2 2 2 ˜ a2 T ˜ a1 T ˜ a1 T /2) KA tan( tan KC = tan K1 = ˜ a1 T /2) 2 2 KB − tan2 ( ˜ KA tan(a2 T /2) K2 = ˜ a2 T /2) − KB tan2 (
B= where
Frame # 32 Slide # 48
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
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
AN (ω) ≥ Aa
Frame # 33 Slide # 49
A. Antoniou
for 0 ≤ |ω| ≤ ωp
for ωa ≤ |ω| ≤ ∞
Part3: IIR Filters – Bilinear Transformation Method
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
AN (ω) ≥ Aa
for 0 ≤ |ω| ≤ ωp
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
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
AN (ω) ≥ Aa
for 0 ≤ |ω| ≤ ωp
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
Formulas for Butterworth Filters LP HP BP
BS
n≥
K = K0 1 K = K0 K1 K = K2 ⎧ 1 ⎪ ⎪ ⎨ K2 K = 1 ⎪ ⎪ ⎩ K1
if KC ≥ KB if KC < KB if KC ≥ KB if KC < KB
log D 100.1Aa − 1 , 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
Formulas for Chebyshev Filters LP HP BP
BS
n≥
K = K0 1 K = K0 K1 if KC ≥ KB K = K2 if KC < KB ⎧ 1 ⎪ ⎪ if KC ≥ KB ⎨ K2 K = 1 ⎪ ⎪ ⎩ if KC < KB K1 √ cosh−1 D 100.1Aa − 1 , D = 100.1Ap − 1 cosh−1 (1/K )
ωp = 1
Frame # 35 Slide # 53
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Formulas for Elliptic Filters k LP HP BP
BS
n≥
Frame # 36 Slide # 54
√
ωp
K = K0 K0 1 1 K = √ K0 K0 √ K K1 if KC ≥ KB √ 1 K = K2 K2 if KC < KB ⎧ 1 1 ⎪ ⎪ if KC ≥ KB √ ⎨ K2 K2 K = 1 1 ⎪ ⎪ √ ⎩ if KC < KB K1 K1 √ cosh−1 D 100.1Aa − 1 , D = 100.1Ap − 1 cosh−1 (1/K ) A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Example – HP Filter An HP filter that would satisfy the following specifications is required: ˜ p = 3.5 rad/s, Ap = 1 dB, Aa = 45 dB, ˜ 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
Example – HP Filter Cont’d Solution
Frame # 38 Slide # 56
Filter type
n
ωp
λ
Butterworth
5
0.873610
5.457600
Chebyshev
4
1.0
6.247183
Elliptic
3
0.509526
3.183099
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Example Cont’d 70 Elliptic (n=3)
60
Passband loss, dB
50
Loss, dB
Chebyshev (n=4)
40 30 20 Butterworth (n=5)
10
1.0
0 0
1
2
Ω, rad/s
1.5
Frame # 39 Slide # 57
A. Antoniou
3
4
5
3.5
Part3: IIR Filters – Bilinear Transformation Method
Example – BP Filter Design an Elliptic BP filter that would satisfy the following specifications: ˜ p 1 = 900 rad/s, ˜ p 2 = 1100 rad/s, Ap = 1 dB, Aa = 45 dB, ˜ a2 = 1200 rad/s, ωs = 6000 rad/s. ˜ a1 = 800 rad/s, Solution
k ωp n ω0 B
Frame # 40 Slide # 58
= 0.515957 = 0.718302 =4 = 1, 098.609 = 371.9263
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Example – BP Filter Cont’d 70 60
Passband loss, dB
Loss, dB
50 40 30 20 1.0
10 0 0
500
1000 800
Frame # 41 Slide # 59
900
1100
A. Antoniou
1500
Ω, rad/s
2000
ψ
1200
Part3: IIR Filters – Bilinear Transformation Method
Example – BS Filter Design a Chebyshev BS filter that would satisfy the following specifications: ˜ p 1 = 350 rad/s, ˜ p 2 = 700 rad/s, Ap = 0.5 dB, Aa = 40 dB, ˜ a1 = 430 rad/s,
˜ a2 = 600 rad/s,
ωs = 3000 rad/s.
Solution ωp = 1.0
n =5 ω0 = 561, 4083 B = 493, 2594
Frame # 42 Slide # 60
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
Example – BS Filter Cont’d 60
50
Passband loss, dB
Loss, dB
40
30
20
0.5
10
0 0
200
400 350 430
Frame # 43 Slide # 61
800
Ω, rad/s
A. Antoniou
600
1000
700
Part3: IIR Filters – Bilinear Transformation Method
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
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
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
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
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
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.
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A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method
This slide concludes the presentation. Thank you for your attention.
Frame # 47 Slide # 68
A. Antoniou
Part3: IIR Filters – Bilinear Transformation Method