warping concept (iir filters-bilinear transformation method)
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Part 3: IIR Filters – Bilinear TransformationMethod
Tutorial ISCAS 2007
Copyright © 2007 Andreas AntoniouVictoria, BC, Canada
Email: aantoniou@ieee.org
July 24, 2007
Frame # 1 Slide # 1 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Introduction
� A procedure for the design of IIR filters that would satisfyarbitrary prescribed specifications will be described.
Frame # 2 Slide # 2 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Introduction
� A procedure for the design of IIR filters that would satisfyarbitrary prescribed specifications will be described.
� The method is based on the bilinear transformation and itcan 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
Introduction Cont’d
Given an analog filter with a continuous-time transfer functionHA(s), a digital filter with a discrete-time transfer function HD(z)
can be readily deduced by applying the bilinear transformationas follows:
HD(z) = HA(s)
∣∣∣∣s= 2
T
(z−1z+1
) (A)
Frame # 3 Slide # 4 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Introduction Cont’d
The bilinear transformation method has the following importantfeatures:
� A stable analog filter gives a stable digital filter.
Frame # 4 Slide # 5 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Introduction Cont’d
The bilinear transformation method has the following importantfeatures:
� A stable analog filter gives a stable digital filter.
� The maxima and minima of the amplitude response in theanalog 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
Introduction Cont’d
Frame # 5 Slide # 7 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Introduction Cont’d
Frame # 6 Slide # 8 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
The Warping Effect
If we let ω and � represent the frequency variable in the analogfilter and the derived digital filter, respectively, then Eq. (A), i.e.,
HD(z) = HA(s)
∣∣∣∣s= 2
T
(z−1z+1
) (A)
gives the frequency response of the digital filter as a function ofthe frequency response of the analog filter as
HD(ej�T ) = HA(jω)
provided that s = 2T
(z − 1z + 1
)
or jω = 2T
(ej�T − 1ej�T + 1
)or ω = 2
Ttan
�T2
(B)
Frame # 7 Slide # 9 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
The Warping Effect Cont’d
· · ·ω = 2
Ttan
�T2
(B)
� For � < 0.3/Tω ≈ �
and, as a result, the digital filter has the same frequencyresponse as the analog filter over this frequency range.
Frame # 8 Slide # 10 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
The Warping Effect Cont’d
· · ·ω = 2
Ttan
�T2
(B)
� For � < 0.3/Tω ≈ �
and, as a result, the digital filter has the same frequencyresponse as the analog filter over this frequency range.
� For higher frequencies, however, the relation between ω
and � becomes nonlinear, and distortion is introduced inthe frequency scale of the digital filter relative to that of theanalog 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
0.1π 0.2 π 0.5π
6.0
T = 2
ω
|HD(e jΩT)|
|HA(jω)| Ω, rad/s
4.0
2.0
0.3 π 0.4 π
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 filterrelative to those of the analog filter in a nonlinear way, asillustrated for the case of a BS filter:
0 1 2 3 4 5 0
5
10
15
20
25
30
35
40 L
oss,
dB
ω, rad/s
Analog filter
Digital filter
Frame # 10 Slide # 13 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Prewarping
� From Eq. (B), i.e.,
ω = 2T
tan�T2
(B)
a frequency ω in the analog filter corresponds to afrequency � in the digital filter and hence
� = 2T
tan−1 ωT2
Frame # 11 Slide # 14 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Prewarping
� From Eq. (B), i.e.,
ω = 2T
tan�T2
(B)
a frequency ω in the analog filter corresponds to afrequency � in the digital filter and hence
� = 2T
tan−1 ωT2
� If ω1, ω2, . . . , ωi , . . . are the passband and stopband edgesin the analog filter, then the corresponding passband andstopband edges in the derived digital filter are given by
�i = 2T
tan−1 ωi T2
i = 1, 2, . . .
Frame # 11 Slide # 15 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Prewarping Cont’d
� If prescribed passband and stopband edges �̃1,
�̃2, . . . , �̃i , . . . are to be achieved, the analog filter must beprewarped before the application of the bilineartransformation to ensure that its band edges are given by
ωi = 2T
tan�̃i T
2
Frame # 12 Slide # 16 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Prewarping Cont’d
� If prescribed passband and stopband edges �̃1,
�̃2, . . . , �̃i , . . . are to be achieved, the analog filter must beprewarped before the application of the bilineartransformation to ensure that its band edges are given by
ωi = 2T
tan�̃i T
2
� Then the band edges of the digital filter would assume theirprescribed values �i since
�i = 2T
tan−1 ωi T2
= 2T
tan−1
(T2
· 2T
tan�̃i T
2
)
= �̃i for i = 1, 2, . . .
Frame # 12 Slide # 17 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 log1
|HN (jω)|(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 inthe literature in normalized form whereby the passband edge istypically of the order of unity.
Frame # 13 Slide # 18 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design Procedure Cont’d
ω
ωa ωp
Ap
Aa
ωc
A(ω)
Frame # 14 Slide # 19 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design Procedure
A denormalized LP, HP, BP, or BS filter that has the samepassband ripple and minimum stopband attenuation as a givennormalized LP filter can be derived from the normalized LP filterthrough the following steps:
1. Apply the transformation s = fX (s̄)
HX (s̄) = HN (s)
∣∣∣s=fX (s̄)
where fX (s̄) is one of the four standard analog-filterstransformations, given by the next slide.
Frame # 15 Slide # 20 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design Procedure
A denormalized LP, HP, BP, or BS filter that has the samepassband ripple and minimum stopband attenuation as a givennormalized LP filter can be derived from the normalized LP filterthrough the following steps:
1. Apply the transformation s = fX (s̄)
HX (s̄) = HN (s)
∣∣∣s=fX (s̄)
where fX (s̄) is one of the four standard analog-filterstransformations, given by the next slide.
2. Apply the bilinear transformation to HX (s̄), i.e.,
HD(z) = HX (s̄)
∣∣∣s̄= 2
T
(z−1z+1
)
Frame # 15 Slide # 21 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design Procedure Cont’d
Standard forms of fX (s̄)
X fX (s̄)
LP λs̄
HP λ/s̄
BP 1B
(s̄ + ω2
0
s̄
)
BSBs̄
s̄2 + ω20
Frame # 16 Slide # 22 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design Procedure Cont’d
� The digital filter designed by this method will have therequired passband and stopband edges only if theparameters λ,ω0, and B of the analog-filter transformationsand the order of the continuous-time normalized LPtransfer function, HN (s), are chosen appropriately.
Frame # 17 Slide # 23 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design Procedure Cont’d
� The digital filter designed by this method will have therequired passband and stopband edges only if theparameters λ,ω0, and B of the analog-filter transformationsand the order of the continuous-time normalized LPtransfer function, HN (s), are chosen appropriately.
� This is obviously a difficult problem but general solutionsare 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 generalsolutions for LP filters is as follows:
1. Assume that a continuous-time normalized LP transferfunction, HN (s), is available that would give the requiredpassband ripple, Ap, and minimum stopband attenuation(loss), Aa.
Let the passband and stopband edges of the analog filterbe ωp and ωa, respectively.
Frame # 18 Slide # 25 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design of LP Filters Cont’d
ω
ωa ωp
Ap
Aa
ωc
A(ω)
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) toobtain a denormalized discrete-time transfer functionHLP(s̄).
Frame # 20 Slide # 27 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) toobtain a denormalized discrete-time transfer functionHLP(s̄).
3. Apply the bilinear transformation to HLP(s̄) to obtain adiscrete-time transfer function HD(z).
Frame # 20 Slide # 28 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) toobtain a denormalized discrete-time transfer functionHLP(s̄).
3. Apply the bilinear transformation to HLP(s̄) to obtain adiscrete-time transfer function HD(z).
4. At this point, assume that the derived discrete-time transferfunction has passband and stopband edges that satisfy therelations
�̃p ≤ �p and �a ≤ �̃a
where �̃p and �̃a are the prescribed passband andstopband edges, respectively.
In effect, we assume that the digital filter has passbandand stopband edges that satisfy or oversatisfy the requiredspecifications.
Frame # 20 Slide # 29 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design of LP Filters Cont’d
Ap
Aa
A(Ω)
Ωa Ωp
Ω
Ωa ~
Ωp ~
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-filtertransformation.
Frame # 22 Slide # 31 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-filtertransformation.
6. Find the minimum value of the ratio ωp/ωa for thecontinuous-time normalized LP transfer function.
The ratio ωp/ωa is a fraction less than unity and it is ameasure of the steepness of the transition characteristic. Itis often referred to as the selectivity of the filter.
The selectivity of a filter dictates the minimum order toachieve the required specifications.
Note: As the selectivity approaches unity, the filter-ordertends to infinity!
Frame # 22 Slide # 32 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design of LP Filters Cont’d
7. The same methodology is applicable for HP filters, exceptthat the LP-HP analog-filter transformation is used in Step2.
Frame # 23 Slide # 33 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design of LP Filters Cont’d
7. The same methodology is applicable for HP filters, exceptthat the LP-HP analog-filter transformation is used in Step2.
8. The application of this methodology yields the formulassummarized by the table shown in the next slide.
Frame # 23 Slide # 34 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Formulas for LP and HP Filters
ωp
ωa≥ K0
LP
λ = ωpT
2 tan(�̃pT /2)
ωp
ωa≥ 1
K0HP
λ = 2ωp tan(�̃pT /2)
T
where K0 = tan(�̃pT /2)
tan(�̃aT /2)
Frame # 24 Slide # 35 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design of LP Filters Cont’d
� The table of formulas presented is applicable to all theclassical 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
� The table of formulas presented is applicable to all theclassical types of analog filters, namely,
– Butterworth
– Chebyshev
– Inverse-Chebyshev
– Elliptic
� Formulas that can be used to design digital versions ofthese 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 generalsolutions for BP filters is as follows:
1. Assume that a continuous-time normalized LP transferfunction, HN (s), is available that would give the requiredpassband ripple, Ap, and minimum stopband attenuation,Aa.
Let the passband and stopband edges of the analog filterbe ωp and ωa, respectively.
Frame # 26 Slide # 38 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) toobtain a denormalized discrete-time transfer functionHBP(s̄).
Frame # 27 Slide # 39 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) toobtain a denormalized discrete-time transfer functionHBP(s̄).
3. Apply the bilinear transformation to HBP(s̄) to obtain adiscrete-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 transferfunction has passband and stopband edges that satisfy therelations
�p1 ≤ �̃p1 �p2 ≥ �̃p2
and�a1 ≥ �̃a1 �a2 ≤ �̃a2
where
– �p1 and �p2 are the actual lower and upper passbandedges,
– �̃p1 and �̃p2 are the prescribed lower and upper passbandedges,
– �a1 and �a2 are the actual lower and upper stopbandedges,
– �̃p1 and �̃p2 are the prescribed lower and upper stopbandedges, respectively.
Frame # 28 Slide # 41 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Design of LP Filters Cont’d
A(Ω)
Ap
Ωa1
Ωp1 Ωa1 ~
Ωp1 ~ Ωa2
Ωp2 Ωa2 ~
Ω
Ωp2 ~
Aa Aa
Attenuation characteristic of required BP digital filterFrame # 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-BPanalog-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-BPanalog-filter transformation.
6. Find the minimum value for the selectivity, i.e., the ratioωp/ωa, for the continuous-time normalized LP transferfunction.
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-BPanalog-filter transformation.
6. Find the minimum value for the selectivity, i.e., the ratioωp/ωa, for the continuous-time normalized LP transferfunction.
7. The same methodology can be used for the design of BSfilters except that the LP-to-BS transformation is used inStep 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-BPanalog-filter transformation.
6. Find the minimum value for the selectivity, i.e., the ratioωp/ωa, for the continuous-time normalized LP transferfunction.
7. The same methodology can be used for the design of BSfilters except that the LP-to-BS transformation is used inStep 2.
8. The application of this methodology yields the formulassummarized in the next two slides.
Frame # 30 Slide # 46 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Formulas for the design of BP Filters
ω0 = 2√
KB
T
BPωp
ωa≥{
K1 if KC ≥ KB
K2 if KC < KB
B = 2KA
T ωp
where KA = tan�̃p2T
2− tan
�̃p1T2
KB = tan�̃p1T
2tan
�̃p2T2
KC = tan�̃a1T
2tan
�̃a2T2
K1 = KA tan(�̃a1T /2)
KB − tan2(�̃a1T /2)
K2 = KA tan(�̃a2T /2)
tan2(�̃a2T /2) − KB
Frame # 31 Slide # 47 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Formulas for the Design of BS Filters
ω0 = 2√
KB
T
BSωp
ωa≥
⎧⎪⎪⎨⎪⎪⎩
1K2
if KC ≥ KB
1K1
if KC < KB
B = 2KAωp
T
where KA = tan�̃p2T
2− tan
�̃p1T2
KB = tan�̃p1T
2tan
�̃p2T2
KC = tan�̃a1T
2tan
�̃a2T2
K1 = KA tan(�̃a1T /2)
KB − tan2(�̃a1T /2)
K2 = KA tan(�̃a2T /2)
tan2(�̃a2T /2) − KB
Frame # 32 Slide # 48 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Formulas for ωp and n
� The formulas presented apply to any type of normalizedanalog LP filter with an attenuation that would satisfy thefollowing 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
Formulas for ωp and n
� The formulas presented apply to any type of normalizedanalog LP filter with an attenuation that would satisfy thefollowing conditions:
0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp
AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞� 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
� The formulas presented apply to any type of normalizedanalog LP filter with an attenuation that would satisfy thefollowing conditions:
0 ≤ AN (ω) ≤ Ap for 0 ≤ |ω| ≤ ωp
AN (ω) ≥ Aa for ωa ≤ |ω| ≤ ∞� However, the values of the normalized passband edge, ωp,
and the required filter order, n, depend on the type of filter.
� 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 K = K0
HP K = 1K0
BP K ={
K1 if KC ≥ KB
K2 if KC < KB
BS K =
⎧⎪⎪⎨⎪⎪⎩
1K2
if KC ≥ KB
1K1
if KC < KB
n ≥ log D2 log(1/K )
, D = 100.1Aa − 1
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 K = K0
HP K = 1K0
BP K ={
K1 if KC ≥ KB
K2 if KC < KB
BS K =
⎧⎪⎪⎨⎪⎪⎩
1K2
if KC ≥ KB
1K1
if KC < KB
n ≥ cosh−1√
D
cosh−1(1/K )
, D = 100.1Aa − 1
100.1Ap − 1
ωp = 1
Frame # 35 Slide # 53 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Formulas for Elliptic Filters
k ωp
LP K = K0√
K0
HP K = 1K0
1√K0
BP K ={
K1 if KC ≥ KB
K2 if KC < KB
√K1√K2
BS K =
⎧⎪⎪⎨⎪⎪⎩
1K2
if KC ≥ KB
1K1
if KC < KB
1√K21√K1
n ≥ cosh−1√
D
cosh−1(1/K )
, D = 100.1Aa − 1100.1Ap − 1
Frame # 36 Slide # 54 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Example – HP Filter
An HP filter that would satisfy the following specifications isrequired:
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 digitalfilter.
Frame # 37 Slide # 55 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
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
Example Cont’d
0 1 2 3 4 5 0
10
20
30
40
50
60
70
Ω, rad/s
Los
s, d
B
Butterworth(n=5)
Elliptic (n=3)
Chebyshev (n=4)
1.0
Pass
band
loss
, dB
3.5 1.5
Frame # 39 Slide # 57 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Example – BP Filter
Design an Elliptic BP filter that would satisfy the followingspecifications:
Ap = 1 dB, Aa = 45 dB, �̃p1 = 900 rad/s, �̃p2 = 1100 rad/s,
�̃a1 = 800 rad/s, �̃a2 = 1200 rad/s, ωs = 6000 rad/s.
Solution
k = 0.515957
ωp = 0.718302
n = 4
ω0 = 1, 098.609
B = 371.9263
Frame # 40 Slide # 58 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Example – BP Filter Cont’d
0 500 1000 1500 2000 0
10
20
30
40
50
60
70
Ω, rad/s
Los
s, d
B
Pass
band
loss
, dB
1.0
800 900 1200 1100
ψ
Frame # 41 Slide # 59 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Example – BS Filter
Design a Chebyshev BS filter that would satisfy the followingspecifications:
Ap = 0.5 dB, Aa = 40 dB, �̃p1 = 350 rad/s, �̃p2 = 700 rad/s,
�̃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
0 200 400
600
800 1000 0
10
20
30
40
50
60
Ω, rad/s
Los
s, d
B
0.5
Pass
band
loss
, dB
350 430 700
Frame # 43 Slide # 61 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
D-Filter
A DSP software package that incorporates the designtechniques 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
� A design method for IIR filters that leads to a completedescription of the transfer function in closed form either interms of its zeros and poles or its coefficients has beendescribed.
Frame # 45 Slide # 63 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Summary
� A design method for IIR filters that leads to a completedescription of the transfer function in closed form either interms of its zeros and poles or its coefficients has beendescribed.
� The method requires very little computation and leads tovery precise optimal designs.
Frame # 45 Slide # 64 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Summary
� A design method for IIR filters that leads to a completedescription of the transfer function in closed form either interms of its zeros and poles or its coefficients has beendescribed.
� The method requires very little computation and leads tovery precise optimal designs.
� It can be used to design LP, HP, BP, and BS filters of theButterworth, Chebyshev, Inverse-Chebyshev, Elliptic types.
Frame # 45 Slide # 65 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
Summary
� A design method for IIR filters that leads to a completedescription of the transfer function in closed form either interms of its zeros and poles or its coefficients has beendescribed.
� The method requires very little computation and leads tovery precise optimal designs.
� It can be used to design LP, HP, BP, and BS filters of theButterworth, Chebyshev, Inverse-Chebyshev, Elliptic types.
� All these designs can be carried out by using DSP softwarepackage D-Filter.
Frame # 45 Slide # 66 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
References
� A. Antoniou, Digital Signal Processing: Signals, Systems,and Filters, Chap. 15, McGraw-Hill, 2005.
� 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
This slide concludes the presentation.Thank you for your attention.
Frame # 47 Slide # 68 A. Antoniou Part3: IIR Filters – Bilinear Transformation Method
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