IIR Filters, Bilinear Transformation Method

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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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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

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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.

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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 =

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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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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.

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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.

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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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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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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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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

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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.

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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¯ ).

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 ) 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.

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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.

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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.

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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.

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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

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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

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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

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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.

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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.

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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

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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

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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.

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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

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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.

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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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Part3: IIR Filters – Bilinear Transformation Method

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

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Part3: IIR Filters – Bilinear Transformation Method