dsp5-iir 1

25-Mar-15

1

CH 5

IIR

ME-4722

Digital Signal Processing

Chapter 5

Design of Digital IIR

Filters(Elective)

Spring 2015, SZABIST, Karachi

CH 5

IIR

Instructor:

Engr. Humera Rafique

Assistant Professor (Mechatronics)

[email protected]

Office: FR-404 (100 Campus )

Course Support

Official: ZABdesk

25-Mar-15 HR DSP S15

2

25-Mar-15

2

CH 5

IIR Chapter Contents

Bilinear Transformation: IIR Digital Filter DesignBilinear Transformation: IIR Digital Filter DesignBilinear Transformation: IIR Digital Filter DesignBilinear Transformation: IIR Digital Filter Design• Introduction• Design summary• Implementation of BLT:

1. 1st order lowpass IIR filter2. 1st order highpass IIR filter3. 2nd order banpass IIR filter (Peaking)4. 2nd order bandstop IIR filter (Notching)5. Higher order lowpass/highpass* filters (Butterworth)6. Higher order bandpass/bandstop* filters (Butterworth)


3

CH 5

IIR

Digital IIR Filter Design:Digital IIR Filter Design:Digital IIR Filter Design:Digital IIR Filter Design:Bilinear TransformationBilinear TransformationBilinear TransformationBilinear Transformation


4

25-Mar-15

3

CH 5

IIR Bilinear Transformation• Simplest & effective method of IIR filter design

• Instead of designing a digital filter directly, this algorithm maps a digital filter into an equivalent analog filter e.g.,

* Butterworth* Chebychev* Elliptical etc

• Designed analog filter is then mapped back into the desired digital filter


5

CH 5

IIR Bilinear Transformation


6

25-Mar-15

4

CH 5


4565789 :;<=><?@A: B C2D:

:E<=>5F89<?7 8?89G6 :;<=><?@A: Ω C 6"B%

I C : J C : <KL

M N CO P NQO

O R NQO

Ω C tanB

2Bilinear transform

• The z-plane design of the digital filter is replaced by s-plane of an equivalent analog filter, using suitable

mapping relationships:

Bilinear transform

• Because of non-linear relationship of analog & digital frequencies, this is called ‘pre-warping

transformation’


7

CH 5


Design Method SummaryDesign Method SummaryDesign Method SummaryDesign Method Summary

1. Given: Magnitude response specification of a desired digital filter2. Transform these specifications using suitable bilinear transformation relationship into an analog filter’s

specification3. Using an analog filter design technique, design an analog filter Ha(s)4. Use bilinear transformation and map analog filter back into desired digital filter H(z)

Z;8?I:<; :>?@75G?: [ J C [\(I)]E ^_(`)

C [\ :(J)

a86?57>4< ;<IbG?I<: [ B C [\(Ω)]c ^d(L)

C [\ 6(B)


8

25-Mar-15

5

CH 5

IIR


5. To design a stable and causal digital filter a causal and stable analog filter is required. Bilinear

transformation maps left hand side of the s-plane to ‘inside of the unit circle’ in z-plane:


9Bilinear Transformation

CH 5

IIR

• All analog filter design methods give rise to stable and causal transfer functions Ha"s%

• This property guarantees that the digital filter H"z% obtained will also be stable and causal




25-Mar-15

6

CH 5

IIR

Implementation ofImplementation ofImplementation ofImplementation of

Bilinear TransformationBilinear TransformationBilinear TransformationBilinear Transformation


11

CH 5

IIR

� 1st order low pass filter

� 1st order high pass filter

� 2nd order band pass filter (peaking)

� 2nd order band stop filter (notching)

� Higher order (Butterworth) low and high pass filters

� Higher order (Butterworth) band pass and band stop filters

� Equalizers*



25-Mar-15

7

CH 5

IIR

Implementation ofImplementation ofImplementation ofImplementation of

Bilinear TransformationBilinear TransformationBilinear TransformationBilinear Transformation

1111stststst Order Order Order Order LowpassLowpassLowpassLowpass Filter DesignFilter DesignFilter DesignFilter Design


13

CH 5

IIR

• A simple filter design of type digital low pass with Nth order can be represented as:

• 1st order:

1st order LPF Design

[ J Cg"J%

h"J%C

ij R ikJQk R ⋯ R imJQm

1 R 8kJQk R ⋯ R 8nJQn

[ J Cg"J%

h"J%C

ij R ikJQk

1 R 8kJQk

Parameters: Parameters: Parameters: Parameters: • ZAb<: 4565789 9Go b8II

• G;4<;: 1I7 • b;<I@;5i<4 :;<=><?@A @>7G:: : :p

• I8qb95?6 :;<=><?@A ∶ :s

25-Mar-15

8

CH 5

IIR

Design Problem:

• To determine the filter’s coefficients in terms of

• Range of frequencies pass through filter:

• Range of frequencies filtered through filter:

• Digital cutoff frequency:

ij, ik, 8k tp , :p , :E

0 v : v :p

:p v : v:E

2

Bp C2D:p

:E



CH 5

IIR

Design Problem:



25-Mar-15

9

CH 5

IIR

Design Problem:Ω C tan

B

2


17

CH 5

IIR

HertzianHertzianHertzianHertzian response parameters:response parameters:response parameters:response parameters:wp: b8II i8?4 685? 87 @>7G:: :;<=><?@Atp: 4x F<;I5G? G: wp:p: @>7G:: :;<=><?@A:E: I8qb95?6 :;<=><?@A

wp: b8II i8?4 685? 87 @>7G:: :;<=><?@Atp: 4x F<;I5G? G: wpBp: @>7G:: :;<=><?@A

wp: b8II i8?4 685? 87 @>7G:: :;<=><?@Atp: 4x F<;I5G? G: wpΩp: 8?89G6 @>7G:: :;<=><?@A

Desired digital Desired digital Desired digital Desired digital lowpasslowpasslowpasslowpass filter’s parameters:filter’s parameters:filter’s parameters:filter’s parameters:

Equivalent analog Equivalent analog Equivalent analog Equivalent analog lowpasslowpasslowpasslowpass filter’s parameters:filter’s parameters:filter’s parameters:filter’s parameters:



25-Mar-15

10

CH 5

IIR

tp C P10 log wpy

wpy C 10Qz{/y|

5: ["0% y C 17}<? ["Bp%y C wp

y

Ωp C tanBp

2C tan

D:p

:E

Analog cutoff

in terms of

digital

prescribed

parameters

Bilinear Transformation:

wpy C

["Bp% y

["0% yC1

2

⟹ tp C P10 log["Bp%

y

["0% yC 34x



CH 5

IIR

1st Order Low Pass Analog Filter Transfer Function:

[\ I C �

I R �

[\ Ω y C�y

Ωy R �y

Determining Filter Parameters:wp , :p i, 8

� Cwp

1 P wpyΩp

wp , :p �, Ωp Bp↔ ↔

digitalfilterspec

analogfilterconstant

analogfrequency

digitalfreq.



dsp5-iir 1

Documents