Lecture Digital signal processing: Lecture 7 - Zheng-Hua Tan

Lecture Digital signal processing - Lecture 7 presents the following content: Filter design, IIR filter design, analog filter design, IIR filter design by impulse invariance, IIR filter design by bilinear transformation.

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Digital Signal Processing, VII, Zheng-Hua Tan, 2006

Lecture 7: Filter Design

2

Course at a glance

Discrete-time signals and systems

Fourier-domain

representation

DFT/FFT

System analysis

System structure System

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3

Part I: Filter design

Filter design

IIR filter design

Analog filter design

IIR filter design by impulse invariance

IIR filter design by bilinear transformation

Filter design process

Filter, in broader sense, covers any system

Three design steps

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5

Specifications – an example

Specifications for a discrete-time lowpass filter

s j

p j

e H

ωω

≤

−

, 001 0

| ) (

|

0 , 01 0 1

| ) (

| 01 0

1

001 0

01 0

6

Specifications of frequency response

Typical lowpass filter specifications in terms of

tolerable

Passband distortion, as smallestas possible

Stopband attenuation, as greatestas possible

Width of transition band: as narrowestas possible

Improving one often worsens others Æ a tradeoff

Increasing filter order improves all

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7

DT filter for CT signals

Discrete-time filter for the processing of

continuous-time signals

Bandlimited input signal

High enough sampling frequency

Then, specifications conversion is straightforward

Fig 7.1

T T

j H e

H

T

T e

H j

H

eff j

T j eff

ωω

ππ

ω

|

| ), ( )

(

/

|

| , 0

/

|

| ), ( )

| ) (

|

) 2000 ( 2 0

, 01 0 1

| ) (

| 01

≤ Ω

≤ +

≤ Ω

eff

) 3000 ( 2

) 2000 ( 2

001 0

01 0

2 1

ππδδ

= Ω

=

s p

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9

Design a filter

Design goal: find system function to make frequency

response meet the specifications (tolerances)

Infinite impulse response filter

Poles insider unit circle due to causality and stability

Rational function approximation

Finite impulse response filter

Linear phase is often required

Polynomial approximation

10

E.g IIR filter design

For rational system function

find the system coefficients such that the

corresponding frequency response

provides a good approximationto a desired response

M

k

k k

z a

z b z

ω

j e z j

z H e

(e jω H desired e jω

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If H(z) is stable and GLP, any non-trivial pole p inside

the unit circle corresponds a pole 1/p outside the unit

circle, so that H(z) cannot have a causal impulse

response (as ROC is a ring including unit circle)

(20- Unrelated to analog filter

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13

Part II: IIR filter design

Filter design

IIR filter design

14

Design IIR filter based on analog filter

The mapping is direct

Advanced analog filter design techniques

Æ Designing DT filter by transforming prototype CT

filter:

Transform (map) DT specifications to analog

Design analog filter

Inverse-transform analog filter to DT

πωω

ππ

(

/

|

| , 0

/

|

| ), ( )

(

T j H e

H

T

T e

H j

H

eff j

T j eff

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15

Transformation method

Transform (map) DT specifications to analog

Design analog filter

Inverse-transform to DT

) (

or )

=

Ω +

=

n

n n

n j j

j

t j st

z n x z

X

e n x e

X

re z

dt e t h j

H

dt e t h s

H

j s

] [ )

(

] [ )

(

) ( )

(

) ( )

(

ω ω

ω

σ

• The imaginary axis of the s-plane Æ

the unit circle of the z-plane

• Poles in the left half of the s-plane Æ

poles inside the unit circle in the

z-plane (stable)

Part III: Analog filter design

Filter design

IIR filter design

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Butterworth lowpass filters

Maximally flat in the passband

Monotonic in both passband and stopband

2 1

FigB

N c

c j

) / ( 1

1

| ) (

|

Ω Ω +

= Ω

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21

Elliptic filters

Equiripple both in stopband and in the passband

22

Part IV: Design by impulse invariance

Filter design

IIR filter design

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23

Filter design by impulse invariance

Impulse invariance: a method for obtaining a DT

of a CT system

In DT filter design, the specifications are provided in

the discrete-time, so T d has no role T dis included for

discussion though T dalso has nothing to do with C/D

and D/C conversion in Fig 7.1

) (jΩ

H c

) (e jωH

interval sampling

design' '

) ( ]

[

d

d c d

T

nT h T n

h =

Relationship btw frequency responses

Impulse response sampling:

πω

(

/

|

| , 0 )

(

)

2 (

)

(

d c j

d c

d

c j

T j H

e

H

T j

H

k T

j T j H e

H

) ( ]

T

k j T j X T e

X( ω) 1 ( ω 2π )

)(][n x nT

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25

Aliasing in the impulse invariance design

)

2 (

)

T

j T j H e

Relationship btw system functions

The transform from CT to DT is easy to carry out as

a transformation on the system function

Rational system function, after partial fraction

n T s k d

N k

nT s k d

d c d

n u e A T

nT h T n h

d k

1

] [ ) (

] [

) ( ]

0

0 ,)

(

)

(

1 1

t

t e A t

h

s s

A s

H

N

k

t s k c

N

k k

k c

z e

A T z

H

d k

)(

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27

Impulse invariance with a Butterworth filter

Specifications

Since the sampling interval Td cancels in the impulse

invariance procedure, we choose Td=1, so

Magnitude function for a CT Butterworth filter

Due to the monotonic function of Butterworth filter

π ω π

π ω

| ) (

|

2 0

|

| 0 , 1

| ) (

| 89125

Ω

=

ω

π π

π

≤ Ω

≤

≤ Ω

≤

≤ Ω

≤

|

| 3 0 , 17783 0

| ) (

|

2 0

|

| 0 , 1

| ) (

| 89125 0

j H

c

17783 0

| ) 3 0 (

|

89125 0

| ) 2 0 (

j H

c c

Squared magnitude function of a Butterworth filter

2 2

) 17783 0

1 ( ) 3

0

(

1

) 89125 0

1 ( ) 2

0

(

1

= Ω

+

= Ω

+

N c

π π

17783 0

| ) 3 0 (

|

89125 0

| ) 2 0 (

|

≤

≥ π

π

j H

j H c

c

N c

c j

) / ( 1

1

| ) (

|

Ω Ω +

= Ω

70474 0

8858 5

(3)

(2)

2

1

) / ( 1

1 )

( ) (

j s s

H s

c c

c

=

Ω +

=

−

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29

12 poles for the

squared magnitude

function

The system function

has the three pole

pairs in the left half of

4945 0 9945 0 )(

4945 0 3640

0

(

12093 0 )

+ +

=

s s

s

H c

) 2570 0 9972 0 1 (

6303 0 8557 1 )

3699 0 0691

.

2

) 6949 0 2971

1

2 1

1

2 1

z z

z

z z

H

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IIR filter design

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33

Bilinear transformation

By using impulse invariance, the relation between

CT and DT frequency is linear (except for aliasing),

thus the shape of the frequency response is

preserved But only proper for bandlimited filters,

problem for e.g highpass

Bilinear transformation between s and z

s T

s T z

z

z T H z H z

z T

s

d d

d c d

) 2 / ( 1

)]

1

1 (

2 [ ) ( ) 1

1 ( 2

1 1 1

/ 1

2 / 2

/ 1 )

T j T s

d d

Ω

= j

s

circle unit the onto

maps axis - the i.e.

axis -

on the

any for , 1

|

| so,

2 / 1

Ω Ω

=

Ω

−

Ω +

=

j j

s z

T j

T j z

d d

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d

d j

T j

T j e

s

Ω

−

Ω +

=

) 2 / arctan(

2

) 2 / tan(

2

) 2 / tan(

2 ] ) 2 / (cos 2

) 2 / sin ( 2 [

2 ) 1

1

(

2

) 1

1

(

2

2 /

1 1

d d

d j

j d j j d

d

T T

T

j e

j e T e

e T

j

z

z T

ω

ω ω

ω

Bilinear transformation

The bilinear transformation maps the entire -axis

in the s-plane to one revolutionof the unit circle in

Compare Ω=ω

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37

Bilinear transformation of a Butterworth filter

Specifications

Magnitude function for a CT Butterworth filter

Due to the monotonic function of Butterworth filter

π ω π

π ω

| ) (

|

2 0

|

| 0 , 1

| ) (

| 89125

≤

≤ Ω

≤

≤ Ω

≤

|

| ) 2

3 0 tan(

2 , 17783 0

| ) (

|

) 2

2 0 tan(

2

|

| 0 , 1

| ) (

| 89125

d c

T j

H

T j

H

17783 0

| )) 15 0 tan(

2 (

|

89125 0

| )) 1 0 tan(

2 (

j H

c c

38

Squared magnitude function of a Butterworth filter

N c

c j

) / ( 1

1

| ) (

|

Ω Ω +

= Ω

305 5

=

N

766 0

| )) 15 0 tan(

2 (

|

89125 0

| )) 1 0 tan(

2 (

j H

c c

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39

Summary

Filter design

IIR filter design

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41

Course at a glance

Discrete-time signals and systems

Fourier-domain

representation

DFT/FFT

System analysis

System structure System

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