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COMM 602: Digital Signal Processing

Lecture 10

Digital Filter Design



Filter Types

:Remember



Practical Filter specifications



IIR Filter

M

M



IIR Filter Design

Design means calculation of the coefficients of the difference

equation or the transfer function.

We use Analog filter design and then we convert the design into

digital domain because:

Analog filter design is a well developed field and highly advanced.

Analog filter design usually give closed form solutions.

Extensive tables are available for analog filter design.



IIR Filter Design

Methods of IIR Filter Design:

- Impulse invariant- Step invariant

- Bilinear transformation.



Impulse Invariant Method

Steps of Design:


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Example: Convert the following analog filter system function into a digital IIR

filter by means of the impulse invariance method:

1

s

s

s



sHth 11 te 5.03

nTnTnt aeth 3325.0

where

Tea 5.0

1

00 1

33z3

az

zaznhnThZHn

nn

n

n

s

5.0

3

ssH

Example:

Convert the following analog filter system function into a digital

IIR filter by means of the impulse invariance method:



Characteristics of Analog Filters Butterworth Filters:

All pole filter (no zeros), with no ripples in both the passband and

stop band. The transfer functions of Butterworth filter withis:

OrderFilter FunctionTransfer

11

s

12

12 ss

122

123 sss

1

2

3

16131.24142.36131.2

1234

ssss

4

filteranatheof

frequencyoffCutc

log

:

1c



Example:

Using the impulse invariant method design a LP digital filter

satisfying the following requirements: cut off frequency 500 Hz,

Sampling rate=500 Hz, 2nd order Butterworth filter.Solution

2/100sin21002/sin2)(

)2/()2/()2/(2

2)2/(

2

1/2/

1)(

)2/100()2/(

22

22

2

2

22

2

2

teteth

s

sss

ss

sH

t

c

t

c

cc

cc

ccc

c

cc

c

cc

a

c



002.0500

11

,2/100sin2100)( )2/100(

s

nT

fT

nTenTh

)(

cos21)sin()sin(

)()(

2

zHThen

zaTzaTznaTzand

zeXenXSinceaTnaT



Step Invariant MethodSteps of Design:

nRealizatio(4)

)(

1

Find(3)

samplingbyFind)3(

)(responsestepFind(2)

filter.analogtheofFind(1)

1

nTaz

z

H(Z)

a(nT)s

sHa(t)

H(s)

stu

1)(



Example:

TT

nTt

ez

z

ez

z

z

z

z

zzH

nTuenTatuetaThen

s

)/(

s

)/(

)s(s

ssHa(t)

)s(ss

H(s)

sH(s)

55

55

11-

11

5

1

15

11)(

)()5

1

5

1()()()

5

1

5

1()(

5

5151

5

1

5)s(s1)(

5

1Solution

filterdigitalit toconvert5

1:T.FanalogGiven the



Bilinear Transformation

See Text book

For the derivation

Of this equation

2tan

2

T



Bilinear Transformation

Derivation of:

(1)



Frequency Warping Effect

2

tan2

T



Properties of Bilinear Transformation

0)(tan2

01

01

01

:havewe(3)and(2)From

)3(1

tan1

tan)arg(

)2(1

1

:bydefinedareangleandradiusthewhere:as

representmaywe(1),equationinPutting.2Let

1

11

21

22

22

for

forr

forr

forr

zand

)(

)(zr

erzz

jsT

/

j



BILINEAR ZTRANSFORMATION(Contd.)

S-Plane

LHS RHSJ-axis

Z-Plane

Unit Circle z =1z = -1



Steps of Bilinear Transformation

)(H

H(z)

2tan

2

T



Example:

ssH )(

)(H

3249.0)1.0tan(2

tan

2tan

2

T

For T=2



H

ssH )(

H(z)



Example:

5.0

3

s

sH

1

1

1

12

z

z

Ts

5.0

1

12

3

1

1

z

z

T

zH

11

1

1

15.01

12

13

zz

z

T

zzH

Convert the following analog filter system function into a digital

IIR filter by means of the bilinear transformation method:

Solution



Example (Contd.)

1

1

1

1

15.01

12

13

zz

z

T

zzH

11

1

15.012

13

zzT

zzH

1

1

25.05.0

2

13

zTT

zzH

1

1

1

1

1

1

5.02

25.0

15.02

13

z

z

z

T

T

T

zzH

s



Example (Contd.)

2

ny

1z

nx

1z

33.0

1

1

33.01

12

z

zzH

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