1

LOGO

iugaza2010.blogspot.commelasmer@gmail.com

Digital Filters - IIR

Part 2

3

Design a digital IIR filter by using

bilinear transformation

IIR digital filter is derived from analog filter

Steps:1. Filter Design Specifications

2. Analog Filter Design

3. Digital Filters from Analog Prototypes

From the given values design

a prototype lowpass filter

4

Design IIR Digital Filter with

the following specifications

H(z) ???

H(p)

Convert the lowpass prototype

filter to the wanted analog filter

H(S)

Design an analog filter with

the following specifications.

H(s) ???

2tan

2T

2 1

1

zs

T z

AnalogDigital

From the given values design

a prototype lowpass filter

5

Design IIR Digital Filter with

the following specific values.

H(z) ???

H(p)

Convert the lowpass prototype

filter to the wanted analog filter

H(S)

Design an analog filter with

the following specific values.

H(s) ???

2tan

2T

2 1

1

zs

T z

AnalogDigital

6

digital filter frequency

analog filter frequency

analog LP prototype filter frequency

7

IIR Filters

-Design a digital lowpass filter to satisfy the following:

(1) monotonic stopband and passband(Butterworth)

(2) -3dB cutoff frequencies of θp= 0.5π rad

(3) magnitude down at least -15dB at 0.75π rad

(4) Max magnitude =0dBTs=1sec

Design this IIR filter by using bilinear transformation

8

| ( ) |dBH j

0dB

15dB

0.5p

0.75s 3dB

digital filter

9

0.5 , 0.75

0.52 tan 2

2

0.752 tan 4.82

2

p s

p

s

2 1

1

zs

T z

2tan

2

2tan

2

T

T

: analog frequency

: digital frequency

change

change

10

| ( ) |dBH j

0dB

15dB

2p

3dB

analog filter

4.82s

11

3

15

p

s

A

A

to find LP prototype filter:

:

4.82 = , = 1 , = 2.41

2

p p sp s

p p p

normalization

12

3dB

0

15dB

LP prototype analog filter

1p 2.41s

| ( ) |dBH j

13

0.1 0.1(3)

0.1

2

0.1(15)

2

For Butterworth filter:

ripple factor:

10 1 10 1 1

10 1log

: N = 2 log( )

10 1log

1 = =2

2log(2.41)

p

s

A

A

s

order

14

(

1/

( 2

(2)1/2

1

( (2) 2

(2)1/2

2

1,2

: = e ,

1 e

1 e

1 1P = ± j

2 2

2k +N 1)j

N 2N

k

2+ 1)j

2

2 + 1)j

2

Poles P k = 1, 2

P

P

15

To restore the magnitude scale

2

(0/20)

2

H(p) = 1.414 1

10 1

11

1

1H(p) =

1.414 1

o

o

o

H

P P

H(p = 0)

H

H

P P

16

2 2

2

1 2

21

21

restor the frequency:

1 4H(S) =H(p) |

2.828 41.414 1

2 2

1 2H(z) =H(s) |

3.414 0.5857

p

S

z

z

to

Sp

S SS S

Z Z

Z

17

1 2

2

1 2

2

1 2H(z)

3.414 0.5857

1 2H( )=H(z) |

3.414 0.5857

:

|H( ) |

j

j j

jz e

Z Z

Z

e e

e

check

plot

18

IIR Filters

-The specified magnitude response of a maximally flat

bandpass digital filter has:

- maximum value of 1.0 in its passband

- cutoff frequencies θ1 = 0.4π and θ 2 = 0.5π.

- the magnitude at cutoff frequencies to be no less

than 0.93

- the stopband frequency θs= 0.7π

-the magnitude at stopband frequency to be no

more than 0.004Ts=1sec

Design this IIR filter by using bilinear transformation

0.4 0.5 0.7

s

| ( ) |H j

1 2

digital filter

s

20

1 2

1

2

0.4 , 0.5 , 0.7

0.42 tan 1.453

2

0.52 tan 2

2

0.72 tan 3.925

2

s

s

2 1

1

zs

T z

2tan

2

2tan

2

T

T

: analog frequency

: digital frequency

change

change

211.453 2 3.925

s

analog filter

| ( ) |H j

22

20log 0.93 0.63

20log 0.004 48

p

s

A dB

A dB

2 1

1 2

bandwidth: 0.547 rad/s

Center freqency: =1.705 rad/so

B

to find LP prototype filter:

:

= , = 1 , =p p s

p s

p p p

normalization

23

1.705 3.925 1.705

0.547 1.705 3.925

5.82

5.82

=

3.9250.6743 rad/sec

5.82

o okk

o k

o s os

o s

s

s

ss

p

sp

s

jj

B

j jj

B

j j

24

0

LP prototype analog filter

1p 5.84s

0.63dB

48dB

| ( ) | |dB

H j

25

0.1 0.1(0.63)

0.1

2

0.1(48)

2

For Butterworth filter:

ripple factor:

10 1 10 1 0.395

10 1log

: N = 2 log( )

10 1log

0.395 = =4

2log(5.82)

p

s

A

A

s

order

26

(

1/

( 4

(4)1/4

1

( (2) 4

(4)1/4

2

( (3) 4

(4)1/4

3

( (4) 4

(4)1/4

4

1,4

: = e , 3,4

0.395 e

0.395 e

0.395 e

0.395 e

P = 0.4827 ±

2k +N 1)j

N 2N

k

2+ 1)j

2

2 + 1)j

2

2 + 1)j

2

2 + 1)j

2

Poles P k = 1,2,

P

P

P

P

2,3

j1.1654

P = 1.1654 ± j0.4827

27

the transfer function of the normalized prototype filter

4 3 2

1H(p) =

3.296 5.4325 5.24475 2.5317P P P P

28

To restore the magnitude scale

4 3 2

(0/20)

(5/20)

4 3 2

H(p) = 3.296 5.4325 5.24475 2.5317

10 1

=10 12.5317

2.5317

2.5317H(p) =

3.296 5.4325 5.24475 2.5317

o

o

o

H

P P P P

H(p = 0)

H

H

P P P P

29

2 2

2 2 2 2

0

1 1.705

0.547

12

1

restor the frequency:

1 1 1.705

0.547

H(S) =H(p) | ...

H(z) =H(s) | ...

S

S

z

z

to

S Sp

B S S

30

31

DIGITAL SPECTRAL TRANSFORMATION

Digital LP prototype filter