TOPIC:-FILTER DESIGNBy

Nehe PradeepkumarDattatraya

Introduction.

IIR Filter Design by Impulse invariance method.

IIR Filter Design by Bilinear transformation method.

FIR Filter Design by Frequency sampling technique.

Advantages and Disadvantages.

Filter:- The systm that modify the spectral contents of the

input signal are termed as filter.

Filters mainly are of two types

1-Analog Filter

2-Digital Filter

Filter

Analog Filter

Digital Filter

IIR Filter

FIR Filter

Performance of digital filter can not influenced due to

component ageing, temp, &power supply variations

Highly immune to noise.

Operated over wide range of frequency.

Multiple filtering possible only in digital filter.

Now we will introduce to the design method of the

FIR filter and IIR filter respectively.

IIR is the infinite impulse response abbreviation.

Digital filters by the accumulator, the multiplier, and

it constitutes IIR filter the way, generally may divide

into three kinds, respectively is Direct form, Cascade

form, and Parallel form.

IIR filter design methods include the impulse

invariance, bilinear transformation.

FIR is the finite impulse response abbreviation,

because its design construction has not returned to the

part which gives.

Its construction generally uses Direct form and

Cascade form.

FIR filter design methods include the window

function, frequency sampling.

Steps:

s z transformation

Step 1: let the impulse response of analog filter is h(t)

Given the transfer function in S-domain, find the impulse response h(t).

Step 2: Sample h(t) to get h(n).

unit sample response can be obtained by putting t=nT

Step 3: Find H(Z), by taking z-transform of h(n).

The most straightforward of these is the impulse

invariance transformation

Let h(t) be the impulse response corresponding to

H(s), and define the continuous to discrete time

transformation by setting h(n)=h(nT)

We sample the continuous time impulse response to

produce the discrete time filter

The impulse invariance transformation does map the

jώ-axis and the left-half s plane into the unit circle

and its interior, respectively.

The most generally useful is the bilinear transformation.

To avoid aliasing of the frequency response as encountered with the impulse invariance transformation.

We need a one-to-one mapping from the s plane to the z plane.

The problem with the transformation is many-to-one.

's Tz e

We could first use a one-to-one transformation from s to s’ , which compresses the entire s plane into the strip

Then s’ could be transformed to z by

with no effect from aliasing.

Im( ')sT T

's Tz e

The transformation from s to s’ is given by

The characteristic of this transformation is seen most

readily from its effect on the jώ axis. Substituting

s=jώ and s’= jώ’ , we obtain

12' tanh ( )

2

sTs

T

12' tan ( )

2

T

T

The discrete-time filter design is obtained from the

continuous-time design by means of the bilinear

transformation

1 1(2/ )(1 )/(1 )( ) ( ) |c s T z z

H z H s

For arbitrary, non-classical specifications of , the calculation of , n=0,1,…,M, via an appropriate approximation can be a substantial computation task.

It may be preferable to employ a design technique that utilizes specified values of directly, without the necessity of determining

' ( )dH

( )dh n

' ( )dH

( )dh n

We wish to derive a linear phase IIR filter with real

nonzero h(n) . The impulse response must be

symmetric

where are real and denotes the integer part

[ /2]

0

1

2 ( 1/ 2)( ) 2 cos( )

1

M

k

k

k nh n A A

M

kA [ / 2]M

It can be rewritten as

where N=M+1 and

Therefore, it may write

where

1/ 2 /

0/2

( )N

j k N j kn N

k

kk N

h n A e e

k N kA A

1

0/2

( ) ( )N

k

kk N

h n h n

/ 2 /( ) j k N j kn N

k kh n A e e

with corresponding transform

where

Hence

which has a linear phase

1

0/2

( ) ( )N

k

kk N

H z H z

/

2 / 1

(1 )( )

1

j k N N

kk j k N

A e zH z

e z

' ( 1)/2 sin / 2( )

sin[( / / 2)]

j T N

k k

TNH A e

k N T

The only nonzero contribution to at is

from , and hence that

Therefore, by specifying the DFT samples of the

desired magnitude response at the

frequencies , and setting

'( )H k

' ( )kH '( )k kH N A

' ( )dH

k

' ( ) /k d kA H N

We produce a filter design from equation for which

The desired and actual magnitude responses are equal

at the N frequencies

''( ) ( )k d kH H

k

In between these frequencies, is interpolated as

the sum of the responses , and its magnitude

does not, equal that of

'( )H ' ( )kH

' ( )dH

FIR advantage:

1. Finite impulse response

2. It is easy to optimalize

3. Linear phase

4. Stable

FIR disadvantage:

1. It is hard to implementation than IIR

IIR advantage :

1. It is easy to design

2. It is easy to implementation

IIR disadvantage:

1. Infinite impulse response

2. It is hard to optimalize than FIR

3. Non-stable