1

Transformation of LPF to other filters

c : is the analog frequency

Butterworth Filter:

Filter Order Transfer Function

1

1

1

s

2

12

12 ss

3

122

123 sss

4

1613.24142.36131.2

1234 ssss

Bilinear Transformation:

The analog frequency ( c ) =

2

)(tan

2 frequencydigital

Ts,

1

1

1

12

z

z

Ts

s

2

Properties of Laplace Transform

Property Signal Laplace Transform Time Shifting )( 0ttx )(0 sXe

tS Shift in S-domain )(txe tS o )( 0ssX

Differentiation in the Time

domain )(txdt

d

)(sXs

Differentiation in the s-

domain )(txt

)(sXsd

d

3

4

5

Use the matrix NW to find the DFT using the relation: NNN xWX .

Where

)1)(1()1(21

)1(242

12

....1

........

....1

....1

1....111

NN

N

N

N

N

N

N

NNN

N

NNN

N

www

www

www

W and

Nj

N ew

2

6