1 Eeng 224 Ideal Filter Magnitude Responses H()H() cc Lowpass Filter H()H() cc Highpass Filter H()H() H()H() 11 22.

1Eeng 224

Ideal Filter Magnitude Responses

H()

c

Lowpass Filter

H()

c

Highpass Filter

H() H()

1 2 21

Bandpass Filter Bandstop Filter

Magnitude responses of different types of ideal filter functions.

2Eeng 224

Actual Filter Magnitude Responses

3Eeng 224

LOW-PASS FILTER

0

2 2 2

1( ) 1

( )1( ) 1

(0) 1and ( ) 0

At the cutof or Rolloff frequency

1 1 1( )

21

i

c

c c

c

V j CH

V j RCRj C

H H

HRCR C

A low-pass filter is designed to pass only frequencies from DC up to the cutoff frequency c.

4Eeng 224

HIGH-PASS FILTER

A high-pass filter is designed to pass all the frequencies above its cutoff frequency c.

0

2 2 2

( )( )

1( ) 1

(0) 0 and ( ) 1

At the cutof or Rolloff frequency

1 1 1( )

21

i

c

c c

c

V R j RCH

V j RCRj C

H H

HRCR C

5Eeng 224

BANDPASS FILTER

A bandpass filter is designed to pass all the frequencies within a band of frequencies, 1 < < 2

0

1 2

2 1

1 2

( )( ) (0) 0 and ( ) 1

1( )

1The center frequency is given by

The Lower and Upper cutoff frequencies are and

The Bandwidth is B= -

1( ) 1and ( ) ( )

2

i

o o

o

V RH H H

V R j L C

LC

H H H

0

6Eeng 224

BANDSTOP FILTER

A Bandstop filter is designed to stop or eliminate all the frequencies within a band of frequencies 1 < < 2 .

0

1 2

2 1

1 2

1( )

( ) (0) 1and ( ) 11( )

1The center frequency is given by

The Lower and Upper cutoff frequencies are and

The bandwidth is B= -

1( ) 1and ( ) ( )

2

i

o o

o

j LV CH H HV R j L C

LC

H H H

7Eeng 224

Active first-order low-pass filter

0 ( ) 1( ) ,

1( ) 1

f

f f fi i f f

i i f f ff

f

R

Z j C RVH Z R Z R

V Z j C j R CRj C

1 1( )

1f

ci f f f f

RH

R j C R R C

Active filters use also active devices such as OP AMPs.

Passive filters use only passive devices such as inductors and capacitors only.

8Eeng 224

Active first-order High-pass filter

0 ( ) 1( ) ,

( )

1( )

1 1

fi i f f

i i i

f i fc

i i i iii

ZVH Z R Z R

V Z j C

R j C RH

j C R C RR j C

9Eeng 224

Active Bandpass Filter A bandpass filter is obtained by cascading a LPF and a HPF together with an inverting amplifier to provide the desired gain.

Active Bandpass Filter Block Diagram. Frequency Response

10Eeng 224

Active Bandpass Filter Example

2 11

o 2

i 1 2

2

1 2

21 2

2

1

1

1 2

1

2

2

1 1,

(

V 1( )

V 1 1

1

1 1

, , ,

( )1 1

)

f

i

f

i

oo

f

oi

i

f

R j C RH

R j C R j C R

R j C R

R j C R j C R

B QB

jR

Hj jR

RC RC

RH

R

Three cascaded stages are used to realize the bandpass filter.

A LPF cascaded with a HPF and an inverter stage.

11Eeng 224

Active Bandreject Filter A bandreject filter may be constructed by parallel combination of a LPF and a HPF filter and a summing amplifier.

Active Bandreject Filter Block Diagram. Frequency Response

12Eeng 224

Active Bandreject Filter Example

o 2

i 1 2

1

2 1

2

1 1 1

2 1

11 2

1 2

V 1( )

V 1 1

1

1 1

2 ( )1

1 1

0 and ( )

2( )

f

i

f

i

f

i

f

i

fo o

i

R j C RH

R j C R j C R

jR

j jR

j jR

j jR

RH

R

RH

R

13Eeng 224

14Eeng 224

15Eeng 224