CHAPTER 6 PART 1 FILTER DESIGN - chebyshev.pdf (1149k)

SKEE 2073

Signals & Systems

1

Chapter 6:

Types of Filter Response

2

Categories of Filters

3

-3dB

{

f₂ f

A_v(dB)

-3dB

{

f₁ f

A_v(dB)

Low-pass response High-pass response

Low Pass Filters:

• pass all frequencies from dc up to the upper cutoff frequency.

• pass low frequencies: ω < ωc

High Pass Filters:

• pass all frequencies that are above its lower cutoff frequency

4

-3dB

{

f₂ f

A_v(dB)

f₁

-3dB

{

f f₂

f₁ A_v(dB)

Band Pass Response Band Stop Response

Band Pass Filters:

• pass only the frequencies that fall between its values of the lower and upper cutoff frequencies

• pass a range of frequencies:

• ω_c1<ω< ω_c2

Band Stop (Notch) Filters:

• eliminate all signals within the stop band while passing all frequencies outside this band

• pass two ranges of frequencies:

Filter Response Characteristics

5

IDEAL FILTER:

•

Bandpass – Gain 1

•

Bandstop – Gain 0

•

No transitional band between bandpass &

bandstop

PRACTICAL FILTER:

•

Bandpass – any constant value in dB

•

Bandstop – Gain

≠

0 (no absolute bandstop)

•

Transitional band exist between bandpass &

Ideal Response of Filter: LPF

pass low frequencies:

|H(jω)|

ω ω_c

Ideal magnitude plot of LPF showing passband and stopband, separated by cutoff frequency

0

Passband _stopband

Cutoff frequency

1

Passband : frequency ranges for which the signal must be passed through

Stopband : the frequency ranges for which the signal must be attenuated

Ideal Filters: HPF

pass high frequencies:

|H(jω)|

ω ω_c

Ideal magnitude plot of HPF showing passband and stopband, separated by cutoff frequency

0

stopband _passband

Cutoff frequency

1

Ideal Filters: BPF

pass a range of frequencies: ω

<ω< ω

|H(jω)|

ω_c1

0

Stopband 1 _{passband Stopband 2}

ω_c2

cutoff frequency

Ideal magnitude plot of BPF showing passband and stopband, separated by two cutoff frequencies

ω

1

Ideal Filters: BSF

pass two ranges of frequencies: ω < ω_c1 and ω > ω_c2

|H(jω)|

ω_c1

0

stopband Passband 1

ω_c2

cutoff frequency

Ideal magnitude plot of BSF showing passband and stopband, separated by two cutoff frequencies

ω

Passband 2

1

Filter Design Process

0 1

1 1

1 ...

) (

a s

a s

a s

K s

H _n

n n

n N

Filter specifications

Find Transfer

Function H

(s)

Circuit realization

• Type of response

• Cut-off frequency, _c

• Passband frequency, _p

• Passband gain/attenuation, _p

• Stopband frequency, _s

• Stopband gain/attenuation, _s

polynomial

L₂

R_S = 1 L4

C₁ C₃

v_i vo RL

Filter Response

Butterworth Transfer Functions

n c

j

H

1

1

|

)

(

|

1

...

1

)

(

1

)

(

1

s

a

s

a

s

s

B

s

H

,

Chebyshev Transfer Functions

c n

T

j

H

1

1

|

)

(

|

...

...

)

(

)

(

a

s

a

s

a

s

a

a

s

a

s

a

s

s

T

K

s

H

n ; for n even

Filter Response: Butterworth Response

p

s

s p

c 1

0.707

n

c

j

H

2

1

1

|

Filter Response: Chebyshev Response

c n

T j

H

2 2

1

1 |

) ( |

A_max

0.707

p

Passband Transition Stopband band

) 1

log(

Butterworth vs Chebyshev

p

s

sC pC

1

0.707

pB sB

Chebyshev

Butterworth

|H(j )|

Passband ripple _flat

Roll off fast slow

Chebyshev vs Butterworth

No. Chebyshev Butterworth

CHEBYSHEV FILTER DESIGN

&

TRANSFER FUNCTION

18

Calculate the order number, n (circular number)

Determine the normalized Transfer Function,

H_n(s) – refer a_n (chebyshev coefficient) from

the table

Determine the real/actual Transfer Function: obtained from step 2 and using the

transformation table.

WHAT IS

CHEBYSHEV

FILTER

CIRCUIT

Based on normalized CLPF

circuit.

Combination of inductor & capacitor produce normalized

circuit

Rs and RL-as input resistor & load resistor

The total of C and L depend on its filter

Transitional band

R

:

input resistor

, fixed =1

R

:

load resistor

[known as Rp in the table]

α

:

gain

ω

:

ripple width

or bandpass angular frequency

ε

:

ripple factor

ω

:

Bandstop angular frequency

A :

ripple amplitude

List of Formula - chebyshev

Filter Gain

Ripple Amplitude

Normalized angular

frequency, ω

2

1

1

1

1

1

A

Value of ripple amplitude

Value of cutoff

frequency 3dB

Transfer function:

Normalized Transfer function

coefficient

List of Formula - chebyshev

Number of filter order

2 1

10 10

1 1

1 10

1 10

cosh ) ( cosh

1

max

A sn

s

n

1

cosh

1

cosh

n

sn : normalized bandstop angular frequency

(depending on types of filter from table )

0.707

A_max

p

EXAMPLE: Chebyshev Filter Design

Design a Chebyshev lowpass filter (LPF) with:

•

A

= 1 dB for 0<ω<10

•

A stopband not to exceed 0.0316 (-30 dB gain)

for

ω≥

20

STEP 1

:

Determine number of filter order,

n

26 2 / 1 10 10 1

1 cosh 10 1 10 1

cosh

1 A_max

sn

s

n

The number of filter order number, n

Use dB values in the calculation, no need to convert into linear values

0.2756 s 7426 . 0 s 4539 . 1 s 9528 . 0 s 0.24 (s)

H_{n 4 3 2}

STEP 2

:

Determine the

normalized

transfer function

H

(

s

) …[see Table F.4]

0 1 1 1 1 1 ... ) ( ) ( 2 0 a s a s a s s T K s H _n n n a n n n

for n even= 4

51 . 0 1 ) 10 ( 1 ) 10 ( 10 1 10 max A

STEP 3

:

(Transfer function Transformation)

Determine the

final

filter transfer function

H

(

s

) …[see Table F.5]

Refer to Table Frequency Transformation.

Depending on type of filter: in this case LPF Replace s with s/ _p = s/10,

0.2756 10 s 7426 . 0 10 s 4539 . 1 10 s 9528 . 0 10 s 0.24

H(s) _{4 3 2}

0.2756 s 7426 . 0 s 4539 . 1 s 9528 . 0 s 0.24 (s)

STEP 4

Determine the normalized circuit, see Table F.2(b) [ for A_max=1dB]

n Rp C1 L2 C3 L4 C5 L6 C7 L8 C9

1 1.00 1.0177

2 0.25 3.7779 0.3001

3 1.00 2.0236 0.9941 2.0236

4 0.25 4.5699 0.5428 5.3680 0.3406

5 1.00 2.1349 1.0911 3.0009 1.0911 2.1349

6 0.25 4.7366 0.5716 6.0240 0.5764 5.5353 0.3486

7 1.00 2.1666 1.1115 3.0936 1.1735 3.0936 1.1115 2.1666

8 0.25 4.7966 0.5802 6.1592 0.6005 6.1501 0.5836 5.5869 0.3515

9 1.00 2.1797 1.1192 3.1214 1.1897 3.1746 1.1897 3.1214 1.1192 2.1797

L₂

R_S = 1 L4

C₁ C₃

Final Element Value: refer Table F.3

STEP 5

Determine the final circuit (Circuit Transformation)

Draw the final circuit include the actual values in the circuit

L₂

R_S = 1 L4

C₁ C₃

v_i vo RL =100

new new

new new

The 3dB cutoff frequency can be written as:

1 cosh 1 cosh 1 n p c

53

.

10

51

.

0

1

cosh

4

1

cosh

10

1

cosh

1

cosh

33

Normalized Chebyshev filter circuit

R_L = 100 6.37F

4.57 F

0.341H 0.543H

Step

5:

EXAMPLE: Chebyshev Filter Design

Design a Chebyshev highpass filter with:

• A_max = 1 dB for ω≥20

• a stopband not to exceed (-22 dB gain) for 0<ω<15

• Sketch the response and find H(s), wc and draw the circuit if RL=1

ohm

Step 1: Determine number of filter order, n

2 / 1 10

10 1

1

cosh

10

1

10

1

cosh

1

sn

s

n

n = 4.901 5

0.1228

s

58

.

0

s

97

.

0

s

69

.

1

s

93

.

0

s

0.1228

(s)

H

Step 2: Determine the normalized transfer function H_n(s) …[see Table F.4]

for n odd

a₀ = 0.1228, a₁ = 0.58, a₂ = 0.97, a₃ = 1.69, a₄ = 0.93

0 1 1 1 1 0

...

)

(

)

(

a

s

a

s

a

s

a

s

T

K

s

H

Step 3: Determine the final filter transfer function for H_HPF(s) …[see Table F.5]

(Transfer function Transformation)

-replace s _p/s = 20/s LPF HPF

0.1228 s 58 . 0 s 97 . 0 s 69 . 1 s 93 . 0 s 0.1228 (s)

H_{n 5 4 3 2}

0.1228 s 20 58 . 0 s 20 97 . 0 s 20 69 . 1 s 20 93 . 0 s 20 0.1228 (s)

H_{HPF 5 4 3 2}

Step 4: Determine the normalized circuit, see Table F.2(b) [A_max=1dB]

n Rp C1 L2 C3 L4 C5 L6 C7 L8 C9

1 1.00 1.0177

2 0.25 3.7779 0.3001

3 1.00 2.0236 0.9941 2.0236

4 0.25 4.5699 0.5428 5.3680 0.3406

5 1.00 2.1349 1.0911 3.0009 1.0911 2.1349

6 0.25 4.7366 0.5716 6.0240 0.5764 5.5353 0.3486

7 1.00 2.1666 1.1115 3.0936 1.1735 3.0936 1.1115 2.1666

8 0.25 4.7966 0.5802 6.1592 0.6005 6.1501 0.5836 5.5869 0.3515

9 1.00 2.1797 1.1192 3.1214 1.1897 3.1746 1.1897 3.1214 1.1192 2.1797

L₂

R_S = 1 L4

C₁ C₃

Step 5: Determine the final circuit (Circuit Transformation-Table F.3)

where R=R_L/R_P

p cC

R L

p cL

R

C 1

LPF HPF

1349

.

2

1

p

C

1 1

p

c

C

Summary: Chebyshev Filter Circuit Design

1. Calculate the order number, n

2. Determine the value of normalized components: from Table

3. Apply the normalized values into this circuit

4. Determine the real components values: refer to the circuit

transformation table, sketch the real circuit with respective

EXAMPLE CEHBYSHEV

40

A Chebyshev low pass filter has the following characteristics:

• The minimum ripple magnitude of 1dB in passband from 0

until 150 Hz

• Bandstop gain weakening/attenuation of at least 40 dB for f

550 Hz

i. Determine the ,

n

ii. Sketch the normalized Chebyshev filter circuit

iii. Sketch the real Chebyshev filter circuit

iv. Sketch the magnitude response of this filter

v. Determine the normalized Transfer function H_n(s)