eec4-4a-ss-lecture-14.pdf

Signals & Systems FEEE, HCMUT

Ch-7: p ng tn s ca h thng LTI v thit k b lc tng t

Lecture-14

7.3. B lc Butterworth

7.4. B lc Chebyshev

7.5. Cc php bin i tn s



Trn thc t ngi ta tm c cc php bin i thit

k b lc thng cao, thng di, chn di da vo b lc

thng thp Tp trung kho st thit k b lc thng thp (xem nh b lc mu Prototype Filter)



p ng bin ca b lc thng thp Butterworth bc n:

c

2n

1|H(j)|=

1+

Ti tn s c, p ng bin bng 1/(2)1/2 hoc -3dB cng

sut suy gim : gi l tn s ct, tn s 3dB hoc tn s

cng sut

Yu cu thit k:

Ch r p

Ch r G( p) Gp

Ch r G( s) Gs

Ch r s



Xc nh bc n ca b lc v c theo cc yu cu thit k:

li (dB) ti tn s x: x

c

2n

x 10 G( ) 10log 1+

li (dB) ti tn s p: p

c

2n

p 10 pG( )= 10log 1+ G

li (dB) ti tn s s: s

c

2n

s 10 sG( ) 10log 1+ G

2/10

10 1s sc

nG

2/10

10 1p pc

nG

/102

/10

10 1

10 1

s

s

p p

Gn

G

/10/10log (10 1) /(10 1)

2log( / )

psGG

s p

n

/10 /10 1/ 21/ 2 (10 1)(10 1)p sp s

cG G nn



Xc nh hm truyn H(s) bc n:

Trong thit k, ta dng p ng chun ha ( c=1) nh sau:

2

1| ( ) |

1 njH

Suy ra H(s) khi bit hm truyn ca p ng chun ha:

( )sH/ cs s

( )H s



Xc nh hm truyn bc n ca b lc chun ha:

Xc nh cc poles ca b lc chun ha:

s j

Cc poles ca H(s)H(-s) phi tha: 2 2( )n ns j

(2 1)1 j ke/ 2jj e

2 (2 1)n j k ns e

2

1( ) ( )

1 nj jH H

2

1( ) ( )

1 ( / ) ns s

s jH H



Re

Im

-1

1

j

-j

(2 1)2 ; 1,2,3,...,2j

k nn

ks e k n

Kt lun: n poles ca H(s):

Vy cc poles ca H(s)H(-s) l:

(2 1)2 ; 1,2,3,...,j

k nn

ks e k n

Re

Im

-1

1

j

-j

n2n

H(-s) H(s) H(-s) H(s)



Hm truyn H(s) c dng:

(2 1)2 ; 1,2,3,...,j

k nn

ks e k n

1 2 3

1( )

( )( )( )...( )ns

s s s s s s s sH

V d: xt trng hp n=4

5 /8

1 0.3827 0.9239js e j

7 /8

2 0.9239 0.3827js e j

9 /8

3 0.9239 0.3827js e j

11 /8

4 0.3827 0.9239js e j

Re

Im

-1

j

-j

s1

s2

s3

s4



1( )

( 0.3827 0.9239)( 0.3827 0.9239)( 0.9239 0.3827)( 0.9239 0.3827)s

s j s j s j s jH

2 2

1( )

( 0.7654 1)( 1.8478 1)s

s s s sH

4 3 2

1( )

2.6131 3.4142 2.6131 1s

s s s sH

Lm tng t ta c th tnh c cho trng hp bc n bt k:

1

1 1

1 1( )

( ) ... 1n nn ns

B s s a s a sH

Bn(s): Gi l a thc Butterworth!!!



Coefficients of Butterworth Polynominal Bn(s)=sn+an-1s

n-1++a1s+1

n1a 2a 3a 4a 5a 6a 7a 8a 9a



Butterworth Polynominal in Factorized Form

n ( )nB s



Cc bc thit k b lc thng thp Butterworth:

V d: Thit k b lc thng thp Butterworth tha mn cc yu cu

sau: li di thng (0



Bc 1:

Bc 2:

2 0.2log (10 1) /(10 1)3.701

2log 2n

0.2 1/8

1010.694

(10 1)c

Bc 3:

Bc 4:

chn n=4

2 1/8

2011.26

(10 1)c

chn c=11

810

10 11( ) 10log 1 1.66 2pG dB dB

820

10 11( ) 10log 1 20.8 20sG dB dB

2 2

1( )

( 0.76536686 1)( 1.84775907 1)s

s s s sH

2 2

11 11 11 11

1( )

[ 0.76536686 1][ 1.84775907 1]s s s sH s

2 2

14641( )

( 8.41903546 121)( 20.32534977 121)H s

s s s s


7.4. B lc Chebyshev

p ng bin ca b lc thng thp Chebyshev:

2 2

1| ( ) |

1 ( )cn

H jC

Trong thit k, ta dng p ng chun ha ( c=1):

2 2

1| ( ) |

1 ( )n

jC

H

Vy khi c H(s) H(s) bng cch:

( )sH/ cs s

( )H s


7.4. B lc Chebyshev

Xt p ng bin ca b lc thng thp chun ha Chebyshev :

2 2

1| ( ) |

1 ( )n

jC

H

1( ) cos cosnC n ;| | 1

1( ) cosh coshnC n ;| | 1

Cn( ) l mt a thc tha tnh cht sau:

1 2( ) 2 ( ) ( ); 2n n nC C C n

0( ) 1CC: v 1( )C2

2 ( ) 2 1C

Mt cch tng t ta c th tnh c bng Cn( )!!!


7.4. B lc Chebyshev

n ( )nC

Chebyshev Polyminals


7.4. B lc Chebyshev

p ng bin b lc Chebyshev: 2 2

1| ( ) |

1 ( )n

jC

H

2

1010log (1 )r

-r Gp (Butterworth)

p c

gn r ( li max/ li min) trong di thng:

(dB)

(dB)

Pass-band Pass-band


7.4. B lc Chebyshev

Xc nh v bc(n) ca b lc Chebyshev tha yu cu thit k:

li ti tn s : 2 21010log [1 ( )]pnG C

li ti tn s s: 2 2

1010log [1 ( )]s

pnC

1/ 2/10

1

/10

10 1cosh cosh

10 1

s

s

p

G

rn

1/ 2/10

1

/101

1 10 1cosh

10 1cosh /

sG

r

s p

n

2

1010log (1 )designr r Xc nh :

sG

/1010 1r

/10

1

10 1

cosh[ cosh ( / )]

sG

s pn


7.4. B lc Chebyshev

1

(2 1) (2 1)sin sinh cos cosh

2 2

1,2,3,...,

1 1sinh

k

k ks x j x

n n

k n

xn

Xc nh hm truyn H(s) ca b lc:

Ngi ta tnh c cc poles ca H(s) nh sau:

600

600

600

600

Re

Im

sinh ; cosha x b x

H(s) H(-s)


7.4. B lc Chebyshev

' 1

1 1 0

( )( ) ...

n n

n n

n n

K Ks

C s s a s a s aH

1 2

( )( )( )...( )

n

n

Ks

s s s s s sH

Kn c la chn bo m li DC:

0

2

0

1

an

a n oddK

n even

vic thit k c n gin, ngi ta thnh lp bng Cn(s)

hoc gi tr ca cc poles vi mt s gn r thng gp Tra bng!!!


7.4. B lc Chebyshev

0.5 dB ripple

0.5r dB

1 dB ripple1r dB

Chebyshev Filter Coefficients of the Denominator Polynominal ' 1 2

1 2 1 0...n n n

n n nC s a s a s a s a

n0a 1a 2a 3a 4a 5a 6a


7.4. B lc Chebyshev

2 dB ripple

2r dB

3 dB ripple3r dB

Chebyshev Filter Coefficients of the Denominator Polynominal ' 1 2

1 2 1 0...n n n

n n nC s a s a s a s a

n0a 1a 2a 3a 4a 5a 6a


7.4. B lc Chebyshev

Chebyshev Filter Poles Locations

n 0.5r dB 1r dB 2r dB 3r dB


7.4. B lc Chebyshev

Cc bc thit k b lc thng thp Chebyshev:

V d: Thit k b lc thng thp Chebyshev tha mn cc yu cu

sau: r trong di thng (0 10) 2dB; li di chn ( 20)

Gs -20dB

Bc 1: Xc nh:

Bc 2: Chn :

1/ 2/10

1

/101

1 10 1cosh

10 1cosh /

sG

r

s p

n

/10/10

1

10 110 1

cosh[ cosh ( / )]

sGr

s pn


7.4. B lc Chebyshev

Bc 3: Xc nh H(s):

Bc 4: Xc nh H(s): ( )sH/ ps s

( )H s

'( )

( )

n

n

Ks

C sH

0

2

0

1

an

a n oddK

n even

Nu sao cho r=0.5dB, 1dB, 2dB hoc 3dB tra bng Cn(s); nu khng tha tnh Cn(s):

(2 1) (2 1)

2 2

11 1

'

1 2

sin sinh cos cosh

1,2,3,..., ; sinh

( ) ( )( )...( )

k k

k n n

n

n n

s x j x

k n x

C s s s s s s s


7.4. B lc Chebyshev

Bc 1:

Bc 2:

1/ 22

1

1 0.2

1 10 1cosh 2.473

cosh (2) 10 1n

Bc 3:

Bc 4:

chn n=3

chn =0.764 (r)design=2dB

20.2

1

10 110 1

cosh[3cosh (2)]

0.382 0.764

Tra bng: ' 3 2( ) 0.7378 1.0222 0.3269nC s s s s

0 0.3269nn odd K a

3 2

0.3269( )

0.7378 1.0222 0.3269s

s s sH

3 2

10 10 10

0.3269( )

0.7378 1.0222 0.3269s s sH s

3 2

326.9( )

7.378 102.22 326.9H s

s s s



B lc thng cao (High-pass Filter):

Prototype Filter High-pass Filter

p ( )sH( )s T s

( )H s ( )p

T ss

V d 1: Thit k b lc thng cao Chebyshev tha mn cc yu cu

sau: r trong di thng ( 200) 2dB; li di chn ( 100)

Gs -20dB?



B lc thng di (Band-pass Filter):

Prototype Filter Band-pass

Filter

2 2

1 2 1 2 1 2

1 2 1 2 2 1

min ;p p s s p p

s

s p p s p p

2

1 2

2 1

( )( )

p p

p p

sT s

sp( )sH

( )s T s( )H s



V d 2: Thit k b lc thng di Chebyshev tha mn cc yu cu

sau: r trong di thng (1000 2000) 1dB; li di chn

( 450 hoc 4000) Gs -20dB?

V d 3: Thit k b lc thng di Butterworth tha mn cc yu cu

sau: li trong di thng (1000 2000) -1dB; li di chn

( 450 hoc 4000) Gs -20dB?



B lc chn di (Band-stop Filter):

Prototype Filter Band-stop

Filter

1 2 1 2 2 1

2 2

1 2 1 2 1 2

min ;s p p s p p

s

p p s s p p

2 1

2

1 2

( )( )

p p

p p

sT s

sp ( )sH( )s T s

( )H s



V d 4: Thit k b lc chn di Butterworth tha mn cc yu cu

sau: li trong di chn (100 150) -20dB; li di thng

( 60 hoc 260) -2.2dB?

eec4-4a-ss-lecture-14.pdf

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