Unformatted text preview: 82 + 1 5,600 0.1 60 1 00 150 200 260 400 Replacing s w ith T (s) in t he r ighthand side of Eq. (7.63) yields t he final bandstop
transfer function 500 1.2312
F ig_ 7 .33 H (8) B utterworth B andstop F ilter Design for Example 7.11. S tep 1 .1: F ind w . f or t he p rototype f ilter.
For t he p rototype lowpass filter transfer function 1 i p ( 8) w ith t he specifications illustrated in F ig. 7.32b, t he frequency w . is found [using Eq. (7.60)J t o b e t he s maller
of
(100)(260  60) = 3.57
(260)(60)  100 2 a nd 150(260  60) = 4.347
150 2  (260)(60) which is 3.57, as shown in Fig. 7.33b. = ( .2+15,600 )2 +1 5692( .2+15,600 ) + 1.
200.
200.
2312
.
84 (8 2 + 15600)2
+ 254.98 3 + 63690.98 2 + (3.977)106 s + (2.433)10 8 T he a mplitude response IH(jw)1 is shown in Fig. 7.33a. • o C omputer E xample C 7.13
Design t he b andstop filter for t he specifications in Example 7.11 using functions from
Signal P rocessing T oolbox in MATLAB. We shall give here MATLAB functions for all t he
four types of filters. S tep 1 .2: D etermine n
F or t he p rototype lowpass filter in Fig. 7.33b, G =  2.2 d B, G. =  20 dB, w p = 1,
p
a nd W s = 3.57. According to Eq. (7.39), t he B utterworth filter order n required to
meet t hese specifications is W p=[60 260J; W s=[lOO ISO); G p=2.2;Gs=20;
Yo B utterworth
[ n,Wn)=buttord(Wp,Ws,Gp,Gs,'s')
[ num,den)=butter(n,Wn,'stop','s')
Yo Chebyshev
[ n,Wn)=cheblord(Wp,Ws,Gp,Gs,'s')
[ num,den)=chebyl(n,Gp,Wn,'stop','s')
Yo I nverse Chebyshev
[ n,Wn)=cheb2ord(Wp,Ws,Gp,Gs,'s')
[ num,denJ=cheby2( n ,Gs, W n , ' stop', ' s')
Yo E lliptic
[ n,Wn)=ellipord(Wp,Ws,Gp,Gs,'s')
[ num,den)=ellip(n,Gp,Gs,Wn, ' stop', ' s') We round up t he value of n t o 2.
S tep 1 .3: D etermine W e
T he h alf power frequency W e for t he p rototype B utterworth filter, using Eq. (7.40)
w ith W p = 1, is 0 1
= 1.1096
(10 022  1):
S tep 1 .4: D etermine t he n ormalized t ransfer f unction
T he t ransfer function of t he secondorder normalized B utterworth filter from t he
T able 7.1 is
1 i(8) 1
(7.62)
 S2 + v'28 + 1 7 .8 Filters t o Satisfy Distortionless transmission Conditions T he p urpose o f a f ilter is t o s uppress u nwanted f requency c omponents a nd
t o t ransmit t he d esired f requency c omponents w ithout d istortion. I n S ec. 4.4, we 534 7 Frequency Response and Analog Filters saw t hat t his requires t he filter amplitude response to be constant a nd t he p hase
response t o b e a linear function of w over t he passband.
T he filters discussed so far have stressed the constancy of t he a mplitude response. T he l inearity of t he p hase response has been ignored. As we saw earlier,
t he h uman e ar is sensitive to amplitude distortion b ut s omewhat insensitive t o
p hase distortion. For this reason filters in audio application are designed primarily for c onstant a mplitude response, and t he phase response is only a secondary
consideration.
We also saw earlier t hat t he h uman eye is sensitive t o phase distortion a nd
relatively in...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.
 Spring '13
 Bayliss
 Signal Processing, The Land

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