Signal Processing and Linear Systems-B.P.Lathi copy

# For b utterworth o n t he o ther h and t he c ritical

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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 ight-hand 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 second-order 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.

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