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

794 2db and g 01 20db we already designed a chebyshev

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Unformatted text preview: t he lowpass prototype filter transfer function 7t p (s). In the second step, the desired bandpass filter transfer function is o btained from 7t p (8) by substituting s with T (s), t he lowpass t o b andpass transformation in Eq. (7.57). S tep 1 : F ind 7t p (s), t he l owpass p rototype f ilter t ransfer f unction. T his goal is accomplished in 5 substeps used in the design of the lowpass Butterworth filter (see Example 7.6): S tep 1 .1: F ind w . f or t he p rototype f ilter. F or t he p rototype lowpass filter transfer function 7t p (s) w ith t he a mplitude response shown in Fig. 7.31b, the frequency w . is found [using Eq. (7.56)J t o be t he smaller of (1000)(2000) - (45W = 3 99 450(2000 _ 1000) . d an (4000)2 - (1000)(2000) = 3.5 4000(2000 - 1000) which is 3.5, as depicted in Fig. 7.31b. S tep 1 .2: D etermine n For a prototype lowpass filter in Fig. 7.29b. p = - 2.4 dB, = - 20 dB, Wp = 1, a nd w . = 3.5. Hence, according to Eq. (7.39), t he B utterworth filter order n required t o m eet these specifications is a 2 1 I [ 10 - 1 ] n = 2 log 3.5 og 100.24 _ 1 which is r ounded up to n = 2. 4 00 0 0-> (7.58) a mplitude response of this prototype filter is depicted in Fig. 7.30h. 8 ~20dB 1000 0.9826 7t p (8) = 82 + 1.09778 + 1.1025 T( ) (b) (a) as = 1.955 5 30 7 7.7 F requency R esponse a nd A nalog F ilters I nverse Chebyshev [ n,Wsj=cheb2ord(Wp,Ws,-Gp,-Gs,'s'); [ num,denj=cheby2( n ,-Gs, W s , ' s') Yo E lliptic f ilter [ n,Wnj=ellipord(Wp,Ws,-Gp,-Gs,'s'); [ num,denj=ellip(n,-Gp,-Gs,Wn,'s') 0 3.5 = (10 2 _ 1)1/4 = 1.10958 S tep 1 .4: D etermine t he n ormalized t ransfer f unction 'J-i(s) T he n ormalized second-order lowpass B utterworth t ransfer function from Table 7.1 is 'J-i(s) = 1 s2 + V 2s + 1 T his is t he t ransfer function of a normalized filter (meaning t hat We 8_ (1.10958)2 p ( ) - 82 + V2(1.10958)8 + (1.10958)2 T o p lot a mplitude r esponse, we c an u se t he l ast t hree f unctions f rom E xample C 7.5. 7 .7-3 Bandstop Filters = 1). S tep 1 .5: D etermine t he p rototype f ilter t ransfer f unction 'J-ip(s) T he p rototype filter transfer function 'J-ip(s) is obtained by substituting s w ith s/w e = s /1.10958 in t he normalized transfer function 'J-i(s) found in s tep 1.4 as 'J-i 531 Yo S tep 1 .3: D etermine W e I n this s tep (which is n ot necessary for the Chebyshev design), we d etermine t he 3 dB cutoff frequency W e for the prototype filter. Use of Eq. (7.41) yields We F requency T ransformations 1.231 82 + 1.56928 + 1.2312 F igure 7 .32a s hows a n a mplitude r esponse o f a t ypical b andstop f ilter. T o d esign s uch a f ilter, we first find 'Hp(s), t he t ransfer f unction o f a p rototype l owpass f ilter, t o m eet t he s pecifications i n F ig. 7.32b, w here W s is given b y t he s maller o f (W p2 - WP1)W S1 (7.59) T he a mplitude response of this prototype filter is i llustrated in Fig. 7.31b. wPl W P2 - or w S1 2 ( W p2 - wPl ) W S2 w s2 2 - W p1 W ( 7.60) P2 T he d esired t ransfer f unction o f t he b andstop f...
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