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

# 740 yields alternately s ubstitution o f n 4 in eq 741

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Unformatted text preview: he passband). For example, f ::; 2 dB specifies t hat t he gain variations of more t han 2 dB cannot be tolerated in the passband. I n t he B utterworth filter Gp == - 2 dB means t he same thing. 6. I f we reduce the ripple, the passband behavior improves, b ut i t does so a t t he cost of s topband behavior. As r is decreased (E is reduced), t he gain in the stopband increases, and vice-versa. Hence, there is a tradeoff between the allowable passband ripple and the desired attenuation in the stopband. Note 7 Frequency Response a nd Analog F ilters 516 7.6 Chebyshev Filters 517 t hat t he e xtreme case f = 0 yields zero ripple, b ut t he filter now becomes a n a llpass filter, as seen from Eq. 7.42, by l etting f = O. 7. Finally, t he C hebyshev filter has a s harper cutoff (smaller t ransition b and) t han t he s ame-order B utterworth f ilter,t b ut t his is achieved a t t he e xpense of inferior p assband b ehavior (rippling). Determination o f n (Filter Order) For a n ormalized C hebyshev filter, t he g ain T he g ain is Gs G in dB [see E q. (7.42)] is a t w s ' T herefore , G s = - lOlog [+f C n2 ws) l2 ( 1 (7.48) or f 2C n 2(ws) = 1 0- 68 / 10 - 1 F ig. 7 .25 Poles of a normalized third-order lowpass Chebyshev filter transfer function and its conjugate. Use o f Eq. (7.43b) a nd Eq.(7.47) in t he above equation yields ' 1 O-G,/10 - -1 cosh [ ncosh H ence n= (ws)l = 1 cosh - 1(w s ) cosh [ -1 1 ] 1 /2 lOf/lO _ 1 [10- 68 / 10 _ 1] 1 /2 - -c::-,-;;;-;:-- IO r / lO - 1 (7.49a) Note t hat t hese e quations are for normalized filters, where wp = 1. F or a general case, we r eplace w . w ith ; :- to o btain n= I 1 cosh - (ws/wp) cosh -1 [1O- G ./ 10 _ 1]1/2 -~,~-- lOr/IO - 1 1. h- (1) I (7.50) -; T he C hebyshev filter poles are s k = - sin [ (2k - 1)11"] 2n s inh x +j cos [(2k - 1)11"] 2n cosh x 1 t(s) (7.49b) Pole Locations We c ould follow t he p rocedure of t he B utterworth filter t o o btain t he pole locations of t he C hebyshev filter. T he p rocedure is straightforward b ut t edious a nd d oes n ot y ield a ny s pecial insight into o ur d evelopment. T he B utterworth filter poles lie on a semicircle. We c an show t hat t he poles of an n th-order n ormalized Chebyshev filter lie on a semiellipse of t he m ajor a nd m inor semi axes cosh x a nd s inh x , r espectively, where 1 x =;:;-sm T he g eometrical construction for determining t he pole location is depicted in Fig. 7.25 for n = 3. A similar procedure applies t o a ny nj i t c onsists of drawing two semicircles of radii a = s inh x a nd b = cosh x . We now draw radial lines along t he c orresponding B utterworth angles a nd l ocate t he n th-order B utterworth poles (shown by crosses) o n t he two circles. T he l ocation of t he k th C hebyshev pole is t he i ntersection of t he h orizontal projection a nd t he v ertical projection from t he c orresponding k th B utterworth poles on t he o uter a nd t he i nner circle, respectively. T he t ransfer f unction 1 t (s) o f t he n ormalized n th-order lowpass Chebyshev filter is k = 1,2"...
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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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