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

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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"...
View Full Document

This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online