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 viceversa. 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 ameorder 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 thirdorder 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 OG,/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 thorder 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 thorder 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 thorder 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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