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Unformatted text preview: s. Moreover, t he phase function (or time-delay) characteristic
of t he inverse Chebyshev filter is b etter t han t hat of the Chebyshev filter.2 Both
the Chebyshev a nd inverse Chebyshev filter require the same order n t o meet a Recall our discussion in Sec. 7.4 t hat placing a zero on t he i maginary axis ( at
8 = jw) causes t he gain (/H(jw)!) t o go t o zero (infinite attenuation). We can realize
a sharper cutoff characteristic by placing a zero (or zeros) near w = W S' B utterworth
a nd Chebyshev filters do not make use of zeros in 11.(8). B ut a n elliptic filter does.
This is t he reason for t he s uperiority of t he elliptic filter.
A Chebyshev filter has a smaller transition band compared to t hat of a B utterworth filter because a Chebyshev filter allows rippling in t he p assband (or stopband).
I f we allow ripple in b oth t he p assband a nd t he s topband, we c an achieve further
reduction in t he t ransition band. Such is t he case with elliptic (or Cauer) filters,
whose normalized amplitude response is given by
1 11t(jw)/ = - -r====
+ f2Rn,2(w) VI where R n (w) is t he n th-order Chebyshev rational function determined from the
specific ripple characteristic. T he p arameter f controls t he ripple. T he gain a t w p
(wp = 1 for t he normalized case) is Jfp-.
T he elliptic filter is t he most efficient if we c an tolerate ripples in b oth t he
p assband a nd t he s topband. For a given transition band, it provides the largest
ratio of t he p assband gain to stopband gain, or for a given ratio of passband to
stopband gain, it requires t he smallest transition band. In compensation, however,
we m ust accept ripples in b oth t he p assband a nd t he s topband. In addition, because
of zeros in t he n umerator of 11. (8), the elliptic filter response decays a t a slower r ate
a t frequencies higher t han W s' For instance, t he a mplitude response of a thirdorder elliptic filter decays a t a r ate of only - 6 d B/octave a t very high frequencies.
This is because t he filter has two zero a nd t hree poles. T he two zeros increase
the amplitude response a t a r ate of 12 d B/octave, a nd t he t hree poles reduce the
amplitude response a t a r ate of - 18 d B/octave, t hus giving a net decay r ate o f-6
d B/octave. For t he B utterworth a nd Chebyshev filters, there are no zeros in 11.(8). 524 7 Frequency Response a nd Analog Filters 525 7.7 Frequency Transformations Therefore, their amplitude response decays a t a r ate of - 18 d B/octave. However,
t he r ate o f d ecay o f t he a mplitude response is s eldom i mportant as long as we m eet
o ur s pecification of a given G s a t W S'
C alculation of pole-zero locations of elliptic filters is much more c omplicated
t han t hat in B utterworth o r even Chebyshev filters. Fortunately, t his t ask is g reatly
simplified by c omputer p rograms a nd extensive ready-made design Tables available
in t he l iterature. 3 T he MATLAB function [z,p,k]=ellipap(n,-Gp,-Gs) in Signal P rocessing T oolbox d etermines poles, zeros, a nd...
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