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

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

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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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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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