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

For t he b utterworth a nd chebyshev filters there

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Unformatted text preview: 000(2000 - 1000) . which is 3 .5. S tep 1 .2: D etermine n now n eed t o design a prototype lowpass filter in Fig. 7.29b with p = - 1 dB, G s = - 20 d B, W p = 1, a nd W s = 3.5, as illustrated in Fig. 7.30b. T he Chebyshev filter order n required t o m eet these specifications is o btained from Eq. (7.49b) (or Eq. (7.49a) because, in this case, W p = 1), as a V:!e 2 n 10 - 1 = cosh 11(3.5) cosh _ 1 [- --100.1 1 a result, which is rounded up to n ! - 2.4 dB 0.7586 0.7 0.5 = 1.904 = 2. S tep 1 .3: D etermine t he p rototype f ilter t ransfer f unction 7t p (8) We c an o btain t he transfer function of t he second-order Chebyshev filter by computing its poles for n = 2 a nd f = 1 (€ = 0.5088) using Eq. (7.51). However, since Table 7.4 lists t he d enominator polynomial for f = 1 a nd n = 2, we need not perform t he c omputations a nd may use the ready-made transfer function directly as Here we u sed Eq. (7.53) t o find t he n umerator K n = ~ = ~ = 0.9826. y l+t:2 , ,1.2589 = 8 2 + 2(10)6 1000s Replacing s w ith T (8) in the right-hand side of Eq. (7.58) yields the final bandpass transfer function 2 H (8) _ 9.826(10)5 8 - 8 4 + 1097.7s3 + 5.1025(1O)6s 2 + 2.195(10)98 + 4(10)12 T he a mplitude response IH(jw)1 of this filter is shown in Fig. 7.30a. F ig. 7 .31 T he S tep 2 : F ind t he d esired b andpass f ilter t ransfer f unction H (s) u sing t he l owpass t o b andpass t ransformation. Finally, t he desired bandpass filter transfer function H(s) is o btained from 7t p (s) by replacing s w ith T (s), where [see Eq. (7.57)J • W e m ay u se a s imilar p rocedure f or t he B utterworth filter. C ompared t o C hebyshev d esign, B utterworth f ilter d esign i nvolves t wo a dditional s teps. F irst, we n eed t o c ompute t he c utoff f requency We o f t he p rototype filter. F or a C hebyshev f ilter, t he c ritical f requency h appens t o t he f requency w here t he g ain is G p • T his f requency is W = 1 i n t he p rototype f ilter. For B utterworth, o n t he o ther h and, t he c ritical f requency is t he h alf p ower ( or 3 d B-cutoff) f requency W e, w hich is n ot n ecessarily t he f requency w here t he g ain is G p' T o find t he t ransfer f unction o f t he B utterworth p rototype f ilter, i t is e ssential t o k now W e' O nce we know W e, t he p rototype f ilter t ransfer f unction is o btained b y r eplacing 8 w ith 8/W e in t he n ormalized t ransfer f unction 7t (8). T his s tep is also u nnecessary i n t he C hebyshev f ilter design. W e s hall d emonstrate t he p rocedure for t he B utterworth f ilter d esign b y a n e xample b elow. o 8000 2 3 ,., Butterworth Bandpass Filter Design for Example 7.10. • E xample 7 .10 Design a Butterworth bandpass filter with t he a mplitude response specifications illustrated in Fig. 7.31a with W P1 = 1000, W P2 = 2000, W S1 = 450, WS 2 = 4000, G p = 0.7586 ( -2.4dB), a nd G . = 0.1 ( -20dB). As in t he previous example, t he s olution is e xecuted in two steps: in the first step, we d etermine...
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