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Unformatted text preview: fy these specifications is given by [see E q. (7.49b)J = C omputer E xample C 12.6
D esign a highpass digital filter t o m eet t he specifications in E xample 12.7, using
bilinear t ransformation w ith p rewarping.
As before, we s hall give here MATLAB functions t o design t he four basic types o f
a pproximations. T he i nput d ata asks for frequencies so normalized t hat t he s ampling
r adian frequency is 2. As explained in E xample C12.5, a ll t he r adian frequencies c an b e
n ormalized b y m ultiplying each o f t hem b y T/7r. I n t he p resent case, T = 7r/100 s o t hat
T/7r = 1 /100. T hus, t o n ormalize any r adian frequency, we j ust d ivide i t b y 100. T he
n ormalized Wp a nd Ws a re 1 5/100 a nd 1 0/100, respectively.
W p=0.15;Ws=O.1;-Gp=1;-Gs=6.3;T=pi/100; i. B utterworth
[ b,a)=butter(n,Wn,'high'); i. I nverse Chebyshev I n t he second step, we d esign a prewarped Chebyshev high pass filter w ith c ritical
frequencies w.' = 0.1584 a nd wp' = 0.24 w ith f = - 1 d B a nd G. = - 6.3 dB (Fig.
12.15b). Following t he p rocedure in Sec. 7.6-1, we first design a p rototype lowpass filter
w ith specifications, as indicated in Fig. 12.15b. O bserve t hat t he c ritical frequencies of
t he p rototype filter are 1 (passband) a nd wp' /w.' = 1.515 (stopband) as explained in Sec. n o [ n,Wn)=cheb1ord(Wp,Ws,-Gp,-Gs);
[ b,a) = cheby1 ( n,-Gp, W n , ' high'); = t an(¥o) = 0.24 - 1 [ (10 0.63 _ 1 ] 1/2
c osh- 1 ( 1515) cosh
-'-110·1 - 1 • i. Chebyshev ¥ = t an(fo) = 0.1584 ¥ T he c ontinuous curve in Fig. 12.15a shows t he a mplitude r esponse o f t his filter. = 1.988 We r ound u p t he value o f n t o 2. From Table 7.4 (Chebyshev filter with f = 1 a nd n = 2)
we o btain t he following p rototype t ransfer function [ n,Wn)=cheb2ord(Wp,Ws,-Gp,-Gs);
[ b,a) = cheby2( n ,-Gs,W n , ' high') i. E lliptic
[ b,a) = ellip ( n,-Gp,-Gs, W n , ' high')
MATLAB r eturns b=O. 6 902 - 1.3804 0 .6902 a nd a =l - 1. 4 678 0 .6298 for Chebyshev option. Therefore
0.6902(z - 1)2
H[zJ = z2 _ 1.4678z + 0.6298
which agrees w ith t he a nswer found in Example 12.6. T o p lot t he a mplitude a nd t he p hase
response, we c an use t he l ast 9 f unctions in E xample C I2.5. 0 12 7 52 12.6 Recursive Filter design: T he Bilinear Transformation Method F requency R esponse a nd D igital F ilters 753 • Bandpass and Bandstop Filters
F or b andpass a nd b andstop filters, we follow a s imilar p rocedure. All t he c ritical
f requencies a re first p rewarped u sing t he s implified form in Eq. (12.64a). N ext, we
d etermine a p rototype low p ass filter, which is t hen c onverted t o t he d esired a nalog
f ilter u sing a ppropriate t ransformations d iscussed i n Sec. 7.6. Finally, we use t he
b ilinear t ransformation i n E q. (12.64b) t o o btain t he d esired d igital f ilter. As i n
t he c ase of t he h ighpass filter (discussed in E xample 1 2.7), we c an c ombine t he t wo
t ransformations i nto a s ingle t ransformation. F or t he b an...
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