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Unformatted text preview: 29 7!- tan ( ¥s) = 8.3623 log[(10Ll - 1)/(1002 - I)J
2Iog(17.7696/8.3623) W e' H[zJ S2 In t he second step, we design a Butterworth filter with critical frequencies w/ = 8.3623
a nd ws' = 17.7696 with Op = - 2 dB and O. = - 11 dB. The value of n (order of t he filter)
is obtained from Eq. (7.39):
n= F ig. 1 2.14 A mplitude response of t he filter in Example 12.6. cies we' = Wi are transformed (prewarped) by the equation
Wi = T tan - 2- ( 12.63a) T he p rewarped c utoff frequency we', d etermined b y u sing p rewarped c ritical frequencies, is u sed t o f ind t he p rewarped a nalog f ilter t ransfer f unction H a(s). F inally, we r eplace s w ith 1. z +1 i n H a(s) t o o btain t he d esired d igital f ilter t ransfer
f unction H[z]
( 12.63b) - ----..,
12 Frequency R esponse a nd D igital F ilters 748 12.6 Recursive Filter design: T he Bilinear Transformation Method 749 w hich is identical t o t he r esult o btained e arlier. o A Simplified Procedure T he a bove p rocedure c an b e s implified by observing t hat t he s caling f actor ~
is irrelevant i n t his m anipulation a nd c an b e i gnored. I nstead o f u sing Eqs. (12.63a)
a nd ( 12.63b), we c an u se t he s implified e quations w/ = w iT
t an - - a nd (12.64a) i = 1 ,2,···, m 2 z -1
z+1 (12.64b) s=-- C omputer E xample C 12.5
Design a lowpass digital filter to meet the specifications in Example 12.6, using bilinear transformation with prewarping.
We shall give here MATLAB functions to design the four types of approximations:
Butterworth, Chebyshev, inverse Chebyshev, and elliptic. The input d ata asks for frequencies so normalized t hat t he sampling radian frequency is 2. This requirement means
the sampling radian frequency, which is 2 n/T, m ust be normalized to 2. Therefore, all t he
radian frequencies can be normalized by multiplying each of them by T /n. I n t he present
case, T = n /35 so t hat T /n = 1 /35. Thus, to normalize any radian frequency, we j ust
divide it by 35. T he normalized Wp and Ws a re 8 /35 a nd 1 5/35, respectively.
W p=8/35;Ws=15/35;Gp=-2;Gs=-11;T=pi/35; T his s implification works b ecause t he f actor 2 /T i n E q. (12.63a) is a f requency
s caling f actor, a nd i gnoring i t i n E q. (12.64a) r esults i n t he p retransformed f ilter
t hat i s s caled by a factor 2 /T i n t he frequency scale. T his s caling is u ndone b y
using Eq. (12.64b) i nstead o f E q. ( 12.63b).
T o d emonstrate t he p rocedure, we shall redo E xample 12.6 using t his simplification. I n t he first s tep, we p rewarp t he c ritical frequencies W p a nd W s a ccording t o
E q. ( 12.64a): wp' = t an ws' = t an % B uttervorth
[ b,a)=butter(n,Wn) %Chebyshev
[ b,a)=chebyl(n,-Gp,Wn) ¥ = t an ( ¥s) = 0 .3753 % I nverse C hebyshev ¥ = t ane;;) = 0.7975 [ n,Wn)=cheb2ord(Wp,Ws,-Gp,-Gs);
[ b,a)=cheby2(n,-Gs,Wn) I n t he s econd s tep, we design a B utterworth f ilter w ith c ritical frequencies
a nd ws' = 0.79...
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