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Unformatted text preview: t he lowpass prototype filter transfer function 7t p (s). In the second step, the
desired bandpass filter transfer function is o btained from 7t p (8) by substituting s with
T (s), t he lowpass t o b andpass transformation in Eq. (7.57).
S tep 1 : F ind 7t p (s), t he l owpass p rototype f ilter t ransfer f unction.
T his goal is accomplished in 5 substeps used in the design of the lowpass Butterworth
filter (see Example 7.6):
S tep 1 .1: F ind w . f or t he p rototype f ilter.
F or t he p rototype lowpass filter transfer function 7t p (s) w ith t he a mplitude response
shown in Fig. 7.31b, the frequency w . is found [using Eq. (7.56)J t o be t he smaller of
(1000)(2000) - (45W = 3 99
450(2000 _ 1000)
an (4000)2 - (1000)(2000) = 3.5
4000(2000 - 1000) which is 3.5, as depicted in Fig. 7.31b.
S tep 1 .2: D etermine n
For a prototype lowpass filter in Fig. 7.29b. p = - 2.4 dB,
= - 20 dB, Wp = 1,
a nd w . = 3.5. Hence, according to Eq. (7.39), t he B utterworth filter order n required
t o m eet these specifications is a 2
I [ 10 - 1 ]
n = 2 log 3.5 og 100.24 _ 1
which is r ounded up to n = 2. 4
00 0 0-> (7.58) a mplitude response of this prototype filter is depicted in Fig. 7.30h. 8 ~20dB 1000 0.9826
7t p (8) = 82 + 1.09778 + 1.1025 T( ) (b) (a) as = 1.955 5 30 7 7.7 F requency R esponse a nd A nalog F ilters I nverse Chebyshev
[ num,denj=cheby2( n ,-Gs, W s , ' s')
Yo E lliptic f ilter
[ num,denj=ellip(n,-Gp,-Gs,Wn,'s') 0 3.5
= (10 2 _ 1)1/4 = 1.10958 S tep 1 .4: D etermine t he n ormalized t ransfer f unction 'J-i(s)
T he n ormalized second-order lowpass B utterworth t ransfer function from Table 7.1
s2 + V 2s + 1
T his is t he t ransfer function of a normalized filter (meaning t hat We 8_
p ( ) - 82 + V2(1.10958)8 + (1.10958)2 T o p lot a mplitude r esponse, we c an u se t he l ast t hree f unctions f rom E xample
C 7.5. 7 .7-3 Bandstop Filters = 1). S tep 1 .5: D etermine t he p rototype f ilter t ransfer f unction 'J-ip(s)
T he p rototype filter transfer function 'J-ip(s) is obtained by substituting s w ith s/w e =
s /1.10958 in t he normalized transfer function 'J-i(s) found in s tep 1.4 as
'J-i 531 Yo S tep 1 .3: D etermine W e
I n this s tep (which is n ot necessary for the Chebyshev design), we d etermine t he 3
dB cutoff frequency W e for the prototype filter. Use of Eq. (7.41) yields We F requency T ransformations 1.231
82 + 1.56928 + 1.2312 F igure 7 .32a s hows a n a mplitude r esponse o f a t ypical b andstop f ilter. T o
d esign s uch a f ilter, we first find 'Hp(s), t he t ransfer f unction o f a p rototype l owpass
f ilter, t o m eet t he s pecifications i n F ig. 7.32b, w here W s is given b y t he s maller o f
(W p2 - WP1)W S1 (7.59) T he a mplitude response of this prototype filter is i llustrated in Fig. 7.31b. wPl W P2 - or w S1 2 ( W p2 - wPl ) W S2 w s2 2 - W p1 W ( 7.60) P2 T he d esired t ransfer f unction o f t he b andstop f...
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