Unformatted text preview: ilter t o s atisfy t he s pecifications
i n F ig. 7 .32a i s o btained f rom 'Hp(s) b y r eplacing s w ith T (s), w here 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(8) is obtained from 'J-i p(8) by
replacing 8 with T(8), where [see Eq. (7.57)] ( 7.61) 2
T ( ) = 8 + 2(10)6
10008 t IJlp Uco) I R eplacing 8 with T(8) in t he r ight-hand side of Eq. (7.59) yields t he final bandpass
1.2312(10)6 82 H8_
( )- I
G pt----''t.. S4 + 1569s 3 + 5.2312(10)6 82 + 3.1384(1O)9 s + 4(10)12 ( b) (a) T he a mplitude response IH(jw)1 of this filter is shown in Fig. 7.31a. G, • o o C omputer E xample C 7.12
Design a bandpass filter for t he specifications in Example 7.10 using functions from
Signal P rocessing Toolbox in MATLAB. We shall give here MATLAB functions for t he
four types of filters.
For b andpass filters, we use t he s ame functions as those used for lowpass filter in
Examples C 7.6, C7.8-C7.10, w ith one difference: W p a nd Ws are 2 element vectors as
W p=[Wp1 W p2], Ws=[Ws1 Ws2].
W p=[lOOO 2 000];Ws=[450 4 000];Gp=-2.4;Gs=-20;
Yo B utterworth
Yo C hebyshev
[ num,den]=cheby1 ( N , -Gp, W n, ' s') COp1 COS1 F ig. 1 .32 COS2 cop2 co ~ 0 COs Frequency transformation for b andstop filters. • E xample 1 .11
Design a B utterworth b andstop filter with the specifications depicted in Fig. 7.33a
with W P1 = 60, W P2 = 260, W S1 = 100, w ' 2 = 150, G p = 0.776 ( -2.2 d B), a nd G s =
0.1 ( -20dB).
I n t he first s tep we shall determine t he p rototype lowpass filter transfer function
'J-ip(s), a nd in t he second step we use t he lowpass t o b andstop transformation in Eq. (7.61)
t o o btain t he desired bandstop filter transfer function H (s).
S tep 1: F ind 'J-ip(s), t he ! owpass p rototype f ilter t ransfer f unction.
T his goal is accomplished in 5 substeps used in t he design of the lowpass B utterworth
filter (see Example 7.6): 532 7 F requency R esponse a nd A nalog F ilters 7 .8 f
0.9 F ilters t o S atisfy D istortionless T ransmission C onditions 533 S tep 1 .5: D etermine t he p rototype f ilter t ransfer f unction 1 i p (8)
T he p rototype filter transfer function 1 i p (8) is o btained by s ubstituting 8 w ith s/w e
8/1.1096 in t he n ormalized transfer function 1 i(s) in s tep 1.4. This move yields I H(jro) I 0.776 ~ 0.776 1 1i s _ 0.7 p( ) - (.L)2
We + v '2.L + 1
We 1.2312 S2 + 1.56928 + 1.2312 (7.63) T he a mplitude response of this p rototype filter is depicted in Fig. 7.33b.
S tep 2 : F ind t he d esired b andstop f ilter t ransfer f unction H (s) u sing t he
l owpass t o b andstop t ransformation
Finally, t he desired transfer function H (8) o f t he b andpass filter w ith specifications
illustrated in Fig. 7.33a is o btained from 1 i p (8) b y r eplacing 8 w ith T (8), where [see
Eq. (7.61)J 0.5 0.3 0.1 o T( ) _
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