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p 7.7 5 27 F requency T ransformations 0.3269 s3 - F requency R esponse a nd A nalog F ilters + 0.7378s 2 + 1.02228 + 0.3269 T he a mplitude response of this prototype filter is depicted in Fig. 7.28b.
S tep 2: S ubstitute s w ith T (s) i n f tp(s)
T he desired highpass filter transfer function H(8) is o btained from f tp(s) by replacing
8 with T(8) = W p/8 = 1 65/s. Therefore H (8) _
- e~5 t+ 0.3269 0.7378 e~5) 2 + 1.0222e~5) + 0.3269
8 83 F ig. 7 .29 Frequency transformation for bandpass filters. 3 + 515.948 2 + 61445.75s + 13742005 T he a mplitude response I H(jw)1 for this filter is i llustrated in Fig. 7.28a. •
• E xample 7 .9
Design a Chebyshev bandpass filter with t he a mplitude response specifications shown
= 4000, G . = 0.1 ( -20dB),
in Fig. 7.30a w ith W p, = 1000, WP2 = 2000,
a nd G p = 0.891 ( -1 dB). Observe t hat for Chebyshev filter, Gp = - 1 dB is equivalent t o
r = 1 dB.
T he solution is executed in two steps: in t he first step, we d etermine t he lowpass
prototype filter transfer function f tp(s). In t he second step, t he desired bandpass filter
transfer function is o btained from f tp(s) by s ubstituting s w ith T (8), t he lowpass t o
b andpass transformation in Eq. (7.57). w., o C omputer E xample C 7.ll
Design t he h ighpass filter for t he specifications in Example 7.8 using functions from
Signal Processing Toolbox in MATLAB. We shall give here MATLAB functions for all
types o f filters.
W s=100;Wp=165;Gp=-2;Gs=-20; %B utterworth w., S tep 1: F ind f tp(s), t he l owpass p rototype f ilter t ransfer f unction.
T his is done in 3 substeps as follows: [ n,Wn]=buttord(Wp,Ws,-Gp,-Gs,'s')
[ num,den]= b utter(n,Wn,'high', ' s')
[ n, W n]=cheb1ord(Wp, W s ,-Gp,-Gs, ' 5')
% I nverse Chebyshev
[n, W n] = che b 2ord(Wp, W s ,-Gp,-Gs,'S')
% E lliptic
[ num,den]=ellip(n,-Gp,-Gs,Wn,'high','s') S tep 1 .1: F ind w . f or t he p rototype f ilter.
T he frequency w . is found [using Eq. (7.56)J, t o b e t he smaller of OdB
- I dB 0.891 To plot amplitude response, we c an use t he l ast three functions in Example C7.5. 7.7-2 (b) 0 0.7 (a) Bandpass Filters F igure 7 .29a s hows a n a mplitude r esponse o f a t ypical b andpass f ilter. T o
d esign s uch a f ilter, we first find J ip(s), t he t ransfer f unction o f a p rototype l owpass
filter, t o m eet t he s pecifications in Fig. 7.29b, w here W s is given b y t he s maller o f 0.5 0.3
W PI wP2 - W S1 2 oc W " (Wp2 - W P ,) W s2 2 - WPI W P2 (7.~ W ' 2 (Wp2 - W P,) N ow, t he d esired t ransfer f unction o f t he b andpass filter t o s atisfy t he specifications i n F ig. 7 .29a is o btained f rom J i p (8) b y r eplacing S w ith T (s), w here T (s) = 2
8 + Wp,WP2 (WP2 - wp,)s 0.1 - 20 dB o 1000 2000 4000 8000 6000 0 2 0 )-> ( 7.57) F ig. 7 .30 Chebyshev Bandpass F ilter Design for Example 7.9. 5 5 28 7 (1000)(2000) - (450)2
450(2000 _ 1000)
= 3.99 F requency R esponse a nd A nalog F ilters a nd 7.7 529 F requency T ransformations
OdB ( 400W - (1000)(2000) = 3 5
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