Signal Processing and Linear Systems-B.P.Lathi copy

T his s caling is u ndone b y using eq 1264b i nstead

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Unformatted text preview: ansfer f unction H[z] c an b e o btained f rom 'H.p(s) b y r eplacing s w ith Tbp[Z] for t he b andpass filter, a nd r eplacing s w ith Tbs[Z] for t he b andstop filter. 1 2 -1] = = 2 log 3.937 log [ 10 10°·21 1 1.8498 which is rounded up t o n = 2. S tep 2 .3: D etermine t he c utoff f requency we' o f t he p rototype f ilter I n this step (which is not necessary for t he Chebyshev design), we determine t he 3-dB cutoff frequency we' for t he p rototype filter using a nyone of the Eqs. (7.40) or (7.41). 12 754 Frequency Response a nd Digital Filters 1 2.6 R ecursive F ilter d esign: The B ilinear T ransformation Method 755 S ubstitution o f t hese values i n E q. (12.67) yields _ z2 + 2 az + 1 _ 6.317(z2 - 1.8042z + 1) b(z2 _ 1) z2 - 1 n p [z] - T he d esired b andpass filter t ransfer f unction H[z] is o btained from 1 i p (8) b y s ubstituting 8 w ith Tbp[z] : 0.8 0.785 Hz _ 0.02964(Z2 - 1)2 [ ] - z4 - 3.119z 3 + 3.926z 2 - 2.354z + 0.576 0.6 T he a mplitude r esponse IH[eiWT]1 o f t his filter is i llustrated i n Fig. 12.16a. o (a) 0.4 0.2 0.1 450 10' 2000 • 10.000 4000 ro -> 0 3.937 C omputer E xample C 12.7 Design a b andpass d igital filter t o m eet t he specifications in E xample 12.8, 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. I n t his case, Wp a nd Ws a re 2-element vectors: Wp= [1000 2000] a nd Ws= [450 4000]. 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 C I2.5, a ll t he r adian frequencies c an b e 4 n ormalized b y m ultiplying each o f t hem b y T/1r. I n t he p resent case, T = 1r/10 so t hat 4 T/1r = 1 /104. T hus, t o n ormalize a ny r adian frequency, we j ust d ivide i t b y 10 . T he n ormalized Wp a nd Ws a re [ 0.1 0 .2] a nd [ 0.045 0 .4], respectively. W p=[O.l O .2];Ws=[O.045 0 .4);Gp=-2.1;Gs=-20; F ig. 1 2.16 pre-warping. B utterworth b andpass filter design using t he b ilinear transformation w ith % B utterworth [ n,Wn)=buttord(Wp,Ws,-Gp,-Gs); [ b,a)=butter(n,Wn) E ach e quation gives a different answer, in general. However e ither answer will satisfy t he specifications. Let us select Eq. (7.41), which yields , We % Chebyshev , = Ws (10 2 _ 1)1/4 [ n,Wn)=cheblord(Wp,Ws,-Gp,-Gs) [ b,a)=chebyl(n,-Gp,Wn) = 1.248 % I nverse Chebyshev [ n,Wn)=cheb20rd(Wp,Ws,-Gp,-Gs) [ b,a)=cheby2(n,-Gs,Wn) S tep 2 .4: F ind t he n ormalized f ilter t ransfer f unction T he n ormalized second-order B utterworth filter t ransfer f unction (from Table 7.1) is %E lliptic 1 1i(8) = -82 -+-..;2=28-+-1 [ n,Wn)=ellipord(Wp,Ws,-Gp,-Gs) [ b,a)=ellip(n,-Gp,-Gs,Wn) S tep 2 .5: F ind t he p rototype f ilter t ransfer f unction 1 ip (8) T he p rototype filter transfer function 1 i p (8) is o btained b y s ubstituting 8 w ith s/wc' = 8/1.248 in t he n ormalized transfer function 1i(8) f ound in s tep 4. This s ubstitution yields 1.5575 1i 8 _ (1.248)2 82 + 1.76498 + 1.5575 p( ) - 82 + ..;2(1.248)...
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