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

56b these are strikingly simple transformations which

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Unformatted text preview: 75 w ith Op = - 2 d B a nd Os = - 11 d B. T he v alue o f n ( order o f t he filter) is found from E q. (7.39): %E lliptic wp' = 0.3753 n= [ n,Wn)=ellipord(Wp,Ws,-Gp,-Gs); [ b,a)=ellip(n,-Gp,-Gs,Wn) log[(10 Ll - 1)/(10°·2 - 1)] = 1.9405 210g(0.7975/0.3753) % P lotting A mplitude a nd P hase R esponse W =O:.OOl:pi;W = W'; H =freqz(b,a,W); w =W/T; m ag=abs(H); p hase=180 / pi*unwrap( a ngle(H»; s ubplot(2,1,1); p lot(w,mag);grid; s ubplot(2,1,2); p lot(w,phase);grid W e r ound u p t he v alue of n t o 2. Also from Eq. (7.41), we find t he 3 -dB c utoff frequency we' as , We = 0.7975 1 = 0 .4322 ( lOLl - 1)4 F rom T able 7.1, we find t he p rewarped filter t ransfer f unction Ha(s) for n = 2 a nd we' = 0.4322 as Ha(s) = 0.1868 1 --......,.-2------ (0.4~22) + V2 (0.4~22) + 1 S2 + 0.6112s + 0.1868 MATLAB returns b=O . 1039 0 .2078 0.1039 a nd a=1 - 0.9045 0 .3201 for Butterworth option. Therefore Hz _ [ I- F inally, we o btain H[z] from Ha(s), u sing t he s implified b ilinear t ransformation i n E q. ( 12.64b): Z2 - 0 .1039(z + 1)2 0.9045z + 0.3201 a result, which agrees with the answer found in Example 12.7. z -1 z+1 0 s=-- T herefore 0.1868 {:; I Z2 0 .1039(z + 1)2 + 0.904z + 0.3201 E xercise E 12.5 D esign a first-order lowpass B utterworth f ilter using t he p rewarping m ethod so t hat t he a nalog a nd d igital gains a re i dentical a t w = 0 a nd a t t he 3 -dB c utoff frequency W e' U se T = n j4we . Answer: H[z] = 0 .8284(z + 1) = 0.2929(z + 1) 2 .8284z - 1.1716 z - 0.4142 - -----------------~-------- 1 750 12 F requency R esponse a nd D igital F ilters 12.6 751 R ecursive F ilter d esign: T he B ilinear T ransformation M ethod 0.982614 'Jt p (8) = 82 (12.65) + 1.09788 + 1.1025 Next, t o o btain t he desired highpass transfer function, we replace 8 w ith wp' / 8 i n t he above p rototype t ransfer f unction [see Eq. (7.55)J. T o o btain t he d esired digital transfer function H[z], we t hen r eplace 8 with ;+~ [the bilinear transformation in Eq. (12.64b)J. T his t wo-step o peration m ay b e c ombined in a single-step transformation as 0.7 wp ' s=---= ( Z-I) z +1 0.3 (12.66) In this case wp ' = 0.24 so t hat we replace s w ith 0.2!~:1) i n t he p rototype t ransfer function in Eq. (12.65) t o o btain t he d esired digital transfer function (a) o w P'(z+l) ( z - 1) 0.6902(z - 1)2 H[zJ = z2 _ 1.4678z + 0.6298 \ 0 15 F ig. 1 2.15 C hebyshev highpass filter design using bilinear transformation w ith p rewarping m ethod. • E xample 1 2.7: H ighpass F ilter D esign Design a I -dB ripple Chebyshev highpass filter w ith t he following specifications (depicted b y t he brick walls in Fig. 12.15a): T he s topband g ain Gs ~ - 6.3 dB (G. ::; 0.484) o ver t he s topband 0 ~ w ::; 10 (ws = 10). T he r ipple f ::; 1 d B (G p ~ 0.891) over a p assband w ~ 15 (w p = 15). T he h ighest frequency t o b e p rocessed is W h = 80 radians/so I n o rder t o select a suitable value o f T , we use Eq. (12.58) t o avoid signal aliasing: =- Let us choose T he c ritical frequencies a re W s = 10 a nd a ccording t o E q. (12.64a), are w.' = t an wp' = t an wp T= ~ 100 = 15. T he p rewarped critical frequencies, 7.6-1 T he value o f n needed t o s atis...
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