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

1214 summary o f t he bilinear transformation method

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Unformatted text preview: dpass filter, we first use t he t ransformation E xample 1 2.8 Design a digital Butterworth bandpass filter with amplitude response specifications illustrated by t he brick walls in Fig. 12.16a with wp, = 1000, wP2 = 2000, ws, = 450, w ' 2 = 4000, G p = 0.7852 ( -2.1 dB), a nd G . = 0.1 ( -20 dB). Take T = 71"/10,000. T he solution is executed in 3 steps: In t he first step, we d etermine t he prewarped critical frequencies. In t he second step, t he lowpass prototype filter transfer function 'Hp(s) is found from t he p rewarped critical frequencies. Finally, t he desired H[z] is found from 'Hp(s) using t he lowpass analog t o b andpass digital transformation by replacing s in 'Hp(s) with np[z]. S tep 1 : F ind p rewarped c ritical f requencies T he prewarped frequencies wp,', WP2', w s,', a nd ws 2 ' corresponding to t he four critical frequencies W Pl> Wp2 , W'l> a nd W' 2 using Eq. (12.64a): a nd t hen u se t he b ilinear t ransformation in E q. ( 12.64b). T hus, i n t he f irst s tep w e r eplace s in t he p rototype t ransfer f unction 'H.p(s) w ith T (s) ( the f requency t ransformation). I n t he s econd s tep we replace s w ith ~~~ ( the s implified bilinear t ransformation). T hus, t he final t ransformation is e quivalent t o r eplacing s w ith Tbp[Z] i n t he p rototype f ilter t ransfer f unction 'H.p(s), w here 45071" ) = t an ( 20,000 wp , , = t an ( 100071") 20,000 = 0.1584 , = t an ( 200071") 20,000 = 0.3249 , = t an ( 400071") 20,000 = 0.7265 WP2 + wp,'wp,' = T(s)ls=~ = ( ' - W P1 ') S Is=~ .1:+1 W z +1 S2 Tbp[Z] P2 w'2 (z - 1)2 + wp,'wp,'(Z + 1)2 (w p ,' - wp , ' )(z2 - 1) _ (wp,'w p,' + l )z2 - + 2(w p,'w p,' - (w p ,' - l)z + (wp,'w p,' + 1) wp ,')(z2 - 1) U sing t he s ame a rgument, we c an s how t hat for t he b andstop f ilter, t he d esired d igital filter t ransfer f unction H[z] c an b e o btained f rom t he c orresponding b andstop p rototype f ilter 'H.p(s) b y r eplacing s w ith Tbs[ZJ, w hich is t he r eciprocal o f Tbp[Z], B oth t hese t ransformations c an b e e xpressed i n a m ore c ompact f orm a s S tep 2 : F ind 'Hp(s), t he p rewarped l owpass p rototype a nalog f ilter T his procedure with 5 substeps is identical t o s tep 1 in t he design of an analog bandpass filter discussed in Example 7.10 (Sec. 7.6-2). T he 5 s ubsteps are: S tep 2 .1: F ind ws' f or t he p rototype f ilter. For t he p rototype lowpass filter transfer function 'Hp(s) w ith amplitude response, as depicted in Fig. 12.16b. T he frequency ws' is found [using Eq. (7.56)] t o be t he smaller of (0.1584)(0.3249) - (0.0708)2 0.0708(0.3249 - 0.1584) _ z2 + 2 az + 1 Tbp [Z] b(z2 _ 1) (12.67b) = 3.939 a nd (0.7265)2 - (0.1584)(0.3249) = 3.937 0.7265(0.3249 - 0.1584) (12.67a) b(z2 - 1) Tbs[Z] = z2 + 2 az + 1 = 0.0708 p which is 3.937. We now have a prototype lowpass filter in Fig. 12.16b with G = - 2.1 dB, G. = - 20 d B, wp' = 1, and ws' = 3.937. S tep 2 .2: D etermine t he f ilter o rder n T he order of t he B utterworth filter from E q. (7.39) is w here n a nd (12.68) T hus, a d igital f ilter t r...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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