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

# 1214 summary o f t he bilinear transformation method

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

Ask a homework question - tutors are online