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

T he idea here is t o begin with a distorted analog

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: fy these specifications is given by [see E q. (7.49b)J = C omputer E xample C 12.6 D esign a highpass digital filter t o m eet t he specifications in E xample 12.7, 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. 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 C12.5, a ll t he r adian frequencies c an b e n ormalized b y m ultiplying each o f t hem b y T/7r. I n t he p resent case, T = 7r/100 s o t hat T/7r = 1 /100. T hus, t o n ormalize any r adian frequency, we j ust d ivide i t b y 100. T he n ormalized Wp a nd Ws a re 1 5/100 a nd 1 0/100, respectively. W p=0.15;Ws=O.1;-Gp=1;-Gs=6.3;T=pi/100; i. B utterworth [ n,Wn)=Buttord(Wp,Ws,-Gp,-Gs); [ b,a)=butter(n,Wn,'high'); i. I nverse Chebyshev I n t he second step, we d esign a prewarped Chebyshev high pass filter w ith c ritical frequencies w.' = 0.1584 a nd wp' = 0.24 w ith f = - 1 d B a nd G. = - 6.3 dB (Fig. 12.15b). Following t he p rocedure in Sec. 7.6-1, we first design a p rototype lowpass filter w ith specifications, as indicated in Fig. 12.15b. O bserve t hat t he c ritical frequencies of t he p rototype filter are 1 (passband) a nd wp' /w.' = 1.515 (stopband) as explained in Sec. n o [ n,Wn)=cheb1ord(Wp,Ws,-Gp,-Gs); [ b,a) = cheby1 ( n,-Gp, W n , ' high'); = t an(¥o) = 0.24 - 1 [ (10 0.63 _ 1 ] 1/2 1 0 c osh- 1 ( 1515) cosh -'-110·1 - 1 • i. Chebyshev ¥ = t an(fo) = 0.1584 ¥ T he c ontinuous curve in Fig. 12.15a shows t he a mplitude r esponse o f t his filter. = 1.988 We r ound u p t he value o f n t o 2. From Table 7.4 (Chebyshev filter with f = 1 a nd n = 2) we o btain t he following p rototype t ransfer function [ n,Wn)=cheb2ord(Wp,Ws,-Gp,-Gs); [ b,a) = cheby2( n ,-Gs,W n , ' high') i. E lliptic [ n,Wn)=ellipord(Wp,Ws,-Gp,-Gs); [ b,a) = ellip ( n,-Gp,-Gs, W n , ' high') MATLAB r eturns b=O. 6 902 - 1.3804 0 .6902 a nd a =l - 1. 4 678 0 .6298 for Chebyshev option. Therefore 0.6902(z - 1)2 H[zJ = z2 _ 1.4678z + 0.6298 which agrees w ith t he a nswer found in Example 12.6. T o p lot t he a mplitude a nd t he p hase response, we c an use t he l ast 9 f unctions in E xample C I2.5. 0 12 7 52 12.6 Recursive Filter design: T he Bilinear Transformation Method F requency R esponse a nd D igital F ilters 753 • Bandpass and Bandstop Filters F or b andpass a nd b andstop filters, we follow a s imilar p rocedure. All t he c ritical f requencies a re first p rewarped u sing t he s implified form in Eq. (12.64a). N ext, we d etermine a p rototype low p ass filter, which is t hen c onverted t o t he d esired a nalog f ilter u sing a ppropriate t ransformations d iscussed i n Sec. 7.6. Finally, we use t he b ilinear t ransformation i n E q. (12.64b) t o o btain t he d esired d igital f ilter. As i n t he c ase of t he h ighpass filter (discussed in E xample 1 2.7), we c an c ombine t he t wo t ransformations i nto a s ingle t ransformation. F or t he b an...
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