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

6902z 12 hzj z2 14678z 06298 0 15 f ig 1 215

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Unformatted text preview: 8 + (1.248)2 S tep 3: F ind H[z] b y u sing t he b ilinear t ransformation Finally, t he d esired transfer function H[z] o f t he b andpass filter is o btained from 1 i p(8) b y replacing 8 with np[z] from Eqs. (12.67) a nd (12.68). From Eq. (12.68), we o btain a= W P ,'WP2' - 1 = - 0.9485 = - 0.9021 1.0515 wp , ' w p , ' + 1 a nd b = wp , ' - w p , ' = 0.1665 wp, ' wp,' + 1 1.0515 = 0.1583 M ATLAB gives b =0.0296 0 - 0.0593 0 0 .0296 a nd a =l - 3.119 3 .9259 - 2.3539 0 .576 for B utterworth o ption. Therefore Hz_ - Z4 - 0.0296(z2 - 1)2 3.119z 3 + 3.9259z 2 - 2.3539z + 0.5760 a r esult, which agrees w ith t he a nswer found in E xample 12.8. To p lot t he a mplitude a nd t he p hase response, we c an use t he l ast 9 functions in E xample C I2.5. 0 o C omputer E xample C 12.S Using b ilinear t ransformation w ith prewarping, design a b andstop d igital filter t o m eet t he following specifications: Wp= [450 4 000], Ws= [1000 2000], Gp = - 2.1 d B , a nd Gs = - 20 dB. Use T = 1r/104. 756 12 F requency Response a nd Digital Filters As before, we give here MATLAB functions t o d esign t he f our t ypes o f a pproximations. T he i nput d ata a sks for frequencies so normalized t hat t he s ampling r adian frequency is 2. As explained i n E xample C12.5, all t he r adian f requencies c an b e n ormalized by multiplying each of t hem b y T/,Tr. I n t he p resent case, T = 11'/104 s o t hat T /lr = 1 /104 • T hus, t o n ormalize a ny r adian frequency, we j ust d ivide i t b y 104 • b nzn B utterworth [ n,Wn]=buttord(Wp,Ws,-Gp,-Gs) [ b,a]=butter(n,Wn,'stop') (12.69a) zn = bn Now, by definition, H[z] bn - bi bn -2 I bo + - z- + - z-2- + ... + -zn---I + z n is t he z -transform of h[k]: 00 Chebyshev [ n,Wn]=cheblord(Wp,Ws,-Gp,-Gs) [ b,a]=chebyl(n,-Gp,Wn,'stop') k=O = h[O] + hIll + h[2] + .,. + h[n] + h[n + 1] + ... z t. I nverse Chebyshev [ n,Wn]=cheb2ord(Wp,Ws,-Gp,-Gs); [ b,a] = cheby2 ( n,-Gs, W n , ' stop ' ) z2 zn zn+1 (12.70) C omparison o f t his e quation w ith Eq. (12.69b) shows t hat h[k] = 0 for k > n, a nd Eq. (12.70) becomes E lliptic [n,W n ]=ellipord ( W p , W s ,-Gp,-Gs); H[z] [ b,a] = ellip( n ,-Gp,-Gs, W n , ' stop') = h[O] h[l] h[2] h[n] + - z + -z 2 + ... + -z n M ATLAB r eturns b =0.3762 - 1.3575 1 .9711 - 1.3575 0 .3762 a nd a =l - 2.2523 2 .0563 - 1. 2 053 0 .4197 for Chebyshev option. Therefore H[z] = 0.3762(z4 - 3.6084z + 5 .2395z - 3.6084z + 1) Z4 - 2.2523z 3 + 2 .0563z 2 - 1.2053z + 0.4197 3 2 0 Nonrecursive Filters T he recursive filters are very sensitive t o coefficient accuracy. Inaccuracies in t heir i mplementation, especially t oo s hort a word length, m ay c hange their behavior drastically a nd e ven make them unstable. Moreover, t he recursive filter designs are well established only for amplitude responses t hat are piecewise constant, such as lowpass, bandpass, highpass, and b andstop filters. I n c ontrast, a nonrecursive filter c an b e designed t o have a n a rbitrarily s haped frequency response. In addition, nonrecursive filters can be designed...
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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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