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

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

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Unformatted text preview: ually t he p rewarping is done a t c ertain critical frequencies r ather t han over t he e ntire band. T he final filter behavior is exactly equal t o t he desired behavior a t these selected frequencies. Such a filter is a dequate for most filtering problems if we choose t he c ritical frequencies properly. I f we require a filter to have gains g1. g2, . .. , gm a t frequencies (critical frequencies) W I, W2, . .. , Wm respectively, t hen we m ust s tart w ith a n analog filter H '(jw) which has gains 9 1, g2, . .. , gm a t frequencies W I', W2', . .. , w m ' respectively, where ., 12 746 2 wiT w/ = - tan-2 T Frequency R esponse a nd D igital F ilters i = 1 ,2,··· , m u 12.6 7 47 Recursive F ilter d esign: T he B ilinear T ransformation M ethod (12.62) T his r esults i n p rewarped filter H '(jw). A pplication o f t he b ilinear t ransformation (12.57) t o t his filter yields t he d esired d igital filter w hich h as g ains 9 1. 92, . .. , 9 m a t f requencies W I, W 2, . .. , W m r espectively. T his is b ecause, a ccording t o E q. ( 12.61a), t he b ehavior o f t he a nalog filter a t a f requency w;' a ppears i n t he d igital f ilter a t f requency 2 T t an- 1 (W/T) -2- T = 2 t an- 1 [ tan (WiT)] - 2- We c larify t hese i deas w ith a n e xample o f a l owpass B utterworth filter. o E xample 1 2.6 Design a lowpass filter with the following specifications: T he gain of unity a t W = 0, a nd t he gain is t o b e no less t han - 2 dB (G p = 0.785) over t he passband 0 $ W $ 10. T he g ain is to be no greater t han - 11 dB (G s = 0.2818) over t he s topband W 2: 15. T he highest frequency to be processed is W h = 35 r ad/s, which yields T $ 1l" /35. Let us use T =7r/35. T he specifications for a B utterworth filter for t his design are wp = 8, w. = 15, Op = - 2 d B, and O. = - 11 dB. In t he first step, we prewarp the critical frequencies W p a nd W s according to Eq. (12.62): 8 20 15 10 0 0--> 30 25 • wp ' = f t an ws' = ~tan ~= ¥ = ~tane;;) = 17.7696 Finally, we obtain H[zJ from Ha(s), using t he bilinear transformation T his substitution yields (~) :~~ = (~) :~~ I s=(~) ::;:: 0.1039(z + 1)2 O.9045z + 0.3201 T he frequency response of this filter is given by H [e jWT 0.1039(e + 1)2 - ej2wT _ 0.9045ejwT + 0.3201 jwTj _ T he a mplitude response IH[ejwTJI, w ith T = 70/7r, is d epicted in Fig. 12.14. • Summary o f t he Bilinear Transformation Method with Prewarping I n t he b ilinear t ransformation m ethod w ith p rewarping, a ll t he c ritical f requen- 17.7696 1 = 9.6308 (lOLl - 1)4 F rom Table 7.1, we find the prewarped filter transfer function Ha(s) for n = 2 a nd 9.6308 as 1 92.7529 )2 ( ) S 2 + 13.62s + 92.7529 Ha(s) = ( 9 .6;08 + v'2 9.6;08 + 1 + 13.62s + 92.7529 Z2 - We round up the value of n to 2. T here are two possible values of w ' e . We shall choose t he o ne given by e quation (7.41), which satisfies the stopband specifications exactly, b ut oversatisfies t hat in t he passband. This choice yields the 3-dB cutoff frequency we' as s= = Ha(s)ls=(~) ~:;:: 92.75...
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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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