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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/ =  tan2 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 3dB 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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