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Unformatted text preview: We for conjugate poles),
as depicted in F ig, 7.15b. At t his p oint, t he o ptimum s hape o f this wall is n ot
obvious because o ur a rguments a re q ualitative a nd i ntuitive. Yet, i t is c ertain t hat
t o h ave enhanced gain ( constant g ain) a t e very frequency over t his r ange, we need
a n infinite number of poles o n t his walL We c an show t hat for a maximaJly f iatt
r esponse over t he frequency range (0 t o we), t he wall is a semicircle with a n i nfinite
n umber o f p oles uniformly d istributed a long t he waJl. 1 I n p ractice, we c ompromise
by using a finite n umber (n) o f poles w ith l ess-than-ideal characteristics. Figure
7.15c shows t he p ole configuration for a fifth-order (n = 5) filter. T he a mplitude
r esponse for various values o f n a re i llustrated i n Fig. 7.15d. As n - ; 0 0, t he filter
response approaches t he ideal. T his family o f filters is known as t he B utterworth
filters. T here a re also o ther families. I n C hebyshev filters, t he waJl s hape is a
semiellipse r ather t han a semicircle. T he c haracteristics of a Chebyshev filter a re
inferior t o t hose of B utterworth over t he p assband (0, we), w here t he c haracteristics
show a rippling effect instead of t he m aximally flat response of B utterworth. B ut in
t he s topband (w > we), C hebyshev behavior is superior in t he sense t hat C hebyshev
filter gain drops faster t han t hat o f t he B utterworth. O Re .... Re .... 1m x .11- ••• ;
" (b) (a) 501 ( c) '. x' " '. IH (joo)1 '" 0 t
IH(joo)1 t jOOo Re .... ..
- jOJ o
OlO (a) (b) F ig. 7 .16 Pole-zero configuration and the amplitude response of a bandpass filter. 7 .4-3 o
(d) F ig. 7 .15 Pole-zero configuration and the amplitude response of a lowpass (Butterworth)
filter. Bandpass Filters T he s haded characteristic in Fig. 7.16b shows t he i deal b andpass filter gain.
I n t he b andpass filter, t he g ain is enhanced over t he e ntire passband. O ur e arlier
tMaximally flat amplitude response means the first 2 n-l derivatives of I H(jwll are zero a t w = o. 502 7 Frequency Response a nd Analog Filters 503 We use the poles and zeros in Fig. 7.17a with wo = 1207r. The zeros are at s = ± jwo.
The two poles are at - wo cos 0 ± jwo sin O. The filter transfer function is (with wo = 1207r) 1m s-plane
.)C • ••• 7.4 Filter Design by Placement of Poles a nd Zeros j Wo .. H (s) = 0 2
8 + w5
= 8 2 + (2wo cos O)s + w5 = R e-. + jwo)
+ Wo cos 0 - jwo sin 0)
8 + 142122.3
s2 + (753.98 cos 0)8 + 142122.3 (s - jwo)(s
(8 + Wo cos 0 + jwo sin 0)(8 and
'. - jro o ''''''
(a) IH(jw)1 = o (b) Fig. 7.17 Pole-zero configuration and the amplitude response of a bandstop (notch) filter.
discussion indicates t hat t his can be realized by a wall o f poles opposite t he imaginary axis in front of t he p assband centered a t wOo ( There is also a wall of conjugate
poles opposite - wo.) Ideally, a n infinite number of poles is required. In practice, we compromise by using a finite number of poles and accepting less-than-...
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