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T his is t he a lternate s olution where t he p assband specifications are exceeded, b ut t he
s topband specifications are satisfied exactly. O n t he o ther h and, t he s olution in Example
C7.5 exceeds t he s topband specifications, b ut satisfies t he p assband specifications exactly
because of use of Eq. (7.40) [ rather t han Eq. (7.41)]. To plot a mplitude response, we can
use t he l ast t hree functions from Example C7.5. 0 {;, E xercise E 7.3
Determine n , the order of the lowpass Butterworth filter to meet the following specifications:
Gp =  0.5dB, Gs =  20 dB, Wp = 100, andws = 200.
Answer: 5. 'V 514 7 .6 7 Frequency Response and Analog Filters 7.6 Chebyshev F ilters Chebyshev Filters 515 t t 1:H(j(j)) I 1:H(j(j)) I T he a mplitude response of a normalized Chebyshev lowpass filter is given by
1 (7.42) 11t(jw)1 =  , = = = = VI + E 2C n 2 b 1 b 1 n =7 (w) where C n(w), t he nthorder Chebyshev polynomial, is given by
Cn(w) = cos ( ncos I w) (7.43a) o o F ig.7.24 A mplitude r esponse o f n ormalized s ixth and s eventhorder l owpass C hebyshev
filters. An alternative expression for Cn(w) is
(7.43b)
The form (7.43a) is most convenient t o c ompute Cn(w) for Iwl < 1 a nd form
(7.43b) is convenient for computing Cn(w) for Iwl > 1. We c an show l t hat Cn(w)
is also expressible in polynomial form, as shownt in Table 7.3 for n = 1 t o 10.
T he normalized Chebyshev lowpass amplitude response [Eq. (7.42)] is depicted
in Fig. 7.24 for n = 6 a nd n = 7. We make the following general observations: 1. T he Chebyshev a mplitude response has ripples in the passband and is smooth
(monotonic) in the stopband. The passband is 0 ::; w ::; 1, a nd there is a t otal
of n m axima and minima over the passband 0 ::; w ::; 1. 2. From Table 1.3, we observe t hat C~(O) == {~ n o dd
n even (7.44) Therefore, the dc gain is
T able 1 .3: C hebyshev P olynomials 11t(O)1 == {I n o dd
I 7 rP' 2
3
4
5
6
7
8
9
10 1
w
2w 2  1
4 w 3  3w
8w 4  8w 2 + 1
16w 5  20w 3 + 5w
32w 6  48w 4 + 18w 2  1
64w 7  112w 5 + 56w 3  7w
128w 8  256w 6 + 160w 4  32w 2 + 1
256w 9  576w 7 + 432w 5  120w 3 + 9w
512w IO  1280w 8 + 1120w 6  400w 4 + 50w 2 t The Chebyshev polynomial C n(w) has the propertyl
C n(w) = 2WCn _l(W)  C n _2(W)
Thus, knowing t hat
a nd
Co(W) = 1
we can construct C n(w) for any value of n . For example, and so on. (7.45) 3. T he p arameter E controls the height of ripples. In the passband, r , t he ratio of
the maximum gain t o t he minimum gain is n o n even r==~ (7.46a) This ratio r , specified in decibels, is
(7.46b)
so t hat
(7.41)
Because all the ripples in the passband are of equal height, the Chebyshev
polynomials Cn(w) are known as e qualripple f unctions.
 1 n> 2 4. T he ripple is p resent only over the passband 0 ::; w ::; 1. A t w == 1, t he amplitude response is 1 /Vl+? == l /r. For w > 1, t he gain decreases monotonically.
5. For Chebyshev filters, the ripple f dB takes the place of G ( the minimum gain
p
in t...
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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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