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

# 907981008 182 1414213568 182 1782013058 182 t ag b

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Unformatted text preview: 2983 + 43382 + 37328 + 16081 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 nth-order 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 eventh-order 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 qual-ripple 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.

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