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

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Unformatted text preview: Example 7.7, using functions from Signal Processing Toolbox in MATLAB. W p=lOjWs=16.5jGp=-2jGs=-20j [ n,Ws]=cheb2ord(Wp,Ws,-Gp,-Gs,'s'); [ num,den]=cheby2(n,-Gs,Ws,'s') MATLAB returns n = 3 and num= 0 5 0 1805.9, den= 1 23.2 256.4 1805.9; t hat is, 58 2 + 1805.9 ( ) - 83 + 23.28 2 + 256.48 + 1805.9 H8_ H(8) = 50.5823 (s + 4.0191 + j 6.8937)(s + 4.0191 - Hint: In this case K 2 = 7.6-1 a. = -20 dB, wp = 10 rad/s, and w. = 28 rad/s. 523 given set of specifications. l B ut t he inverse Chebyshev realization requires more elements a nd t hus is less economical t han t he Chebyshev filter. B ut t he inverse Chebyshev does require fewer elements t han a c omparable performance B utterworth filter. R ather t han give complete development of inverse Chebyshev filters, we shall solve a problem using MATLAB functions from t he Signal P rocessing Toolbox. E xercise E 7.5 Determine n (the order) and the transfer function of a Chebyshev filter to meet the following specifications: f Answer:n = 2 Chebyshev Filters ~ y '1+e 2 50.5823 j6.8937) s2 + 8.03818 + 63.6768 To plot amplitude response, we can use the last three functions from Example C7.5. 0 \l 7.6-2 Elliptic Filters Inverse Chebyshev Filters T he p assband behavior of t he Chebyshev filters exhibits ripples a nd t he stopband is s mooth. Generally, passband behavior is more i mportant a nd we would prefer t hat t he p assband have smooth response. However, ripples can be tolerated in t he s topband as long as they meet a given specification. T he i nverse C hebyshev filter does exactly t hat. B oth, t he B utterworth a nd the Chebyshev filters, have finite poles a nd no finite zeros. T he inverse Chebyshev has finite zeros a nd poles. I t e xhibits maximally flat passband response a nd equal-ripple s topband response. T he inverse Chebyshev response can be obtained from t he Chebyshev in two steps as follows: Let 1tc{w) b e t he Chebyshev amplitude response given in Eq. (7.42). In t he first step, we s ubtract /1tc{w)/2 from 1 to o btain a highpass filter characteristic where t he s topband (from 0 t o 1) has ripples a nd t he p assband (from 1 t o 0 0) is s mooth. I n t he second step, we i nterchange the stopband a nd p assband by frequency transformation where w is replaced by l /w. T his step inverts the passband from t he range 1 t o 0 0 t o t he range 0 t o 1, a nd t he s topband is now from 1 t o 0 0. Moreover, t he p assband is now smooth a nd t he s topband has ripples. This is precisely the inverse Chebyshev amplitude response /1t(w)/ given by /1t(W)/2 = 1 -/1tc{I/w)/2 = 1 ~~~~{~}w) where Cn(w) a re t he n th-order Chebyshev polynomials listed in Table 7.3. T he inverse Chebyshev filters are preferable t o t he Chebyshev filters in many ways. For example, the passband behavior, especially for small w , is b etter for t he inverse Chebyshev t han for t he Chebyshev or even for t he B utterworth filter of t he s ame order. T he inverse Chebyshev also has t he smallest transition band of t he t hree filter...
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