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

However e ither answer will satisfy t he

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Unformatted text preview: uency response reduces t o (12.77) 7 60 12 Frequency Response and Digital Filters 'rvwM h lk} (a) 6 o 12 3 4 5 7 8 k_ o 7t 2n 12.8 N onrecursive F ilter D esign ( e) Q- 761 Using the same argument, the reader can show t hat an nth-order comb filter transfer function i s zn - 1 (12.79) H[z]=~ and ( b) 1 2.8 Fig. 12.18 Comb filter: Its impulse and frequency response. T he phase response in this case is also a linear function of w. T he system has the ti~e delay (the neg~tive slope of LH[e jwT ] w ith respect t o w) of 2 T seconds (2 u m ts), the same as I II t he symmetric case. The only difference is t hat t he phase response has a constant term 7r / 2. We c an obtain similar results for odd values of n (see Prob. 12.7-1). This result c an b e generalized for an nth-order case t o show t hat t he phase response is linear, a nd t he time delay is seconds (or ~ units). ni • E xample 12.9: C omb F ilter Determine the transfer function and the frequency response of a sixth-order comb filter whose impulse response is given by h[k] = 6[k] - 6[k - 6] This impulse response is illustrated in Fig. 12.18a. Its canonical realization is depicted in Fig. 12.18b. Observe t hat h[k] is antisymmetric about k = 3. Also 6 00 H[z] = " " f [k]z-k = 1 - z-6 = ~ ~ z6 (12.78) T he frequency response is given by H[e jD ] Nonrecursive Filter Design As in t he c ase of recursive (IIR) filters, nonrecursive filters c an b e designed by using t he t ime-domain a nd t he frequency-domain equivalence criteria. I n t he t imedomain equivalence criterion, t he d igital filter impulse response is m ade identical t o t he s amples of t he desired (analog) filter impulse response. I n t he frequency-domain equivalence criterion, t he d igital filter frequency response samples a t u niform frequency intervals a re m atched t o t he desired analog filter frequency response samples. T his m ethod is also known as t he f requency s ampling o r t he s pectral s ampling m ethod. 12.8-1 Time-Domain Equivalence Method o f FIR Filter Design T he t ime-domain equivalence method (also known as t he F ourier s eries m ethod) o f design of F IR filters is identical t o t hat for I IR filters discussed in Sec . 12.5, except t hat F IR filter impulse response must b e of finite duration. Therefore, t he desired impulse response must be t runcated t o have finite duration. Truncating t he impulse response a bruptly will result i n oscillatory frequency response because o f t he Gibbs phenomenon discussed in Sec. 3.4-3. In some filtering applications t he oscillatory frequency response (which decays slowly as l /w) i n t he s topband m ay not be acceptable. B y using a t apered window function for t runcation o f h[kJ, t he oscillatory behavior can b e r educed or even eliminated a t t he c ost of increasing t he t ransition b and a s discussed in Sec. 4.9. Note t hat t he impulse response of a n n th-order F IR filter h as n + 1 samples. Hence, for t runcating h[k] for a n n -th o rder filter, we must use a n No-point window, where No = n + 1. Several window functions a nd t heir tradeoffs a ppear i n Table 12.2. Design Pro...
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