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

1048 or exercise ei22 therefore a pole a zero a t t

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Unformatted text preview: der nonrecursive filter transfer function is given by ( ao = a l = a 2 = ... = a n-l = 0). (12.27a) (12.27b) T he inverse z-transform o f t his equation yields (a) h[k] = bn 6[k] + bn - 1 6[k - 1] + ... + h6[k - n + 1] + bo6[k - n] (12.28) Observe t hat h[k] = 0 for k > n . Because nonrecursive filters are a special case o f recursive filters, we e xpect t he p erformance of recursive filters t o b e superior. This expectation is t rue in t he sense t hat a given amplitude response can be achieved by a recursive filter o f a n o rder smaller t han t hat r equired for t he corresponding nonrecursive filter. However, nonrecursive filters have t he a dvantage of having linear phase characteristics. Recursive filters can realize linear phase only approximately. F [zl (b) F ig. 1 2.7 Digital filter realization: 1 2.4 (a) recursive filter ( b) nonrecursive filter. Nonrecursive Filters I f t he recursive coefficients a o, a l, a nd a 2 a re zero, H[z] in Eq. (12.23) reduces to (12.25a) (12.25b) T he difference equation corresponding t o t his system now reduces t o y[k] = b3f[k] + b2/[k - 1] + b d[k - 2] + bof[k - 3] (12.26) N ote t hat y[k] is now computed from t he present a nd t he t hree p ast values of t he i nput f[k]. Such filters are called n onrecursive f ilters. F igure 12.7b shows a canonical realization of H[z], which is identical t o t he r ealization in Fig. 12.7a, with all t he feedback connections eliminated. I f we a pply a n i nput t [k] = 6[k] t o this Filter Design Criteria A digital filter processes discrete-time signals t o yield a discrete-time o utput. D igital filters can also process analog signals by converting them into discrete-time signals. I f t he i nput is a c ontinuous-time signal f (t), i t is c onverted into a discretetime signal f[k] = f (kT) by a C jD (continuous-time t o discrete-time) converter. T he signal f [k] is now processed by a "digital" (meaning discrete-time) system with transfer function H[z]. T he o utput y[k] of H[z] is t hen converted into a n "analog" (meaning continuous-time) signal y(t). T he s ystem in Fig. 12.8a, therefore, acts as a continuous-time (or " analog") s ystem. O ur objective is t o d etermine t he "digital" (discrete-time) processor H [z] t hat will make t he s ystem in Fig. 12.8a equivalent t o a desired "analog" (continuous-time) system w ith t ransfer function H a (s ), shown in Fig. 12.8b. We may strive t o m ake t he t wo systems behave similarly in t he t ime-domain or in t he frequency-domain. Accordingly, we have two different design procedures. Let us now determine t he equivalence criterion of t he two systems in t he t ime-domain a nd in t he frequency-domain. 1 2.4-1 The Time-Domain Equivalence Criterion By time-domain equivalence we m ean t hat for t he same input f (t), t he o utput y(t) of t he s ystem in Fig. 12.8a is equal t o t he o utput y(t) of t he s ystem in Fig. 732 12 Frequency Response a nd Digital Filters 1 12.4 Filter Design C riteria 73...
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