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

1 15 f ig 1 17 a c anonical r ealization o f hz r

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Unformatted text preview: ontinuous-time u nit impulse 6(t). T hus, a t t he i nput of a discrete-time processor, a discrete-time signal I [k] is j ust a sequence of numbers. B ut a t t he i nput of a continuous-time system, I [k] is a sequence o f impulses. There are appropriate converters a t t he interface o f discrete-time a nd continuous-time systems t o c arry o ut signal conversion t o a ppropriate forms. To begin with, consider a basic continuous-time system (Fig. l1.lOa) with transfer function H (s). T he i nput I {t) is sampled a nd t he sampled signal I [k] is applied t o t he i nput o f H (s). A lthough y(t), t he o utput o f t his system, is continuous, we shall endeavor t o find t he values of y(t) only a t t he discrete i nstants t = kT. Such a n analysis is relatively simple using t he m ethod of z-transform. For this purpose, we consider as if t he o utput is sampled by a n h ypothetical sampler shown d otted in Fig. 11.10a. Now, we shall relate t he i nput samples I [k] a nd t he o utput samples y[k]. L et h[k] b e t he u nit impulse response relating t he o utput samples to t he i nput samples. In o ther words, y[k] = h[k] * J[k]. Recall also t hat a n u nit impulse 6[k] is 6(t) when considered in conjunction with a continuous-time system. Hence, h[k], t he u nit impulse response is t he s ampled version o f t he s ystem's u nit impulse response h(t). T hus, h[k] = h(kT) where T is t he sampling interval. For instance, if H (s) = s~A' t hen h(t) = e At a nd Therefore, t he equivalent discrete-time transfer function H [z] of this system is given by H[z] = Z {h[k]} = Z [e AkT z = z - eAT (11.49) 700 11 Discrete- Time S ystems Analysis U sing t he Z -Transform 11.6 S ampled-Data ( Hybrid) Systems 1 T[z] = - ( - Ylz] = H[z]F[z] z z -1 2 - z ) f. G[z]H[z] z -e -2T I n t his case, we use t he n otation GH[z] for T[z]. T hus, GH[z] f. G[z]H[z], b ut is t he d iscrete-time t ransfer f unction which corresponds t o G(s)H(s). For t he s ystem i n Fig. 11.1Oc, Ylz] = GH[z]F[z] Y[z] = H [z]X[z] = H[z]G[z]F[z] so t hat T[z] = G[z]H[z] For t he s ystem i n Fig. 11.1Od, Ylz] = G[z]H[z]F[z] E[z] = F [z]- W[z] Moreover, y (t) T .:>' y [k] W[z] = H[z]Y[z] Y[z] = G[z]E[z] = G[z] ( F[z]- W[z]) = G [z](F[z]- H[z]Y[z]) I --i---t-. ..... .. . ~ .... G[z] Ylz] = I + G[z]H[z] F[z] Hence F ig. 1 1.10 G[z] Y[z] = 1 + G [z]H[z{[z] C omputing t he o utput i n h ybrid o r s ampled-data s ystems. T hus, H[z] is t he d iscrete-time t ransfer f unction of H (s) = s~>. t hat r elates y[k] ( the o utput s amples) t o t he d iscrete-time i nput J[k].t I f we have two systems w ith t ransfer f unctions G (s) a nd H (s) in cascade (Fig. 1 l.lOb), t he equivalent transfer T[z] f. G[z]H[z], b ut is GH[z], where G[z], H[z] a nd G H [z] c orrespond t o d iscrete-time t ransfer functions of G (s ), H (s) a nd G (s ) H (s ), respectively. For instance, if 1 G(s) = s+2 a nd z - 2T z -e G[z] T[z] = 1 + G[z]H[z] • E xample 1 1.9 F ind t he o utput s amples y[k] for t he s ampled-data s ystem i llustrated i n F ig. I lolla w hen t he i nput i s a u n...
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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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