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Unformatted text preview: ontinuoustime u nit impulse 6(t). T hus, a t t he i nput of a discretetime
processor, a discretetime signal I [k] is j ust a sequence of numbers. B ut a t t he i nput
of a continuoustime system, I [k] is a sequence o f impulses. There are appropriate
converters a t t he interface o f discretetime a nd continuoustime systems t o c arry
o ut signal conversion t o a ppropriate forms.
To begin with, consider a basic continuoustime 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 ztransform. 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 continuoustime 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 discretetime 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 ampledData ( 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 iscretetime 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 it. ..... .. . ~ .... 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 ampleddata s ystems. T hus, H[z] is t he d iscretetime t ransfer f unction of H (s) = s~>. t hat r elates y[k]
( the o utput s amples) t o t he d iscretetime 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 iscretetime 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 ampleddata 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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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