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Unformatted text preview: e, • E xample 1 1.10
F ind t he o utput y (t) for all t in Example 11.9.
In Example 11.9, we found t he response y[kJ only a t t he sampling instants. To
find t he o utput values between sampling instants, we use t he modified ztransform. The
procedure is t he same as before, except t hat we use modified ztransform corresponding
t o continuoustime systems a nd signals. For t he system G (s) = 1 / s + 4 w ith T = 0.5, t he
modified ztransform [Eq. (11.50b) w ith>' =  4, a nd T = 0.5J is Z3 Y[zJ = T[zJF[zJ = 7(zI"')('z0=.394')"(z0.174""')
Moreover t o find the modified ztransform corresponding t o f (t) = u (t) [>' = 0 in Eq.
(11.50a)], we have F[z, J.iJ = z /(z  1). S ubstitution of these expressions in those found in
Example 11.9, we o btain a nd z (z  1)(z  0.394)(z  0.174)
1 0.583
= z  1  z  0.394 + 0.083
z  0.174 Y[ J z, J.i  e 2" [Z 0.583z
z _ 1  z _ 0.394 0.083z ]
0.174 +z 7 04 11 D iscreteTime Systems Analysis Using t he Z Transform 1 From Eqs. (11.50), we obtain the inverse (modified) ztransform of thi~ equation as y[(k + JL)TJ = e  2" [1  0.583(0.394)k t
 y k u [ (k+ I )] + 0.083(  0.174)k]
7 T he complete response is also shown in Fig. U .ll. 705 11. 7 T he B ilateral Z  Transform 5 3 \ k • D esign o f S ampledData S ystems
As with continuoustime control systems, s ampleddata systems are designed t o
m eet c ertain transient (PO, t r , ts, e tc.) and s teadystate specifications. T he design
procedure follows along the lines similar to those used for continuoustime systems.
W e begin with a general secondorder system. t he r elationship between closedloop pole locations a nd t he corresponding transient p arameters PO, t r , t s ," . are
determined. Hence, for a given t ransient specifications, a n a cceptable region in the
z plane where the dominant poles of t he closedloop transfer function T[z] should
lie is d etermined. Next, we s ketch t he root locus for t he system. T he rules for
s ketching t he r oot locus are t he s ame as those for continuoustime systems. I f t he
r oot locus passes through t he a cceptable region, t he t ransient specifications can be
m et b y simple adjustment of t he gain K . I f not, we m ust use a compensator, which
will s teer t he r oot locus in t he a cceptable region. Fig. 1 1.12 _ ·/u[(k + I}J and the region of convergence of its ztransform. = = 1 1. 7 T he Bilateral Z Transform 00 L + ...J +( 1 [1 + ~ + (~r + (~r + ...J
I~I < 1 'Y z Izl < hi Therefore f [k]zk Izl < k =oo T he inverse ztransform is given by
f [k] = r ~r 1
=11 ~ S ituations involving noncausal signals or systems c annot b e handled b y t he
(unilateral) ztransform discussed so far. Such cases c an b e analyzed by t he b ilateral (or twosided) z  transform defined by
F[z] == [~ + ( ~ ~fF[Z]zkldZ
21rJ T hese e quations define t he b ilateral ztransform. T he u nilateral ztransform discussed so far is a special case, where t he i nput signals a re r estricted t o b e causal.
R estricting signals in this way results in considerable simplification in t he region of
convergen...
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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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