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Unformatted text preview: onding continuoustime system H [esT] o n t he impulse sequence i ( t ) 11 696 DiscreteTime Systems Analysis Using t he Z  Transform Also F(s) = I:I lk] f (t) 697 11.6 SampledData (Hybrid) Systems [~f[k16(t  kT)] Now, because t he Laplace transform o f 6(t  kT) is e  l' skT 00 F(s) L f[kle SkT
k=O y es) t = = L y[kle SkT
k=O (11.46) Similarly
00 y [k] f lk]
H [z] )!< f (t) S ubstitution of Eqs. (11.46)and (11.47) in Eq. (11.45) yields ~ )!< ~Y[kleSkT =
, (t) y [kJ (11.47) H[eSTl [~t[kleSkT] B y introducing a new variable z = esT, t his equation can be expressed as
00 00 L y[klzk
k=O
or = H[zl L f[klzk
k=O Y[zl = H[zlF[zl t_ where (b) (a) 00 00 F[zl = L f[klzk a nd Y[zl = L y[klzk k=O F ig. 1 1.8 Connection between the Laplace and ztransform.
( Fig. 11.8b). T he delaying of a sequence in H[zl would amount t o t he delaying
of a n impulse t rain in H[eST1. T he case of adding a nd multiplying operations is
similar. I n o ther words, onetoone correspondence of t he two systems is preserved
in every aspect. Therefore, if y[kl is t he o utput of t he discretetime system in Fig.
I1.Sa, t hen yet), t he o utput of t he continuoustime system in Fig. 11.8b, would be
a sequence of impulses whose k th impulse s trength is y[kl· T hus k=O I t is clear from this discussion t hat t he z transform can be considered as a Laplace
transform with a change of variable z = esT or s = ( liT) In z . O n t he o ther h and,
we m ay consider t he z transform as a n i ndependent transform in its own right.
Note t hat t he t ransformation z = esT t ransforms t he i maginary axis in t he splane
jwT
(s = j w) i nto a u nit circle in t he z plane (z = esT = e
, or Izl = 1). T he L HP
a nd R HP in t he s plane m ap i nto t he inside a nd t he outside, respectively, of t he u nit circle in t he zplane. 00 yet) = (11.44) L y[k16(t  kT)
k=O T he s ystem in Fig. 11.8b, being a continuoustime system, can be analyzed by using
t he Laplace transform. I f jet) ¢ =} F(s) a nd yet) ¢ =} y es) t hen
(11.45) 1 1.6 Sampleddata (Hybrid) Systems
S ampleddata systems are hybrid systems consisting of discretetime as well
as continuoustime subsystems. Consider, for example, a fire control system. In
this case, t he p roblem is t o search a nd t rack a moving t arget a nd fire a projectile
for a direct hit. T he d ata o btained from t he search a nd t racking r adar is discretetime d ata because of a scanning operation, which results in sampling of azimuth,
elevation, a nd t he t arget velocity. This d ata is now fed t o a digital (discretetime)
processor, which performs extensive computations. T he c omputer o utput is t hen
fed t o a c ontinuoustime plant, such as a gun mount, which accordingly positions 698 11 D iscreteTime Systems Analysis Using t he Z Transform
Discretetime
controller ~_AlD_, Continuoustime
plant GIZ1H D/AH G(~~ jtl... e (k) Y...,(,..;t)_ y(t~t'__________________________·.....Jr (.,
Ideal
S ampler y (t) T 699 11.6 SampledDa...
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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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