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Unformatted text preview: onding continuous-time system H [esT] o n t he impulse sequence i ( t ) 11 696 Discrete-Time Systems Analysis Using t he Z - Transform Also F(s) = I:I lk] f (t) 697 11.6 Sampled-Data (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[kle-SkT =
, (t) y [kJ (11.47) H[eSTl [~t[kle-SkT] B y introducing a new variable z = esT, t his equation can be expressed as
00 00 L y[klz-k
or = H[zl L f[klz-k
k=O Y[zl = H[zlF[zl t_ where (b) (a) 00 00 F[zl = L f[klz-k a nd Y[zl = L y[klz-k k=O F ig. 1 1.8 Connection between the Laplace and z-transform.
( 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, one-to-one correspondence of t he two systems is preserved
in every aspect. Therefore, if y[kl is t he o utput of t he discrete-time system in Fig.
I1.Sa, t hen yet), t he o utput of t he continuous-time 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 s-plane
(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 z-plane. 00 yet) = (11.44) L y[k16(t - kT)
k=O T he s ystem in Fig. 11.8b, being a continuous-time 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 Sampled-data (Hybrid) Systems
S ampled-data systems are hybrid systems consisting of discrete-time as well
as continuous-time 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 (discrete-time)
processor, which performs extensive computations. T he c omputer o utput is t hen
fed t o a c ontinuous-time plant, such as a gun mount, which accordingly positions 698 11 D iscrete-Time Systems Analysis Using t he Z -Transform
controller ~_AlD_--, Continuous-time
plant GIZ1H D/AH G(~~ jt---l... e (k) Y...,(,..;t)_ y(t~t'-__________________________·.....Jr (.,
S ampler y (t) T 699 11.6 Sampled-Da...
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