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8 z Y[zJ = - z - 0.5 +:3 W e have _!i. (_Z_) + ~ (_Z_)
3 z - 0.5 3 Izl > 0 .8 z - 0.8 Y2[ZJ = 5 ( _Z_) _ 6 ( _Z_)
z - 0.5
z - 0.6 0.5 < Izl < 0.6 Therefore
yt[kJ = [_~(0.5)k
Y2[kJ = 5(0.5)ku[kJ + ~(0.8)kl u[kJ
+ 6(0.6)kU[-(k + I)J 711 7 10 P roblems a.nd s ignals are r estricted t o t he c ausal type, t he z -transform analysis is greatly simplified; t he region o f convergence o f a s ignal becomes irrelevant t o t he analysis process.
T his s pecial case of z -transform (which is restricted t o c ausal signals) is called t he
u nilateral z-transform. Much o f t he c hapter d eals w ith t his transform. Section
11.7 discusses t he g eneral variety of t he z -transform ( bilateral z-transform), which
c an h andle c ausal a nd n oncausal signals a nd s ystems. I n t he b ilateral transform,
t he inverse t ransform o f F[z] is n ot u nique, b ut d epends o n t he region of convergence of F[z]. T hus, t he region of convergence plays a crucial role in t he b ilateral
z-transform. y[kJ = ydkJ + Y2[kJ
+ ~(O.S)k] u[kJ + 6 (0.6)ku[-(k + I)J = [~(0.5)k £::,. • E xercise E ll.14 For a causal system in Example 11.13, find the zero-state response to input
j[k] = (u k u[k] + 5(3)ku[-(k + 1)] Answer: Ht)k+3(~)k]u[k]+6(3)ku[-(k+l)] \l Problems
1 1.8 Summary
I n this c hapter we discuss t he a nalysis o f l inear, time-invariant, discrete-time
(LTID) systems by z-transform. T he z -transform is a n e xtension of t he D TFT
w ith t he frequency variable j n generalized t o a + j n. S uch a n e xtension allows us
t o synthesize discrete-time signals by using exponentially growing (discrete-time)
sinusoids. The relationship o f t he z -transform t o t he D TFT is identical t o t hat o f
t he Laplace transform t o t he Fourier. Because o f t he g eneralization of t he frequency
variable, we c an a nalyze all kinds o f LTID systems a nd a lso handle exponentially
T he z -transform changes t he difference e quations o f LTID systems into algebraic equations. Therefore, solving these difference equations reduces t o solving
T he t ransfer function H[z] o f a n LTID s ystem is equal t o t he r atio o f t he z transform of t he o utput t o t he z -transform o f t he i nput w hen all initial conditions
a re zero. Therefore, if F[z] is t he z -transform o f t he i nput f[k] a nd Y [z] is t he
z -transform o f t he c orresponding o utput y[k] ( when all initial conditions are zero),
t hen Y[z] = H[z]F[z]. For a system specified by t he difference equation Q[E]y[k] =
P [E]f[k], t he t ransfer function H[z] = P [z]/Q[z]. Moreover, H[z] is t he z -transform
o f t he s ystem impulse response h[k]. We also showed in C hapter 9 t hat t he s ystem
response t o a n e verlasting exponential z k is H [z]zk.
LTID systems can b e realized by scalar multipliers, summers, a nd t ime delays.
A given transfer function can be synthesized in m any different ways. Canonical,
cascade and parallel forms of realization are discussed. T he r ealization procedure
is identical t o t hat for continuous-time systems.
In Sec. 11.5, we...
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