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circle (Fig. 9.8d). T he s ystem is unstable. •
E xercise E9.13
Find and sketch the location in the complex plane of the characteristic roots of the system
specified by the following equation:
(E + 1)(E2 + 6E + 25)y[k] = 3Ef[k]
f::, Is this a stable, unstable, or marginally stable system? Answer: unstable.
f::, E xercise E9.14
Repeat Prob. E9.13 for
(E  1)2(E + O.5)y[k] = (E2 Answer: unstable. 9.61 \l (d) ( e) * f [k] 00 = L h [m]f[k  m] m =oo a nd 00 ::; L I h[m]llf[kmJl m =oo + 2E + 3)f[k] \l System Response t o Bounded Inputs A s i n t he c ase o f c ontinuoustime s ystems, a symptotically s table d iscretetime
s ystems h ave t he p roperty t hat e very b ounded i nput p roduces a b ounded o utput. I f f [k] is b ounded, t hen I /[k  mJl < K 1 < 0 0, a nd 00 m =oo C learly t he o utput is b ounded i f t he s ummation o n t he r ighthand s ide i s b ounded;
t hat is, if 608 9 TimeDomain Analysis o f DiscreteTime Systems
00 L /h[kJl < K 2 < 00 (9.75) 609 9.7 Appendix 9.1: Determining t he Impulse Response passed through a discretetime system. T he a mount of dispersion (or spreading
out) is e qual t o t he s ystem t ime c onstant (or w idth o f h[kJ). k =oo 9 .7 For a n a symptotically stable system
h[k] = 9 .62 For a discretetime system specified by Eq. (9.27), we have demonstrated t hat ( cnf + C2/~ + " . + cn)'~)u[k] w ith t he c haracteristic roots ) '1> ) '2, . . . ) 'n lying within t he u nit circle. Under these
conditions h[k] satisfies Eq. (9.75).t Therefore, t he response of a n a symptotically
s table system to a bounded input is also bounded. Moreover, we c an show t hat
for a n u nstable or marginally stable system, t he response is u nbounded for some
bounded input. As seen in Sec. 2.61, these results lead t o t he formulation of an
a lternate definition of system stability. A system is said t o b e stable in t he sense of
having bounded o utput for every bounded i nput (BIBD s tability) if a nd only if its
impulse response h[k] satisfies Eq. (9.75). Condition (9.75) is sufficient for a system
t o produce bounded o utput for any bounded input. I t is relatively easy to show
t hat i t is also a necessary condition; t hat is, for a system t hat violates (9.75), there
exists a bounded input t hat p roduces unbounded o utput (see P rob. 9.62). Note
t hat a n a symptotically stable system is always BIBDstable.j: Appendix 9.1: Determining t he Impulse Response h[k] = Ao8[k] + Yn[k]u[k] (9.76) We now show t hat
Ao = bo
ao (9.77) To prove this point, we s ubstitute Eq. (9.76) in Eq. (9.28) t o o btain
Q [E] (Ao8[k] + Yn[k]u[kJ) (9.78) = P [E]8[k] Because Yn[k]u[k] is a s um o f c haracteristic modes,j: [see Eq. (9.12a)]
Q [E] (Yn[k]u[kJ) = 0 k~0 (9.79) E quation (9.78) now reduces t o
A oQ[E]8[k] = P [E]8[k] Intuitive Insights Into S ystem Behavior
or T he i ntuitive insights into t he behavior o f continuoustime systems a nd their
qualitative proofs, discussed in Sec. 2.7, also apply t o d iscretetime systems. F...
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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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