Signal Processing and Linear Systems-B.P.Lathi copy

1 a constant input fkj c t his is a special case of

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Unformatted text preview: nit 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. 9.6-1 \l (d) ( e) * f [k] 00 = L h [m]f[k - m] m =-oo a nd 00 ::; L I h[m]llf[k-mJl m =-oo + 2E + 3)f[k] \l System Response t o Bounded Inputs A s i n t he c ase o f c ontinuous-time s ystems, a symptotically s table d iscrete-time 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 ight-hand s ide i s b ounded; t hat is, if 608 9 Time-Domain Analysis o f Discrete-Time Systems 00 L /h[kJl < K 2 < 00 (9.75) 609 9.7 Appendix 9.1: Determining t he Impulse Response passed through a discrete-time 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 .6-2 For a discrete-time 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. 2.6-1, 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. 9.6-2). Note t hat a n a symptotically stable system is always BIBD-stable.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 continuous-time systems a nd their qualitative proofs, discussed in Sec. 2.7, also apply t o d iscrete-time 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.

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