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

This would be like pouring gasoline on a fire in a d

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Unformatted text preview: pulse response h(t) is t he s ystem response t o a n impulse i nput 8(t) a pplied a t t = 0 w ith all t he i nitial conditions zero a t t = 0 -. A n impulse i nput 8(t) is like lightning, which strikes instantaneously a nd t hen vanishes. B ut in its wake, in t hat single moment, lightning rearranges things where it strikes. Similarly, a n impulse i nput 8(t) a ppears momentarily a t t = 0, a nd t hen i t is gone forever. B ut in t hat m oment it generates energy storages; t hat is, it creates nonzero initial conditions instantaneously within t he s ystem a t t = 0+. A lthough t he impulse i nput 8(t) vanishes for t > 0 so t hat t he s ystem has no i nput a fter t he impulse has been applied, t he s ystem will still have a response generated by these newly created initial conditions. T he impulse response h(t), therefore, must consist of the system's characteristic modes for t 2: 0+. As a result t 2: 0+ h(t) = c haracteristic m ode t erms T his response is valid for t > O. B ut w hat h appens a t t = 07 A t a single moment t = 0, t here c an a t most b e a n i mpulset, so t he form of t he c omplete response h(t) is given by t I t m ight b e possible for t he derivatives o f 6(t) t o a ppear a t t he origin. However, if m :::: n, i t is impossible for h (t) t o have any derivatives o f 6(t). T his conclusion follows from Eq. (2.17b) w ith 1 ft) = 6 (t) a nd y (t) = h (t). T he coefficients o f t he i mpulse a nd all o f i ts derivatives must b e m atched o n b oth sides of t his e quation. I f h (t) c ontains 6 (1) ( t), t he first derivative o f 6 (t), t he l eft-hand side o f Eq. (2.17b) will contain a t erm 6 (n+1)(t). B ut t he h ighest-order derivative t erm o n t he r ight-hand side is 6(n)(t). T herefore, t he t wo sides c annot m atch. Similar a rguments c an b e m ade a gainst t he presence of t he i mpulse's higher-order derivatives in h(t). 2 T ime-Domain Analysis of Continuous-Time Systems 116 h (t) = Ao8(t) + c haracteristic m ode t erms t~O + [ P(D)Yn(t)]u(t) T he U nit I mpulse Response h (t) (2.18) T he d etailed d erivation of h (t) is neither illuminating nor necessary for o ur f uture development, so t o p revent needless distraction, t his d erivation is placed in Appendix 2.1 a t t he e nd o f t he c hapter. There, we show t hat for an L TIC s ystem specified b y Eq. (2.17), t he u nit impulse response h (t) is given by h (t) = bn 8(t) 2.3 (2.19) where bn is t he coefficient of t he n th-order t erm in P (D) [see Eq. (2.17b)], a nd Yn(t) is a l inear c ombination of t he c haracteristic modes of t he s ystem s ubject t o t he following i nitial conditions: Cl =1 117 and C2 = -1 Therefore (2.25) = D, so t hat P(D}Yn(t} = DYn(t} = Yn(t} = _ e- t + 2 e- 2t Also in this case, bn = b2 = 0 [the second-order term is absent in P(D}]. Moreover, according to Eq. (2.22), P(D} h(t} = bn8(t} + [P(D}Yn(t}]U(t} = ( _e- t + 2 e- 2t )u(t} Therefore • (2.26) C omment y~n-l)(O) = 1, a nd Yn(O) = Yn(O) = iin(O) = ... = y~n-2)(0) = 0 (2.20) where...
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