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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 efthand side o f Eq. (2.17b) will contain a t erm 6 (n+1)(t). B ut t he h ighestorder derivative t erm
o n t he r ighthand 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 higherorder derivatives in h(t). 2 T imeDomain Analysis of ContinuousTime 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 thorder 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 secondorder 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~nl)(O) = 1, a nd Yn(O) = Yn(O) = iin(O) = ... = y~n2)(0) = 0 (2.20) where...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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