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

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 &gt; 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 &gt; 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...
View Full Document

Ask a homework question - tutors are online