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

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Unformatted text preview: Time S ystem Analysis Using t he Laplace Transform .. F (s) 3 97 y ( s) = F ( s)H ( s) ... H (s) F ig. 6 .9 The transformed representation of an LTJC system. if the input f (t) = 3 e- St u(t) and all the initial conditions are zero; t hat is, the system is in zero-state. The system equation is (D2 + 5 D + 6) y(t) = ( D + 1) f (t) ~ '-v---' Q(D) P(D) Therefore Hs () I t is t he r atio o f Y (8) t o F (8) when all t he i nitial conditions are zero (when t he s ystem is in zero state). T hus H (8) == Y (s) = C [zero-state response] F(8) C[input] We can n ow r epresent t he t ransformed version of the system, as depicted in Fig. 6.9. T he i nput F (s) is t he Laplace transform of f (t), a nd t he o utput Y (s) is t he L aplace t ransform o f ( the z ero-input response) y (t). T he s ystem is d escribed by t he t ransfer function H (s). T he o utput Y (s) is t he p roduct F (s)H(s). T he r esult Y (s) = H (s)F(s) g reatly facilitates derivation of t he s ystem response t o a g iven input. We shall d emonstrate t his assertion this by a n example. (s + 6y(t) = df dt + f (t) _ 3_ s +5 3(8 + 1) + 5)(s + 2)(s + 3) -2 1 3 = s+5-s+2+s+3 The inverse Laplace transform of this equation is y (t) • = ( _2e- St - e- 2t + 3 e- 3t ) u(t) • E xample 6 .12 Show t hat the transfer function of ( a) an ideal delay of T seconds is e - sT; ( b) an ideal differentiator i s 8; (c) an ideal integrator is 1/ s. ( a) I deal D elay For an ideal delay of T seconds, the input f (t) and output y(t) are related by y (t) = f (t - T) or Y (s) dy + 5 dt = Y (s) - F(8)H(8) _ 3(s + 1) - (s + 5)(8 2 + 58 + 6) 3. A n LTIG s ystem is m arginally stable if a nd only if there a re no poles o f H (8) in t he R HP, a nd t here are some u nrepeated poles on t he i maginary axis. d2y dt2 S+1 + 5s + 6 F(s) = [, [3e- St u(t)] (6.53) 2. A n LTIG s ystem is u nstable if a nd o nly if either one or b oth o f t he following conditions exist: (i) a t l east one pole of H (s) is in t he R HPj (ii) t here a re r epeated p oles of H (s) o n t he i maginary axis. Find the response y(t) of an LTJC system described by the equation 82 Also 1. An LTIG s ystem is a symptotically stable if a nd only if all t he poles of its transfer function H (8) a re in t he LHP. T he poles may b e r epeated or unrepeated. E xample 6 .11 Q (s) - and We have s hown t hat Y (8), t he L aplace transform of the zero-state response y (t), is t he p roduct o f F (s) a nd H (s), w here F (8) is t he Laplace transform of t he i nput f (t) a nd H (s) is the system transfer function [relating the p articular o utput y (t) t o t he i nput f (t)]. Observe t hat t he d enominator of H (8) is Q (s), t he c haracteristic polynomial of t he system. Therefore, the poles o f H (8) a re the c haracteristic roots o f the system. C onsequently, t he s ystem stability criterion can b e s tated in terms of t he poles of t he t ransfer function of a system, as follows: • = P(8) _ = F (s)e- sT [see Eq. (6.29a)] Therefore H( ) = Y (s) = - sT S F (s) e (6.54) 6 398 C ontinuous-Time S ystem A nalysis Using t he L aplace Tra...
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