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

W hen we i ntegrate 18 aje st along this p ath t he r

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Unformatted text preview: TO satisfies t he c ondition (6.19). Thus, if for some M a nd (To, (6.20) t2 we c an choose (T > (TO t o satisfy (6.19).t T he signal e , in contrast, grows a t a r ate t2 faster t han e<7ot, a nd consequently e is n ot Laplace transformable. Fortunately such signals (which are not Laplace transformable) are of little consequence from either a practical or a theoretical viewpoint. I f (TO is t he s mallest value of (T for which the integral in (6.19) is finite, (TO is called t he a bscissa of c onvergence a nd t he region o f convergence o f F (s) is R e s > (TO. T he abscissa of convergence for e -atu(t) is - a ( the region o f convergence is Re s > - a). • E xample 6 .2 D etermine t he L aplace t ransform o f ( a) oCt) ( b) u (t) ( c) cos w atu(t). ( a) £. [8(t)] = 1~ 8 (t)e- 8t dt U sing t he s ampling p roperty [Eq. (1.24a)], we o btain £. [8(t)] = 1 for all s t hat is o (t)=l for all s (6.21) ( b) T o find t he L aplace t ransform o f u (t), r ecall t hat u(t) 1 00 £.[u(t)] = a- 1 s u (t)e- 8 'dt 1 00 = 0- Re s > 0 e -stdt = 1 for t :::: O. T herefore = _ ~e-s'l~ s a (6.22) t Condition (6.20) is sufficient b ut not necessary for the existence of t he Laplace transform. For example f(t) = 1 /0 is infinite a t t = 0 and, (6.20) cannot be satisfied, b ut t he t ransform of 1 /0 exists and is given by Rs. 6 C ontinuous-Time S ystem Analysis Using t he L aplace Transform 370 f (1) :I f (t) 2 :I I- n 4 2 (a) I- F ig. 6 .3 Signals for Exercise E6.1. (c) Because cos w otu(t) = £ [cos wotu(t)] = !£ [eJwo'u(t) 6 .1-3 = O. ! [eiWo ' + e -iwo') u(t) (6.23) + e-Jwo'u(t») From Eq. (6.16), it follows t hat 1 - -.-+-+-.1] £ [coswotu(t)]=-2 [ 1 S - JWo S = 82 +W02 S Re JWo S >0 T he Laplace Transform 371 p ath o f integration is restricted to t he i maginary axis. Because of this restriction, t he Fourier integral for t he s tep f unction does n ot converge in t he o rdinary sense as Example 4.7 demonstrates. We h ad t o use a generalized function (impulse) for convergence. T he Laplace integral for u(t), in contrast, converges in t he o rdinary sense, b ut o nly for Re 8 > 0, a region forbidden to t he F ourier transform. Another interesting fact is t hat a lthough t he L aplace transform is a g eneralization of t he F ourier transform, t here a re signals (e.g., periodic signals) for which t he Laplace transform does n ot exist, although t he Fourier transform exists ( but n ot in t he o rdinary sense). (b) We also could have obtained this result from (6.16b) by letting a 6.1 Re(s±jw)=Re8>O (6.24) • F or t he u nilateral L aplace transform, t here is a unique inverse transform of F (s); c onsequently, there is no need t o specify t he region of convergence explicitly. For t his r eason, we shall generally ignore any mention of t he region of convergence for unilateral transforms. Recall, also, t hat in t he u nilateral Laplace transform it is u nderstood t hat every signal f (t) is zero for t < 0, a nd it is a ppropriate t o i ndicate this fact by multiplying t he signal by u(t). Finding the Inverse Transform F inding t he inverse Laplace transform by using t he definition (6.8a) requires inte...
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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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