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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 = _ ~es'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 ontinuousTime 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 .13 = O. ! [eiWo ' + e iwo') u(t) (6.23) + eJwo'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.
 Spring '13
 Bayliss
 Signal Processing, The Land

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