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Unformatted text preview: ) (b) (e) splane
splane 1m cr + jOQ 1m (j0l) (j0l) o R e R e 0 o jOQ
0>= ~ 00 ~
(c) ( f) j ro t os=a+jw. o
6 .12 Analytical Development o f t he Bilateral Laplace Transform 0 00 Region of convergence Because we are unifying t he Fourier a nd Laplace transforms, we need t o use
here t he unified notation F (jw) i nstead of F(w) for t he Fourier transform, which is
given by
(6.1) (g) F ig. 6 .1 H euristic e xplanation o f e xtension o f t he F ourier t ransform t o L aplace transform. 3 64 6 ContinuousTime System Analysis Using the Laplace Transform a nd 1 211" F(jw)ejwtdw (6.2)  00 Consider now t he Fourier transform of f(t)e<7t I:
I: F[J(t)e<7t] =
= ( 0' f(t)e<7t ei wt dt == lim 00 F(O' 211" T hus + jW) (6.5) + jw )e Jwt dw
. + jw)e(<7+jw)t dw (6.7)  00 T he q uantity ( 0' + j w) is t he complex frequency s . We now change t he variable
in this integral from w to s. Because s = 0' + jw, dw = ( 1/j) ds. T he limits of
integration from w =  ao t o ao now become from (0'  j ao) t o (0' + j ao) for t he
variable s. Recall, however, t hat for a given f (t) , 0' has a certain minimum value
0 '0, a nd we c an select any value of 0' > 0 '0. In general, we c an let the limits of
integration r ange from s = c  jao t o s = c + j ao, where c > 0 '0. T hus, Eq. (6.7)
becomes
c jOO
+
(6.8a)
f (t) = 1.
F (s lest ds
211") From Eqs. (6.4) a nd (6.5), we o btain F (s) = l c joo I: f (t)e st dt (6.8b) This pair of equations is known as the b ilateral L aplace t ransform p air (or the
t wosided L aplace t ransform p air). T he bilateral Laplace transform will be
denoted symbolically as
a nd F (s) = 'c[J(t)] f (t) = , el[F(s)] or simply as (6.9) f 1 F(O' [F(nL:.s)L:.s]e(nL':.s)t
211"j y (t) = lim
[F(nL:.s)H(nL:.s)l:,.s] e(nC.s)t
c's_O n =oo
211") (6.6) =~
1rJ 1 f Clearly, t he Laplace transform expresses f (t) as a sum of everlasting exponentials
of t he form e(nL':.s)t along t he p ath c  jao t o c + j ao w ith c > 0 '0. These are
exponentially growing (if c > 0) o r decaying (if c < 0) sinusoids. We c an readily
determine a n LTIC s ystem response t o t he input f (t) by observing t hat if the system
transfer function is H (s), then, as demonstrated in Eq. (2.47), t he s ystem response
t o t he (everlasting) exponential e(nC.s)t is H(nL:.s )e(nC.s)t. From Eq. (6.9), it follows
t hat t he system response t o t he i nput f (t) is Multiplying b oth sides with e<7t yields f (t) = 1 F (s)e st ds (6.4)  00 00 + L':.s_O n =oo f(t)e(<7+ j w)t dt + jw). c ioo c joo (6.3) T he inverse Fourier transform of the above equation yields 1 l f (t) = 1 .
2
11") F[J(t)e<7t] = F(O' 211" 365 E quation (6.8a) expresses f (t) as a weighted s um of exponentials of t he form
est. T his point becomes clear when we write t he integral in Eq. (6.8a) as a sum real) I t follows from E q. (6.1) t hat t he above integral is F(O' f(t)e<7t =  1 T he Laplace Transform Response o f LT IC Systems 00 f (t)=1 6.1 l c '+jOO F (s)H(s)e st ds (6.10) c 'joo T he p a...
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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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