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

# F irst t he f ourier t ransform e xists only for a r

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Unformatted text preview: ) (b) (e) s-plane s-plane 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 .1-2 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 Continuous-Time 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 e-i 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 wo-sided 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) = , e-l[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.

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