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

Oliver heaviside 1850 1925 a lthough laplace

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Unformatted text preview: he s trict language o f t he e stablishment. 6 .2 Some Properties o f t he Laplace Transform Because it is a generalized form of the Fourier transform, we e xpect t he Laplace transform t o have properties similar t o those of t he Fourier transform. However, we a re discussing here mainly t he p roperties of t he u nilateral Laplace transform, and they differ somewhat from those of t he Fourier transform (which is a b ilateral transform) . Properties of t he Laplace transform are useful not only in t he derivation of t he Laplace transform of functions b ut also in t he solutions of linear integro-differential equations. A glance a t Eqs. (6.8a) a nd (6.8b) shows t hat, as in t he case of t he Fourier transform, there is a c ertain measure of symmetry in going from f (t) t o F (s), a nd 6 Continuous-Time System Analysis Using t he L aplace Transform 382 6.2 vice versa. T hjs s ymmetry o r duality is also carried over t o t he p roperties of t he L aplace t ransform. T his fact will b e e vident in t he following development. 383 Some P roperties o f t he L aplace Transform f (l) f (t) f {l) + 1. Time Shifting j (t) j (t - to) F (s) ¢ ==> F (s)e- 8to ¢ ==> Proof: [ , [J(t - to)u(t - to)] = x, = F (s)e- sto l'o 2 0 " (b) to :2: 0 (6.29b) E xample 6 .4 Find the Laplace transform of j (t) depicted in Fig. 6.5a. Describing mathematically a function such as the one in Fig. 6.5a is discussed in Sec. 1.4. T he function j (t) in Fig. 6.5a can be described as a sum of two components shown in Fig. 6.5b. The equation for t he first component is t - lover 1 :s; t :s; 2, so t hat this part of j (t) can be described by (t - I)[u(t - I) - u(t - 2)]. T he second component can be described by u (t - 2) - u(t - 4). Therefore j (t - to)u(t - to)e-st dt + [u(t - 2) I)u(t - 2) + u (t - j (t) = (t - I) [u(t - I) - u(t - 2)] = (t - I)u(t - I) - (t - [ , [ j(t - to)u(t - to)] = [ : j (x)u(x)e-s(x+to) d x B ecause u (x) = 0 for x < 0 a nd u (x) = 1 for x :2: 0, t he l imits of i ntegration a re from 0 t o 0 0. T hus loco j(x)e-s(x+to) dx = e - 8to _ 1 T he t ime-shifting p roperty proves very convenient in finding t he L aplace t ransform o f functions w ith different descriptions over different intervals, as t he following example demonstrates. we o btain [ , [J(t - to)u(t - to)] = 4 2 0 1- • F (s) t hen j (t - to)u(t - to) _ 1 F ig. 6 .5 Finding a mathematical description of a function j (t) in Fig. 6.5a. (6.29a) If ¢ ==> 4 I -I Observe t hat f (t) s tarts a t t = 0, a nd, t herefore, j (t - to) s tarts a t t = to· T his f act is i mplicit, b ut is n ot e xplicitly indicated in Eq. (6.29a). T his o ften leads t o i nadvertent e rrors. To avoid such a pitfall, we should restate t he p roperty a s follows: j (t)u(t) 3 (a) ¢ ==> t hen for to :2: 0 S etting t - to 2 0 T his p roperty s tates t hat if N ote t hat j (t - to)u(t - to) is t he s ignal j (t)u(t) delayed by to seconds. T he timeshifting p roperty s tates t hat delaying a signal b y to seconds amounts t o m ultiplying its transform e - sto. T his p roperty o...
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