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

In general we c an let the limits of integration r

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Unformatted text preview: r the values of s in the shaded region in Fig. 6.2a. For other values of s, the integral in Eq. (6.14) does not converge. For this reason the shaded region is called the region of convergence (or the region of e xistence) for F (s). Recall t hat the Fourier transform of e -atu(t) does not exist for negative values of a. In contrast, the Laplace transform exists for all values of a, and its region of convergence is t o the right o f the line Re s = - a. • T he region of convergence is required for evaluating t he inverse Laplace transform f {t) from F (s) as defined by Eq. (6.8a). T he o peration of finding t he inverse transform requires a n i ntegration in t he complex plane, which needs some explanation. T he p ath o f i ntegration is along c + j w, w ith w varying from - 00 t o 0 0. Moreover, t he p ath of integration must lie in t he region of convergence (or existence) for F {s). For t he signal e -atu{t), t his is possible if c > - a. O ne possible p ath of integration is shown (dotted) in Fig. 6.2a. Thus, t o o btain f {t) from F {s), t he i ntegration in (6.8a) is p erformed along this p ath. W hen we i ntegrate [1/{8 + a)Je st along this p ath, t he r esult is e -atu(t). Such integration in t he complex plane requires a background in t he t heory of functions of complex variables. We can avoid this integration by compiling a table of Laplace transforms (Table 6.1), where the Laplace transform pairs are t abulated for a variety of signals. To find t he inverse Laplace transform of say, 1 /(8 + a), i nstead of using t he complex integral (6.8a), we look up t he t able a nd find t he inverse Laplace transform t o b e e -atu{t). A lthough the table given here is r ather s hort, it comprises t he functions of most practical interest. A more comprehensive t able a ppears in Doetsch. 1 T he Unilateral Laplace Transform I n order t o u nderstand t he need for defining a unilateral transform, let us find the Laplace transform of signal I (t) i llustrated in Fig. 6.2b f {t) = _ e-atu{_t) T he Laplace transform of this signal is 368 6 Because u ( - t) = 1 for Continuous-Time S ystem Analysis Using t he Laplace Transform t < 0 a nd u ( - t) = 0 for t > 0, F (s) = j O _ e-ate- st dt = -1° - 00 T he variable s in t he Laplace transform is complex in general, a nd i t can be expressed as s = (T + j w. B y definition s +a - 00 E quation (6.15) shows t hat F (s) = lim e -(s+a)t = 0 Hence 369 Existence o f the Laplace Transform e -(s+a)t dt = _ l_ e -(s+a)tIO -IX) 6.1 T he Laplace Transform t --oo R e(s+a)<O = 1 F (s)=Re s < - a (6.17) s+a T he signal - e-atu(-t) a nd its region of convergence (Re s < - a) a re depicted in Fig. 6.2b. Note t hat t he Laplace transforms for t he signals e -atu(t) a nd - e-atu( - t) a re identical e xcept for their regions of convergence. Therefore, for a given F (s), t here may be m ore t han one inverse transform, depending on t he region of convergence. In other words, there is no one-to-one correspondence between F (8) a nd f (t), unless t he r egion of convergence...
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