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) = _ eatu{_t) T he Laplace transform of this signal is 368 6 Because u (  t) = 1 for ContinuousTime S ystem Analysis Using t he Laplace Transform
t < 0 a nd u (  t) = 0 for t > 0, F (s) = j O _ eate 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  eatu(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  eatu(  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 onetoone correspondence between F (8) a nd f (t),
unless t he r egion of convergence...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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