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Unformatted text preview: th o f integration (from c '  jao t o c ' + j ao) in Eq. (6.10) may be different
from t hat in Eq. (6.9) t o allow for t he convergence of t he integral in Eq. (6.10). This
subtle point will b e discussed later in Sec. 6.81. I f y (t)  $=} Y (s), t hen according
t o Eq. (6.10) it follows t hat
Y (s) = F (s)H(s)
(6.11)
Here, we have expressed the input f (t) as a sum of exponentials components of
the form est. T he system response is t hen o btained by adding the responses t o all
these exponential components. T he procedure followed here is identical t o t hat in
chapter 2 (where the input is expressed as a sum of impulses) or chapter 4 (where
the input is expressed as a sum of exponentials of the form e jwt ).
I n conclusion, we have shown t hat for an LTIC s ystem with transfer function
H (s), if the input a nd t he o utput a re f (t) a nd y (t), respectively, and if f (t)  $=} F (s), y (t)  $=} Y (s) t hen Y (s) = F (s)H(s) (6.12) We shall derive this result more formally later.
Linearity o f the Laplace Transform T he Laplace transform is a linear operator, and t he principle of superposition
applies; t hat is, if
then f (t)  $=} F(s) (6.13) 6 366 C ontinuousTime System Analysis Using t he Laplace Transform 6.1 T he Laplace Transform T he p roof is t rivial a nd follows directly from t he definition of t he Laplace transform.
This result can be extended t o a ny finite number of terms.
T he Region o f Convergence
t Earlier, we discussed t he i ntuitive meaning of t he region of convergence (or
region of existence) of t he Laplace transform F (s). Mathematically, t he region of
convergence for F (s) is t he s et of values of s ( the region in t he complex plane)
for which t he i ntegral in Eq. (6.8b) defining t he direct Laplace transform F (s)
converges. T his c oncept will become clear in t he following example .
• (a) E xample 6 .1 For the signal f (t) = e atu(t), find the Laplace transform F (s) and its region of
convergence.
By definition
F (s) Because u (t) = 1: e atu(t)estdt = 0 for t < 0 and u (t) = 1 for t ~ o ( I _eOl u ( t) 0, (b) F ig. 6 .2 Signals e atu(t) and _ eatu(  t) have the same Laplace transform but different regions of convergence.
F (s)= { OOeateS'dt= (OOe<s+a)t dt = _ __ e (s+a)'jOO
l io io s+ a (6.14) 0 Role o f the Region of Convergence Note t hat s is complex and as t  + 0 0, the term e (s+a)' does not necessarily vanish. Here
we recall t hat for a complex number z = a + j/3, Now lej~tl = 1 regardless of the value of /3t. Therefore, as
 > 0 0 if a < O. Thus
Re z > 0
lim e zt =
t oo
00
Re z < 0
Clearly t + 0 0, e  zt + 0 only if a > 0, and e  zt {O lim e(s+a)t = {O
00 t _oo (6.15) R e(s+a»O
R e (s + a) < 0 Use of this result in Eq. (6.14) yields
1
F (s)=s+a + a) > 0 (6.16a) Re s >  a (6.16b) R e(s or
e atu(t) = _+a
1_
s The region of convergence of F (s) is Re s >  a, as shown in the shaded area in Fig. 6.2a.
This fact means that the integral defining F (s) in Eq. (6.14) exists only fo...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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