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3 3 LAPLACE TRANSFORMS OF SOME COMMON SIGNALS
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Unit Impulse Function S( t ):
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Using Eqs. (3.3) and (1.20), we obtain
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B. Unit Step Function u ( t 1:
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C. Laplace Transform Pairs for Common Signals:
The Laplace transforms of some common signals are tabulated in Table 3-1. Instead of having to reevaluate the transform of a given signal, we can simply refer to such a table and read out the desired transform.
3.4 PROPERTIES OF THE LAPLACE TRANSFORM
Basic properties of the Laplace transform are presented in the following. Verification of these properties is given in Probs. 3.8 to 3.16.
CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
Table 3-1 Some Laplace Transforms Pairs
All s
cos wotu(t) sin wotu(t
e-"' cos wotu(t)
(s+a12+w;
Re(s) > - Re(a)
Linearity:
The set notation A I B means that set A contains set B, while A n B denotes the intersection of sets A and B, that is, the set containing all elements in both A and B. Thus, Eq. (3.15) indicates that the ROC of the resultant Laplace transform is at least as large as the region in common between R , and R 2 . Usually we have simply R' = R , n R , . This is illustrated in Fig. 3-4.
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
[CHAP.
Fig. 3-4
ROC of a , X , ( s ) + a , X , ( s ) .
B. Time Shifting:
) '-*X(S)
then
x ( t - t o ) -e-"[)X ( s 1
R'=R
(3.16)
Equation (3.16)indicates that the ROCs before and after the time-shift operation are the same.
C. Shifting in the s-Domain:
then
e+"x(t) X ( s -
-so)
R' = R
+ Re(so)
(3.17)
Equation (3.17 ) indicates that the ROC associated with X ( s - so) is that of X ( s ) shifted by Re(s,,).This is illustrated in Fig. 3-5.
D. Time Scaling:
x ( t )+ + X ( S )
then
x(at)- -X
la l
R1=aR
CHAP- 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
(4 (b) Fig. 3-5 Effect on the ROC of shifting in the s-domain. ( a ) ROC of X(s); ( b ) ROC of X ( s - so).
Equation (3.18) indicates that scaling the time variable t by the factor a causes an inverse scaling of the variable s by l / a as well as an amplitude scaling of X ( s / a ) by I/ Jal.The corresponding effect on the ROC is illustrated in Fig. 3-6. E. Time Reversal: If
Fig. 3-6 Effect on the ROC of time scaling. (a) ROC of X ( s ) ; ( b ) ROC of
X(s/a).
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
[CHAP. 3
then
x ( - t ) *X(-S)
(3.19)
'Thus, time reversal of x ( t ) produces a reversal of both the a- and jw-axes in the s-plane. Equation (3.19) is readily obtained by setting a = - 1 in Eq. (3.18). F. Differentiation in the Time Domain:
~ ( t++X(S) ) then Equation (3.20) shows that the effect of differentiation in the time domain is multiplication of the corresponding Laplace transform by s. The associated ROC is unchanged unless there is a pole-zero cancellation at s = 0.
ROC = R
Differentiation in the s-Domain:
41) ++X(S)
then
ROC = R
H. Integration in the Time Domain:
then Equation (3.22) shows that the Laplace transform operation corresponding to time-domain integration is multiplication by l/s, and this is expected since integration is the inverse operation of differentiation. The form of R' follows from the possible introduction of an additional pole at s = 0 by the multiplication by l/s.
I. Convolution:
xdl) * w ) ~ 2 ( ++XZ(S) 4
ROC= R, ROC = R 2
CHAP. 31
LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS
Table 3-2 Properties of the Laplace Transform
Property
Signal
x(t) x,(t) x2W a , x , ( t )+ a 2 x 2 ( l ) x(t - t o ) es"'x(t x( at
Transform
X(s)
R R1 R2 R'IR, nR2 R' = R R' = R Re(s,)
Linearity Time shifting Shifting in s Time scaling Time reversal Differentiation in t Differentiation in s Integration Convolution
a ,X , ( s )+ a , X 2 ( s ) e-""X(s) X ( s - so) 1 -X(s) la l
x,w x,w
= aR
R'= - R
-t x ( t )
dX( s ) ds
Rf=R
then
%(t) * ~
2 0H )
X I ( ~ ) X ~ ( ~ ) R'IR, nR2
(3.23)
This convolution property plays a central role in the analysis and design of continuous-time LTI systems. Table 3-2 summarizes the properties of the Laplace transform presented in this section.
THE INVERSE LAPLACE TRANSFORM
Inversion of the Laplace transform to find the signal x ( t ) from its Laplace transform X(s) is called the inverse Laplace transform, symbolically denoted as
Inversion Formula:
There is a procedure that is applicable to all classes of transform functions that involves the evaluation of a line integral in complex s-plane; that is,
In this integral, the real c is to be selected such that if the ROC of X(s) is a,< Re(s) < a 2 , then a, < c < u2.The evaluation of this inverse Laplace transform integral requires an understanding of complex variable theory.
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