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X(s)is in VS .NET
X(s)is QR Code Scanner In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. QR Code Creator In .NET Using Barcode creation for .NET framework Control to generate, create Quick Response Code image in .NET applications. 3 3 LAPLACE TRANSFORMS OF SOME COMMON SIGNALS
Read QRCode In Visual Studio .NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications. Bar Code Creation In .NET Framework Using Barcode drawer for .NET Control to generate, create bar code image in VS .NET applications. Unit Impulse Function S( t ): Scan Barcode In .NET Using Barcode reader for .NET Control to read, scan read, scan image in .NET applications. Encode QR Code 2d Barcode In Visual C#.NET Using Barcode creation for VS .NET Control to generate, create QR Code image in .NET applications. Using Eqs. (3.3) and (1.20), we obtain
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Barcode Maker In None Using Barcode printer for Microsoft Excel Control to generate, create bar code image in Microsoft Excel applications. Make EAN13 In Java Using Barcode encoder for Android Control to generate, create EAN13 image in Android applications. B. Unit Step Function u ( t 1: Decoding Bar Code In C# Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. Recognizing Barcode In .NET Framework Using Barcode Control SDK for ASP.NET Control to generate, create, read, scan barcode image in ASP.NET applications. where O + = lim, , "(0 Code39 Generator In None Using Barcode maker for Excel Control to generate, create USS Code 39 image in Excel applications. Recognizing USS Code 39 In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. + 1. Creating GS1  13 In None Using Barcode drawer for Font Control to generate, create UPC  13 image in Font applications. Recognizing Barcode In VS .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications. C. Laplace Transform Pairs for Common Signals: The Laplace transforms of some common signals are tabulated in Table 31. 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 CONTINUOUSTIME LTI SYSTEMS
Table 31 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. 34. LAPLACE TRANSFORM AND CONTINUOUSTIME LTI SYSTEMS
[CHAP.
Fig. 34 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 timeshift operation are the same.
C. Shifting in the sDomain: 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. 35. D. Time Scaling: x ( t )+ + X ( S ) then
x(at) X
la l
R1=aR
CHAP 31 LAPLACE TRANSFORM AND CONTINUOUSTIME LTI SYSTEMS
(4 (b) Fig. 35 Effect on the ROC of shifting in the sdomain. ( 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. 36. E. Time Reversal: If Fig. 36 Effect on the ROC of time scaling. (a) ROC of X ( s ) ; ( b ) ROC of
X(s/a). LAPLACE TRANSFORM AND CONTINUOUSTIME 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 jwaxes in the splane. 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 polezero cancellation at s = 0. ROC = R
Differentiation in the sDomain: 41) ++X(S) then
ROC = R
H. Integration in the Time Domain: then Equation (3.22) shows that the Laplace transform operation corresponding to timedomain 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 CONTINUOUSTIME LTI SYSTEMS
Table 32 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 continuoustime LTI systems. Table 32 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 splane; 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.

