TABLE 6.1 Laplace Transform Pairs Time Signal Laplace Transform Region of 2.re-"u(-n) Re(s)>-Refa) 3. Re(s<-Re(a 4.(-n)e-"a(- Re(s)<-Re(a) 5. un Re(s) 6.() 8.r(t) 2 9. 11. cos o,t u(r Re(s)>0 12. ea sin o,t u(r) Re(s)>-Re(a (s+a)2+ 13. e-a cos t u(n) s+a Re(s)>-Re(a) (s+a)2+o Source: JA Cadzow and H.E. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J Prentice-Hall, 1985, P. 133. with permission. Properties of Laplace Transform Li Let us obtain the Laplace transform of a signal, xd n), that is composed of a linear combination of two other x(1)=1x(t)+2x2(t) The linearity property indicates that [α1x1(1)+a2x2(]=a1X1(s)+2X2(s)© 2000 by CRC Press LLC Properties of Laplace Transform Linearity Let us obtain the Laplace transform of a signal, x(t), that is composed of a linear combination of two other signals, x(t) = a1x1(t) + a2x2(t) where a1 and a2 are constants. The linearity property indicates that + [a1x1(t) + a 2x2(t)] = a 1X1(s) + a2X2(s) and the region of absolute convergence is at least as large as that given by the expression TABLE 6.1 Laplace Transform Pairs Time Signal Laplace Transform Region of x(t) X(s) Absolute Convergence 1. e –atu(t) Re(s) > –Re(a) 2. tke –atu(–t) Re(s) > –Re(a) 3. –e –atu(–t) Re(s) < –Re(a) 4. (–t)ke –atu(–t) Re(s) < –Re(a) 5. u(t) Re(s) > 0 6. d(t) 1 all s 7. sk all s 8. tk u(t) Re(s) > 0 9. Re(s) = 0 10. sin w0t u(t) Re(s) > 0 11. cos w0t u(t) Re(s) > 0 12. e –at sin w0t u(t) Re(s) > –Re(a) 13. e –at cos w0t u(t) Re(s) > –Re(a) Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 133. With permission. 1 s a + k s a k ! ( ) + +1 1 ( ) s a + k s a k ! ( ) + +1 1 s d t dt k k d( ) k s k ! +1 sgnt t t = ³ < Ï Ì Ó 1 0 1 0 , – , 2 s w w 0 2 0 2 s + s s 2 0 2 + w w ( ) s a + +w 2 0 2 s a s a + ( ) + +2 0 2 w