Inverse Laplace Transform Given a transform function X(s) and its region of convergence, the procedure for finding the signal x(r)that generated that transform is called finding the inverse Laplace transform and is symbolically denoted as x(t)=[X( The signal x(t) can be recovered by means of the relationshi (t) X(s)e" ds In this integral, the real number c is to be selected so that the complex number c+ jo lies entirely within the region of convergence of X(s)for all values of the imaginary component o For the important class of rational Laplace transform functions, there exists an effective alternate procedure that does not necessitate directly evaluating this integral. This procedure is generally known as the partial-fraction expansion method Partial Fraction Expansion Method just indicated, the partial fraction expansion method provides a convenient technique for reacquiring the ignal that generates a given rational Laplace transform. Recall that a transform function is said to be rational if it is expressible as a ratio of polynomial in s, that X(s)= b(s b m-+…+bs+b A(s) s"+a-S"-+.+a,s+a The partial fraction expansion method is based on the appealing notion of equivalently expressing this rational transform as a sum of n elementary transforms whose corresponding inverse Laplace transforms (i.e. generating ignals)are readily found in standard Laplace transform pair tables. This method entails the simple five-step process as outlined in Table 6.3. A description of each of these steps and their implementation is now given . Proper Form for Rational Transform. This division process yields an expression in the proper form as given by A(s) Q(s)+ A(s) TABLE 6.3 Partial Fraction Expansion Method for Determining the Inverse Laplace Transform L. Put rational transform into proper form whereby the degree of the numerator polynomial is less than or equal to that of Il. Factor the denominator polynomial. lll. Perform a partial fraction expansion. IV. Separate partial fraction expansion terms into causal and anticausal components using the associated region of absolute convergence for this purpose Source: J.A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N J. Prentice-Hall, 1985, P. 153. with permission. c 2000 by CRC Press LLC© 2000 by CRC Press LLC Inverse Laplace Transform Given a transform function X(s) and its region of convergence, the procedure for finding the signal x(t) that generated that transform is called finding the inverse Laplace transform and is symbolically denoted as The signal x(t) can be recovered by means of the relationship In this integral, the real number c is to be selected so that the complex number c + jw lies entirely within the region of convergence of X(s) for all values of the imaginary component w. For the important class of rational Laplace transform functions, there exists an effective alternate procedure that does not necessitate directly evaluating this integral. This procedure is generally known as the partial-fraction expansion method. Partial Fraction Expansion Method As just indicated, the partial fraction expansion method provides a convenient technique for reacquiring the signal that generates a given rational Laplace transform. Recall that a transform function is said to be rational if it is expressible as a ratio of polynomial in s, that is, The partial fraction expansion method is based on the appealing notion of equivalently expressing this rational transform as a sum of n elementary transforms whose corresponding inverse Laplace transforms (i.e., generating signals) are readily found in standard Laplace transform pair tables. This method entails the simple five-step process as outlined in Table 6.3. A description of each of these steps and their implementation is now given. I. Proper Form for Rational Transform. This division process yields an expression in the proper form as given by TABLE 6.3 Partial Fraction Expansion Method for Determining the Inverse Laplace Transform I. Put rational transform into proper form whereby the degree of the numerator polynomial is less than or equal to that of the denominator polynomial. II. Factor the denominator polynomial. III. Perform a partial fraction expansion. IV. Separate partial fraction expansion terms into causal and anticausal components using the associated region of absolute convergence for this purpose. V. Using a Laplace transform pair table, obtain the inverse Laplace transform. Source: J. A. Cadzow and H.F. Van Landingham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 153. With permission. x t( ) = + [X(s)] ±1 x t j X s e ds st c j c j ( ) = ( ) - • + • Ú 1 2p X s B s A s b s b s b s b s a s a s a m m m m n n n ( ) ( ) ( ) = = + + × × × + + + + × × × + + - - - - 1 1 1 0 1 1 1 0 X s B s A s Q s R s A s ( ) ( ) ( ) ( ) ( ) ( ) = = +