The coefficient oo may be expediently evaluated by letting s approach infinity, whereby each term on the right side goes to zero except a. Thus, do lim X(s)=0 The aa coefficient is given by the ent ression aa =(s-pi)X(s)ap B(P) (6.3) (p2) The remaining coefficients a1, a2,...,a-l associated with the multiple root P, may be evaluated by solving Eq(6.3)by setting s to a specific value. IV Causal and Anticausal Components. In a partial fraction expansion of a rational Laplace transform X(s) whose region of absolute convergence is given by σ+<Re(s)<o it is possible to decompose the expansions elementary transform functions into causal and anticausal fur (and possibly impulse-generated terms). Any elementary function is interpreted as being(1)causal if component of its pole is less than or equal to o, and (2)anticausal if the real component of its pole is g than or equal to o_ The poles of the rational transform that lie to the left(right) of the associated region of absolute convergence correspond to the causal (anticausal)component of that transform Figure 6.4 shows the location of causal and anticausal poles of rational transform. V. Table Look-Up of Inverse Laplace Transform. To complete the inverse Laplace transform procedure, one need simply refer to a standard Laplace transform function table to determine the time signals that generate each of the elementary transform functions. The required time signal is then equal to the same linear combi nation of the inverse Laplace transforms of these elementary transform functions Re(s) Anticausal URE 6.4 Location of causal and anticausal poles of a rational transform. Source: J.A. Cadzow and H.F. Van Landing- Signals, Systems, and Transforms, Englewood Cliffs, N J. Prentice-Hall, 1985, P. 161. with permission. c 2000 by CRC Press LLC© 2000 by CRC Press LLC The coefficient a0 may be expediently evaluated by letting s approach infinity, whereby each term on the right side goes to zero except a0. Thus, The aq coefficient is given by the convenient expression (6.3) The remaining coefficients a1, a 2 , … , aq–1 associated with the multiple root p1 may be evaluated by solving Eq. (6.3) by setting s to a specific value. IV. Causal and Anticausal Components. In a partial fraction expansion of a rational Laplace transform X(s) whose region of absolute convergence is given by it is possible to decompose the expansion’s elementary transform functions into causal and anticausal functions (and possibly impulse-generated terms). Any elementary function is interpreted as being (1) causal if the real component of its pole is less than or equal to s+ and (2) anticausal if the real component of its pole is greater than or equal to s– . The poles of the rational transform that lie to the left (right) of the associated region of absolute convergence correspond to the causal (anticausal) component of that transform. Figure 6.4 shows the location of causal and anticausal poles of rational transform. V. Table Look-Up of Inverse Laplace Transform. To complete the inverse Laplace transform procedure, one need simply refer to a standard Laplace transform function table to determine the time signals that generate each of the elementary transform functions. The required time signal is then equal to the same linear combi￾nation of the inverse Laplace transforms of these elementary transform functions. FIGURE 6.4 Location of causal and anticausal poles of a rational transform. (Source: J.A. Cadzow and H.F. Van Landing￾ham, Signals, Systems, and Transforms, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 161. With permission.) a0 = = 0 Æ+• lim ( ) s X s aq q s p s p X s B p A p = - = = ( ) ( ) ( ) ( ) 1 1 1 1 1 s s + - < Re(s) <