Laplace Transform Properties - Signals and Systems - PDF

Sec. 9.5 Properties of the Laplace Transform 691 Example 9.16 The initial- and final-value theorems can be useful in checking the correctness of the Laplace transform calculatio...

Sec. 9.5 Properties of the Laplace Transform 691 Example 9.16 The initial- and final-value theorems can be useful in checking the correctness of the Laplace transform calculations for a signal. For example, consider the signal x(t) in Example 9.4. From eq. (9.24), we see that x(O+) = 2. Also, using eq. (9.29), we find that.. 2s3 + 5s 2 + 12s hm sX(s) = hm = 2, S->OC S-->00 S3 + 4S 2 + 14S + 20 which is consistent with the initial-value theorem in eq. (9.110). 9.5.11 Table of Properties In Table 9.1, we summarize the properties developed in this section. In Section 9. 7, many of these properties are used in applying the Laplace transform to the analysis and characterization of linear time-invariant systems. As we have illustrated in several exam- ples, the various properties of Laplace transforms and their ROCs can provide us with TABLE 9.1 PROPERTIES OF THE LAPLACE TRANSFORM Laplace Section Property Signal Transform ROC x(t) X(s) R x, (t) X1(s) R, x2(t) X2(s) R2 ---------- ----------- ------------------ 9.5.1 Linearity ax 1(t) + bx2(t) aX 1(s) + bX2(s) At least R 1 n R2 9.5.2 Time shifting x(t- to) e-sto X(s) R 9.5.3 Shifting in the s-Domain esot x(t) X(s- s0 ) Shifted version of R (i.e., s is in the ROC if s - s0 is in R) 9.5.4 Time scaling x(at) Scaled ROC (i.e., s is in the 1 x(s) Ia! a ROC if s/a is in R) 9.5.5 Conjugation x*(t) X*(s*) R 9.5.6 Convolution x 1(t) * x 2(t) X1(s)X2 (s) At least R 1 n R2 d 9.5. 7 Differentiation in the dix(t) sX(s) At least R Time Domain d 9.5.8 Differentiation in the -tx(t) dsX(s) R s-Domain 1 9.5.9 Integration in the Time foo X(T)d(T) -X(s) s At least R n { 0} Domain Initial- and Final-Value Theorems 9.5.1 0 If x(t) = 0 for t < 0 and x(t) contains no impulses or higher-order singularities at t = 0, then x(O+) = lim sX(s) s~oo If x(t) = 0 fort< 0 and x(t) has a finite limit as t -7 oc, then lim x(t) = lim sX(s) 1-+-:xo S--t-:xo