Page 147 - Schaum's Outline of Theory and Problems of Signals and Systems

136 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS [CHAP. 3

Using the linearity property (3.15) and the shifting in the s-domain property (3.17), we
obtain

(g) Applying the shifting in the s-domain property (3.17) to the result from part (f), we
obtain

3.14. Verify the convolution property (3.23), that is,

m
y(t) =x,(t) * x2(t) = j x,(r)x2(t - r) dr
- m
Then, by definition (3.3)

Noting that the bracketed term in the last expression is the Laplace transform of the shifted
signal x2(t - 71, by Eq. (3.16) we have

with an ROC that contains the intersection of the ROC of X,(s) and X2(s). If a zero of one
transform cancels a pole of the other, the ROC of Y(s) may be larger. Thus, we conclude that

3.15. Using the convolution property (3.23), verify Eq. (3.22), that is,

We can write [Eq. (2.601, Prob. 2.21

From Eq. (3.14)