Page 146 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

Applying the differentiation property (3.20), we obtain
X(s) =sF(s)
Thus,

The form of the ROC R' follows from the possible introduction of an additional pole at s = 0
by the multiplying by l/s.

Using the various Laplace transform properties, derive the Laplace transforms of the
following signals from the Laplace transform of u(t).

(a) 6(t) (b) 6'(t)
(c) tu(t) (d) e-"'u(t)
(e) te-"'u(t) (f) coso,tu(t)
(g) e-"'cos w,tu(t)
(a) From Eq. (3.14)we have
1
u(t) H - for Re( s) > 0
S
From Eq. ( 1.30) we have

Thus, using the time-differentiation property (3.20), we obtain
1
S(t) HS- = 1 all s
S
(b) Again applying the time-differentiation property (3.20) to the result from part (a), we
obtain
a'(!) HS all s ( 3.78)

(c) Using the differentiation in s property (3.211, we obtain

(dl Using the shifting in the s-domain property (3.17), we have

1
e-a'u(t) w - Re(s) > -a
s+a
(el From the result from part (c) and using the differentiation in s property (3.21), we obtain

(f) From Euler's formula we can write