Page 172 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 31 LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS

Am.
(a) x(r) = (1 - e-' - te-'Mt)
(b) x(t) = -4-t) -(l + t)e-'dt)
(c) ~(t) (- 1 + e-' + te-'Id-t)
=
(d) x(t) = e-2'(~os3t - f sin 3t)u(t)
(el x(t) = at sin 2tu(t)
(f) x(t)=(- $e-2'+ Acos3t-t &sin3t)u(t)

3.50. Using the Laplace transform, redo Prob. 2.46.
Hint: Use Eq. (3.21) and Table 3-1.

3.51. Using the Laplace transform, show that

(a) Use Eq. (3.21) and Table 3-1.
(b) Use Eqs. (3.18) and (3.21) and Table 3-1.

3.52. Using the Laplace transform, redo Prob. 2.54.
Hint:
(a) Find the system function H(s) by Eq. (3.32) and take the inverse Laplace transform of
H(s).
(6) Find the ROC of H(s) and show that it does not contain the jo-axis.

3.53. Find the output y(t) of the continuous-time LTI system with

for the each of the following inputs:

(a) x(t) = e-'u(t)
(b) x(t) = e-'u(-t)

Ans.
(a) y(r) = (e-'-e-")u(t)
(b) y(t) = e-'u(-f)+e- 2'~(t)

3.54. The step response of an continuous-time LTI system is given by (1 - e-')u(t). For a certain
unknown input x(t), the output y(t) is observed to be (2 - 3e-' + e-3')u(r). Find the input
x(t).