Page 157 - Schaum's Outline of Theory and Problems of Signals and Systems

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

3.27. Using the Laplace transfer, redo Prob. 2.14.

(a) Using Eqs. (3.36) and (3.40, we have
Y(s) =X(s)H,(s)H,(s) =X(s)H(s)
where H(s) = H,(s)H,(s) is the system function of the overall system. Now from Table
3-1 we have

Hence,

Taking the inverse Laplace transfer of H(s), we get
h(t) = 2(ep' - e-2')u(t)

(b) Since the ROC of H(s), Re(s) > - 1, contains the jo-axis, the overall system is stable.

3.28. Using the Laplace transform, redo Prob. 2.23.
The system is described by

Taking the Laplace transform of the above equation, we obtain

sY(s) + aY(s) = X(s) or (s + a)Y(s) = X(s)
Hence, the system function H(s) is

Assuming the system is causal and taking the inverse Laplace transform of H(s), the impulse
response h(t ) is
h(t) = e-"'u(t)
which is the same as Eq. (2.124).

3.29. Using the Laplace transform, redo Prob. 2.25.

The system is described by
yf(t) + 2y(t) =x(t) +xl(t)
Taking the Laplace transform of the above equation, we get
sY(s) + 2Y(s) = X(s) + sX(s)

or (s + 2)Y(s) = (s + l)X(s)