Page 173 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Fig. 3-18

Determine the overall system function H(s) for the system shown in Fig. 3-18.
Hint: Use the result from Prob. 3.31 to simplify the block diagram.
S
Am. H(s)=
s3+3s2+s-2

If x(t) is a periodic function with fundamental period T, find the unilateral Laplace transform
of x(t).

Find the unilateral Laplace transforms of the periodic signals shown in Fig. 3-19.

Using the unilateral Laplace transform, find the solution of

y"(t) - yl(t) - 6y(t) =et
with the initial conditions y(0) = 1 and y'(0) = 0 for t 2 0.
Am. y(t)= -;el+ fe-2'+ ;e3', tzO

Using the unilateral Laplace transform, solve the following simultaneous differential equations:
yl(t) + y(t) +xf(r) +x(t) = 1
yl(t) -y(t) -2x(t) =O
with x(0) = 0 and y(0) = 1 for t 1 0.
Ans. x(t) = e-' - 1, y(t) = 2 - e-', t r 0

Using the unilateral Laplace transform, solve the following integral equations:

Am. (a) y(t) =eat, t 2 0; (6) y(t) =e2', t 2 0