Page 155 - Schaum's Outline of Theory and Problems of Signals and Systems

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

In this case, the system function H(s) is an improper fraction and can be rewritten as

Since the system is causal, taking the inverse Laplace transform of H(s), the impulse
response h(t ) is

Note that we obtained different system functions depending on the different sets of input
and output.

3.24. Using the Laplace transform, redo Prob. 2.5.

From Prob. 2.5 we have
h(t) = e-"'u(t) ~(t) =ea'u(-t) a>O

Using Table 3-1, we have
1
H(s) = - Re(s) > -a
s+a
1
X(s) = - - Re(s) <a
s-a
Thus,

and from Table 3-1 (or Prob. 3.6) the output is

which is the same as Eq. (2.67).

3.25. The output y(t) of a continuous-time LTI system is found to be 2e-3'u(t) when the
input x( t is u(t ).

(a) Find the impulse response h(t) of the system.
(6) Find the output y(t) when the input x(t) is e-'u(f).
(a) x(f) = u(t), y(t) = 2e-3'u(t)
Taking the Laplace transforms of x(t) and we obtain