Page 118 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS 107

2.53. The system shown in Fig. 2-31 is formed by connection two systems in parallel. The impulse
responses of the systems are given by

h ) =e2() and h,(t) = 2ee'u(t)

(a) Find the impulse response h(t) of the overall system.
(b) Is the overall system stable?
Ans. (a) h(t) = (e-2' + 2e-')u(t)
(b) Yes

2.54. Consider an integrator whose input x(t) and output y(t) are related by

(a) Find the impulse response h(t) of the integrator.
(b) Is the integrator stable?
Ans. (a) h(t) = u(t)
(b) No

2.55. Consider a discrete-time LTI system with impulse response h[n] given by

h[n] =S[n - 11
Is this system memoryless?
Am. No, the system has memory.

2.56. The impulse response of a discrete-time LTI system is given by
h[n] = (f )"u[n]

Let y[nl be the output of the system with the input
x[n] = 2S[n] + S[n - 31
Find y[l] and y[4].
Am. y[ll = 1 and y[4] = 5.