Page 98 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS 87

2.22. Consider the system in Prob. 2.20. Show that the initial rest condition y(0) = 0 also
implies that the system is time-invariant.
Let y,(t) be the response to an input x,(t) and
xI(t) = 0 t10 (2.114)

Then

and Y do) = 0 (2.116)
Now, let y2(t) be the response to the shifted input x2( t ) = x ,(t - T). From Eq. (2.114) we have
x2(t) = 0 tsr (2.117)

Then y ,( t must satisfy

and ~~(71
0
=
Now, from Eq. (2.115) we have

If we let y2(t) = yl(t - T), then by Eq. (2.116) we have

yz(7) = Y,(T - 7) = ~1(0) = 0
Thus, Eqs. (2.118) and (2.119) are satisfied and we conclude that the system is time-invariant.

2.23. Consider the system in Prob. 2.20. Find the impulse response h(r) of the system.
The impulse response h(t) should satisfy the differential equation

The homogeneous solution hh(t) to Eq. (2.120) satisfies

To obtain hh(t) we assume
hh(t) = ceS'
Substituting this into Eq. (2.121) gives

sces' + aces' = (s + a)ces' = 0
from which we have s = -a and
hh(t) = ce-"'u(t)

We predict that the particular solution hp(t) is zero since hp(t) cannot contain Nt). Otherwise,
h(t) would have a derivative of S(t) that is not part of the right-hand side of Eq. (2.120). Thus,

h(t) =ce-"'u(t) (2.123)