Page 91 - Schaum's Outline of Theory and Problems of Signals and Systems

2
80 LINEAR TIME-INVARIANT SYSTEMS [CHAP.

Then, using Eq. (2.10), we have

since lx(t - 711 1 k, from Eq. (2.77). Therefore, if the impulse response is absolutely inte-
grable, that is,

then l y(t )I s k,K = k, and the system is BIBO stable.

2.14. The system shown in Fig. 2-17(a) is formed by connecting two systems in cascade. The
impulse responses of the systems are given by h,(t) and h2(0, respectively, and
h,(t) = e-2'u(t) h,(t) = 2e-'u(t)

(a) Find the impulse response h(t) of the overall system shown in Fig. 2-17(b).
(b) Determine if the overall system is BIBO stable.

(b)
Fig. 2-17

(a) Let w(t) be the output of the first system. By Eq. (2.6)
~(t) * (2.78)
= ~(t) h,(t)
Then we have
~(t) w(t) * h2(t) = [x(O * h,( t)] * h2( t) (2.79)
=
But by the associativity property of convolution (2.81, Eq. (2.79) can be rewritten as

~(t) =x(t) *[hl(t)*h2(t)] =~(t)*h(O (2.80)
Therefore, the impulse response of the overall system is given by
h(t) =hdt) *h2(t) (2.81)
Thus, with the given h ,( t ) and h2(t ), we have