Page 53 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

Then

Signals x(t) and xYt) are sketched in Fig. 1-31(a).
(b) Using the rule for differentiation of the product of two functions and the result from part
(a), we have
xl(t) = [u(t) - u(l -a)] + t[6(t) - S(t - a)]

But by Eqs. (1.25) and (1.26)

t6(t) = (O)S(t) = 0 and t6(t -a) =a6(t -a)
Thus,

Signals x(t) and xf(t) are sketched in Fig. 1-31(b).
(c) x(t) = sgn t can be rewritten as

~(t) =sgnt =u(t) -u(-t)
Then using Eq. (1.301, we obtain

xl(t) =ul(t) -ul(-t) =6(t) - [-6(t)] =2S(t)

Signals x(t) and x'(t) are sketched in Fig. 1-31(c).

(b)
Fig. 1-31