Page 51 - Schaum's Outline of Theory and Problems of Signals and Systems

SIGNALS AND SYSTEMS [CHAP. 1

1.28. Verify Eq. (1.30):

From Eq. (1.28) we have

where 4(t) is a testing function which is continuous at t = 0 and vanishes outside some fixed
interval. Thus, +I(!) exists and is integrable over 0 < t < rn and +(a) = 0. Then using Eq. (1.98)
or definition (1.18), we have

Since 4(t) is arbitrary and by equivalence property (1.991, we conclude that

1.29. Show that the following properties hold for the derivative of 6(t):

(6) ts'(t) = -6(r)
(a) Using Eqs. (1.28) and (1.201, we have

00
4(1)S.(t) dl = - /- m.(t)6(t) dl = -4'(O)
1- rn - m
(b) Using Eqs. (1.101) and (1.201, we have

Thus, by the equivalence property (1.99) we conclude that
t6'(t) = -6(t)